CYCLOTOMIC SOLOMON ALGEBRAS

ANDREW MATHAS AND ROSA C. ORELLANA

Abstract. This paper introduces an analogue of the Solomon descent algebra for the complex reflection groups of type G(r, 1, n). As with the Solomon descent algebra, our algebra has a basis given by sums of ‘distinguished’ coset representatives for certain ‘reflection subgroups’. We explicitly describe the structure constants with respect to this basis and show that they are polynomials in r. This allows us to define a deformation, or q-analogue, of these algebras which depends on a parameter q. We determine the irreducible representations of all of these algebras and give a basis for their radicals. Finally, we show that the direct sum of cyclotomic Solomon algebras is canonically isomorphic to a concatenation Hopf algebra.

1. Introduction

In a seminal paper [27], Solomon showed that the group algebra of any finite Coxeter group has a remarkable subalgebra, the Solomon descent algebra. In this paper we construct a similar subalgebra of the complex reflection group of type G(r, 1, n) and show that this algebra shares many of properties of the Solomon descent algebras.

Solomon showed that each descent algebra has a distinguished basis for which he gave an explicit description of the structure constants. This distinguished basis is given by the sums of the distinguished coset representatives of the parabolic subgroups. Solomon gave a basis for the radical of the descent algebra and he constructed a natural homomorphism from the descent algebra into the parabolic Burnside ring of the associated Coxeter group. As a consequence, it follows that the irreducible representations of the Solomon descent algebras are all one dimensional and that, in characteristic zero, they are naturally indexed by the conjugacy classes of the parabolic subgroups.

There has been an explosion of research into the descent algebras of Coxeter groups since Solomon discovered them; see, for example, [2, 5, 6, 7, 8, 10, 25]. The study of the Solomon descent algebras of the symmetric groups has been even more intense because of connections between these algebras and free Lie algebras, 0-Hecke algebras, noncommutative and quasi-symmetric functions [1, 13, 15, 22], the representation theory of the symmetric group, and card shuffling and associated random walks [4, 17].

The algebra that we construct in this paper is in many ways a natural generalization of the Solomon algebra of the symmetric groups. The cyclotomic Solomon algebra Sol(G_{r,n}) is a subalgebra of the group algebra of the complex reflection group G_{r,n} of type G(r, 1, n). Like Solomon, we define our algebra to be the subalgebra of the group algebra of G_{r,n} with basis the ‘distinguished’ coset representatives of a natural class of subgroups of G_{r,n}. It turns out that many natural choices of subgroups, and coset representatives for these subgroups, do not yield a subalgebra of the group algebra (see Remark 8.10). We show, however, that with respect to the ‘right’ length function, the sums of the minimal length coset representatives of the standard reflection subgroups of G_{r,n} give rise to a subalgebra of ℤG_{r,n} which is free of rank 2 · 3^{n−1}. We give an explicit formula for the structure constants for this basis which is similar to Solomon’s formula for the structure constants of the descent algebra of the symmetric group 𝔖_{n}.

2000 Mathematics Subject Classification. 16W30, 20C05, 05E15.

Key words and phrases. Solomon descent algebra, complex reflection groups, Hopf algebra.

One surprising feature of the cyclotomic Solomon algebras Sol(G_{r,n}) is that the structure constants of these algebras for n ≥ 0 are polynomials in r which are independent of n. As a consequence, these algebras admit a simultaneous deformation Sol_{q}(n) which depends on a parameter q. For fixed n ≥ 0, we show that the algebras Sol_{q}(n) are free of rank 2 · 3^{n−1}. We construct and classify the irreducible representations of these algebras over an arbitrary field, and hence give a basis for the radical of Sol_{q}(n).

A remarkable result of Gessel [16] shows that there is a natural duality between the Hopf algebra of quasi-symmetric functions and the descent Hopf algebra. This led Malvenuto and Reutenauer [22] to show that the direct sum of these algebras under the shuffle (or convolution) product is a Hopf subalgebra of the Hopf algebra of permutations. This Hopf algebra is dual to the Hopf algebra of quasi-symmetric functions and it is isomorphic to the Hopf algebra of non-commutative symmetric functions [15]. These results are important because they relate the coproduct of the quasi-symmetric functions with the product in the descent algebras.

Baumann and Hohlweg[3] showed that there is a similar Hopf algebra structure under the shuffle product on the space 𝒢 (r) = ⨁_{n≥0} ℤG_{r,n} of coloured permutations. We prove that the direct sum of the cyclotomic Solomon algebras Sol(r) = ⨁_{n≥0} ℤ Sol(G_{r,n}) is a Hopf subalgebra of 𝒢 (r). We show that Sol(r) is a concatenation Hopf algebra and that Sol(r) has a second bialgebra structure which has the same coproduct as 𝒢 (r) but where the product map is induced by group multiplication. We expect that the Hopf algebra Sol(r) is dual to the Hopf algebra of quasisymmetric functions of type B considered by Hsiao and Petersen [18].

Different generalizations of the Solomon algebras have been considered by other authors, the most striking of which are the Mantaci-Reutenauer algebras [23]. It is natural to ask whether the cyclotomic Solomon algebras and the Mantaci-Reutenauer algebras are isomorphic, at least for type B_{n}, since they are both free of rank 2 · 3^{n−1}. We show in Remark 8.10 that, in general, these two algebras are not isomorphic. Example 8.9 shows that, in stark contrast to the Solomon descent algebra, there is no map from Sol(G_{r,n}) into the character ring of G_{r,n}.

This paper is organized as follows. In the second section we introduce the complex reflection groups G_{r,n} and set our notation. In section 3 we define and classify the standard reflection subgroups of G_{r,n} and section 4 shows that every coset of a reflection subgroup has a unique element of minimal length. Sections 4 and 5 give combinatorial descriptions of the coset and double representatives of the reflection subgroups. This combinatorics turns out to be closely related to the structure constants of the cyclotomic Solomon algebras, which are finally introduced in section 6. The first main result of the paper, Theorem 6.8, determines the structure constants of the cyclotomic Solomon algebras, hence showing that they are in fact subalgebras of G_{r,n}. In section 7 we investigate the ‘generic’ cyclotomic Solomon algebras and in section 8 we construct and classify the irreducible representations of the cyclotomic algebras and their deformations. In section 9 we show that the direct sum of the cyclotomic algebras gives rise to a concatenation Hopf algebra which is a Hopf subalgebra of the Hopf algebras of coloured permutations. Finally, in section 10 we give a second combinatorial interpretation of the structure constants of the cyclotomic Solomon algebras. We use this to show that the direct sum of the cyclotomic Hopf algebras comes equipped with a second bialgebra structure which has the same coproduct but where the product map is induced by group multiplication.

2. Complex reflection groups of type G(r, 1, n)

This paper is concerned with certain subalgebras of the group algebra of the complex reflection groups of type G(r, 1, n), in the Shephard–Todd classification of the finite subgroups of GL_{n}(ℂ) which are generated by (pseudo) reflections. In this section we introduce these groups and study a length function on them.

Fix positive integers r and n. The complex reflection group of type G(r, 1, n) is the group G_{r,n} which is generated by elements s_{0}, s_{1}, …, s_{n−1} subject to the relations

s_{0}^{r} = 1 = s_{i}^{2}  s_{0}s_{1}s_{0}s_{1} = s_{1}s_{0}s_{1}s_{0}
s_{i}s_{j} = s_{j}s_{i},  s_{i}s_{i+1}s_{i} = s_{i+1}s_{i}s_{i+1},

where 1 ≤ i ≤ j − 1 ≤ n − 1. This presentation is very similar to the presentation of a Coxeter group; indeed, if r ≤ 2 then G_{r,n} is a Coxeter group. Accordingly, we encode this presentation in the following “cyclotomic Dynkin diagram”:

[figure: r; …; s_{0}; s_{1}; s_{2}; s_{n−1}]

The node labeled by r indicates that the generator s_{0} has order r; otherwise, this graph gives the presentation of G_{r,n} in exactly the same way as a Dynkin diagram gives the presentation of the corresponding Coxeter group. If r = 1 then G_{1,n} is isomorphic to the symmetric group of degree n.

From the presentation of G_{r,n} it is evident that there is a homomorphism from the symmetric group 𝔖_{n} into G_{r,n} which is determined by mapping each transposition (i, i+1) to s_{i}, for i = 1, …, n − 1. In fact, this map is injective so we can – and do – identify 𝔖_{n} with the subgroup ⟨s_{1}, …, s_{n−1}⟩ via this homomorphism.

The symmetric group 𝔖_{n} acts on {1, 2, …, n} from the right. We write this action exponentially. Thus, w ∈ 𝔖_{n} sends the integer i to i^{w}, for 1 ≤ i ≤ n.

Define t_{1} = s_{0} and t_{i+1} = s_{i}t_{i}s_{i}, for 1 ≤ i < n. Using the relations it is easy to see that t_{i}t_{j} = t_{j}t_{i}, for all i, j. It follows that the subgroup T = ⟨t_{1}, …, t_{n}⟩ is abelian and, further, one can show that T ≅ (ℤ/rℤ)^{n}. It is easy to see that

(2.1)  t_{i} w = w t_{i^{w}}, for all w ∈ 𝔖_{n} and 1 ≤ i ≤ n,

Hence, T is a normal subgroup of G_{r,n}. With a little more work we obtain the following description of G_{r,n} as an (internal) semidirect product, or wreath product:

(2.2)  G_{r,n} = T ⋊ 𝔖_{n} = ⟨s_{0}⟩ ≀ ⟨s_{1}, …, s_{n−1}⟩ ≅ (ℤ/rℤ) ≀ 𝔖_{n}.

Let ℤ_{r}^{n} = { α = (α_{1}, …, α_{n}) : 0 ≤ α_{i} < r }. For α ∈ ℤ_{r}^{n} let t^{α} = t_{1}^{α_{1}} … t_{n}^{α_{n}}. Then, as a set, G_{r,n} = { t^{α}w : α ∈ ℤ_{r}^{n} and w ∈ 𝔖_{n} } and |G_{r,n}| = r^{n}n!.

Let Π = Π_{r,n} = {t_{1}, …, t_{n}, s_{1}, …, s_{n−1}}. Then Π generates G_{r,n} because {s_{0} = t_{1}, s_{1}, …, s_{n−1}} generates G_{r,n}.

2.3. Definition. The Π–length function on G_{r,n} is the function ℓ = ℓ_{Π} : G_{r,n} ⟶ℕ given by ℓ(g) = min { k ≥ 0 : g = r_{i} … r_{k}, for somer_{i} ∈ Π }.

2.4. Remark. Let S_{0} = {s_{0}, s_{1}, …, s_{n−1}}. Bremke and Malle [11] have studied the length function ℓ_{0} : G_{r,n} ⟶ℕ which is defined by

ℓ_{0}(g) = min { k ≥ 0 : g = r_{i} … r_{k}, for some r_{i} ∈ S_{0} } .

By definition, ℓ(g) ≤ ℓ_{0}(g), for all g ∈ G_{r,n}. Furthermore, it is not hard to see that ℓ(g) ≡ ℓ_{0}(g) (mod 2). Moreover, if w ∈ 𝔖_{n} then

ℓ(w) = ℓ_{0}(w) = # { (i, j) : 1 ≤ i < j ≤ n and i^{w} > j^{w} } .

(The last equality is well–known; see, for example, [24, Prop. 1.3].) Hence, Proposition 2.5 below gives an effective way of computing the Π–length function on G_{r,n}.

For α = (α_{1}, …, α_{n}) ∈ ℤ_{r}^{n} we set |α| = α_{1} + ⋯ + α_{n}.

2.5. Proposition. Suppose that α ∈ ℤ_{r}^{n} and w ∈ 𝔖_{n}. Then ℓ(t^{α}w) = |α| + ℓ(w).

Proof. By definition ℓ(t^{α}w) ≤ |α| + ℓ(w). Conversely, suppose that t^{α}w = r_{1} … r_{k}, for some r_{i} ∈ Π. Using (2.1) we can move each t_{i} ∈ {r_{1}, …, r_{k}} to give a new word in which all of the elements of T appear on the left. As every element of G_{r,n} can be written uniquely in the form t^{β}v, for β ∈ ℤ_{r}^{n} and v ∈ 𝔖_{n}, this new word must be t^{α}w. By (2.1), this rewriting process does not increase the Π–length of the word, however, it may decrease the Π–length if some cancellation occurs. Hence, k ≥ |α|+ℓ(w), completing the proof. □

2.6. Corollary. Suppose that α ∈ ℤ_{r}^{n} and that w ∈ 𝔖_{n}. Then

ℓ(t_{j} · t^{α}w) = \begin{cases}ℓ(t^{α}w) + 1, & ifα_{j} < r − 1, \\ ℓ(t^{α}w) − r + 1, & ifα_{j} = r − 1, \end{cases}

for 1 ≤ j ≤ n and ℓ(s_{i} · t^{α}w) = |α| + ℓ(s_{i}w), for 1 ≤ i < n − 1.

Note that t^{α}w·t_{j} = t_{j^{w^{−1}}} ·t^{α}w by (2.1) and ℓ(t^{α}w·s_{i}) = |α|+ℓ(ws_{i}), for 1 ≤ i < n−1 and 1 ≤ j ≤ n. Hence, Corollary 2.6 can be used to compute ℓ(g · t^{α}w) and ℓ(t^{α}w · g), for any g ∈ G_{r,n}.

It is sometimes convenient to describe G_{r,n} combinatorially as a set of ‘words’. Fix a primitive r^{th} root of unity ζ = exp(2πi/r) ∈ ℂ and set

n = {1, 2, …, n} and n_{ζ} = { mζ^{i} : m ∈ n and 0 ≤ i < r } .

Recall that if z ∈ ℂ then |z| is the complex modulus of z. In particular, if mζ^{i} ∈ n_{ζ} then |mζ^{i}| = m. Define a word in n_{ζ} to be an element of the set

G_{r,n} = \{ \underline{ω} = (ω_{1}, …, ω_{n}) : ω_{i} ∈ n_{ζ} and \{|ω_{1}|, …, |ω_{n}|\} = n \} .

If ω = (ω_{1}, …, ω_{n}) is a word then we abuse notation and write ω = ω_{1} … ω_{n}.

There is a faithful right action of G_{r,n} on G_{r,n} given by

ω_{1} … ω_{n} · t^{α}w = ζ^{α_{1}}ω_{1^{w}} … ζ^{α_{n}}ω_{n^{w}},

for α ∈ ℤ_{r}^{n} and w ∈ 𝔖_{n}. Consequently, there is a natural bijection G_{r,n} → G_{r,n} given by t^{α}w ↦ 1 … n · t^{α}w, so that |G_{r,n}| = r^{n}n! = |G_{r,n}|. Thus, we have described the regular representation of G_{r,n} as the permutation representation on the set of words G_{r,n}. Equivalently, G_{r,n} is the group of permutations of n_{ζ} such that (mζ^{i})^{g} = m^{g}ζ^{i}, for all m ∈ n, 0 ≤ i < r and g ∈ G_{r,n}.

3. Reflection subgroups

Recall that Π = {t_{1}, …, t_{n}, s_{1}, …, s_{n−1}}. In this section we define the reflection subgroups of G_{r,n} and show that every coset of a reflection subgroup contains a unique element of minimal Π–length.

3.1. Definition. A (standard) reflection subgroup of G_{r,n} is a subgroup which is generated by a subset of Π.

Geometrically, a reflection subgroup of G_{r,n} should be any subgroup which is generated by elements which act by (pseudo) reflections in the reflection representation of G_{r,n}. All of the elements of Π act as reflections in the reflection representation of G_{r,n}, so every standard reflection subgroup is a reflection subgroup in this geometric sense. If r > 2 then it is not difficult to see that there are ‘geometric reflection subgroups’ of G_{r,n} which are not standard reflection subgroups.

If J ⊆ Π let G_{J} = ⟨J⟩ be the corresponding (standard) reflection subgroup of G_{r,n}. This notation is inherently ambiguous because it can happen that G_{J} = G_{K} even though J ≠ K, for J, K ⊆ Π. For example, G_{Π} = G_{r,n} = G_{S_{0}} (recall that S_{0} = {s_{0}, s_{1}, …, s_{n−1}}), and yet Π ≠ S_{0} if n > 1. We start our study of the reflection subgroups by resolving this ambiguity.

A composition of n is a sequence μ = (μ_{1}, …, μ_{k}) of positive integers which sum to n. A signed composition of n is a sequence of non–zero integers μ = (μ_{1}, …, μ_{k}) such that |μ| = |μ_{1}| + ⋯ + |μ_{k}| = n. Let Λ_{n}^{±} be the set of signed compositions of n and let Λ_{n} be the set of compositions of n. Then Λ_{n} ⊆ Λ_{n}^{±}.

If μ = (μ_{1}, …, μ_{k}) ∈ Λ_{n}^{±} let μ^{+} = (|μ_{1}|, …, |μ_{k}|) and −μ = (−μ_{1}, …, −μ_{k}). Then μ^{+} ∈ Λ_{n} is a composition of n and −μ ∈ Λ_{n}^{±}. We set |μ|^{+} = \frac{1}{2} ∑_{i=1}^{k}(μ_{i}^{+} + μ_{i}), so that |μ|^{+} is the sum of the positive parts of μ. Similarly, let |μ|^{−} = \frac{1}{2} ∑_{i=1}^{k}(μ_{i}^{+} − μ_{i}) be the absolute value of the sum of the negative parts of μ. Then |μ| = |μ|^{−} + |μ|^{+} = n. Finally, set \overline{μ}_{0} = 0 and \overline{μ}_{i} = |μ_{1}| + ⋯ + |μ_{i}|, for i ≥ 1.

3.2. Definition. Suppose that μ = (μ_{1}, …, μ_{k}) ∈ Λ_{n}^{±} is a signed composition. Define

Π_{μ} = ⋃_{1≤i≤k}\{s_{\overline{μ}_{i−1}+1}, …, s_{\overline{μ}_{i}−1}\} ∪ ⋃_{\begin{matrix}1≤i≤k \\ μ_{i}>0 \end{matrix}}\{t_{\overline{μ}_{i−1}+1}, …, t_{\overline{μ}_{i}}\}.

Then Π_{μ} ⊆ Π so we set G_{μ} = G_{Π_{μ}}.

Let S = {s_{1}, …, s_{n−1}} ⊆ Π. Suppose that μ ∈ Λ_{n}^{±}. Then Π_{μ} ⊆ S if and only if −μ ∈ Λ_{n}. In general, Π_{μ} ⊆ Π and the reflection subgroup G_{μ} is conjugate to the reflection subgroup

G_{μ} ≅ ∏_{μ_{i}>0} G_{r,μ_{i}} × ∏_{μ_{j}<0} 𝔖_{−μ_{j}}

of G_{r,n}. Moreover, { G_{μ} : μ ∈ Λ_{n}^{±} } is the complete set of reflection subgroups of G_{r,n}.

3.3. Proposition. Suppose that n ≥ 1, r ≥ 2 and that J ⊆ Π. Then G_{J} = G_{μ}, for a unique signed composition μ ∈ Λ_{n}^{±}. Consequently, G_{r,n} has 2 · 3^{n−1} distinct reflection subgroups.

Proof. We prove both statements in the Proposition by induction on n. If n = 1 then G_{∅} = G_{(1)} and G_{Π} = G_{(−1)} are the only reflection subgroups of G_{r,1} so the Proposition holds. In particular, G_{r,1} has |Λ_{1}^{±}| = 2 reflection subgroups.

Suppose then that n > 1 and observe that Π_{r,n} = Π_{r,n−1} ∪ {s_{n−1}, t_{n}}. Let G′ = G_{r,n−1}. Then G_{μ} ∩ G′_{r,n} is a reflection subgroup of G′ and, by induction on n, every reflection subgroup is of the form G′_{μ}, for some μ = (μ_{1}, …, μ_{k}) ∈ Λ_{n−1}^{±}. Now, t_{n−1} ∈ G′_{μ} if and only if μ_{k} > 0, so one can check that

⟨G′_{(μ_{1},...,μ_{k})}, s_{n−1}, t_{n}⟩ = \begin{cases}⟨G′_{(μ_{1},...,μ_{k−1},μ_{k})}, s_{n−1}⟩, & if μ_{k} > 0, \\ ⟨G′_{(μ_{1},...,μ_{k−1},−μ_{k})}, s_{n−1}⟩, & if μ_{k} < 0. \end{cases}

Consequently, the reflection subgroups of G_{r,n} are precisely the groups G′_{μ}, ⟨G′_{μ}, t_{n}⟩, and ⟨G′_{μ}, s_{n−1}⟩, where μ = (μ_{1}, …, μ_{k}) ∈ Λ_{n−1}^{±}. Moreover, by (2.2) the groups arising this way for different μ ∈ Λ_{n−1}^{±} are all distinct. Applying the definitions,

G_{(μ_{1},...,μ_{k},−1)} = G′_{(μ_{1},...,μ_{k})},
G_{(μ_{1},...,μ_{k},1)} = ⟨G′_{(μ_{1},...,μ_{k})}, t_{n}⟩,
G_{(μ_{1},...,μ_{k}+ε_{k})} = ⟨G′_{(μ_{1},...,μ_{k})}, s_{n−1}⟩,

where ε_{k} = 1 if μ_{k} > 0 and ε_{k} = −1 if μ_{k} < 0. Hence, the reflection subgroups of G_{r,n} are naturally indexed by the signed compositions of n. Consequently, by induction, G_{r,n} has 3|Λ_{n−1}^{±}| = 2 · 3^{n−1} reflection subgroups. □

It follows from the definitions and Proposition 3.3that Π_{μ} is the unique maximal subset of Π (under inclusion) which generates the reflection subgroup G_{μ}. In contrast, if μ ∈ Λ_{n}^{±} then the reader can check that there are ∏_{i:μ_{i}>0} μ_{i} distinct minimal subsets of Π which generate G_{μ}. Thus, the (minimal) subsets of Π which generate the reflection subgroups are, in general, not unique.

4. Distinguished coset representatives.

In this section we describe, both algebraically and combinatorially, a set of ‘distinguished’ coset representatives for the reflection subgroups of G_{r,n}.

Fix a composition λ = (λ_{1}, …, λ_{k}) of n. Then 𝔖_{λ} = 𝔖_{λ_{1}} × ⋯ × 𝔖_{λ_{k}} is a parabolic, or Young subgroup of 𝔖_{n}. According to our conventions 𝔖_{λ} = G_{−λ}, so 𝔖_{λ} is a reflection subgroup of G_{r,n}. Let

𝒟_{λ} = { d ∈ 𝔖_{n} : ℓ(d) ≤ ℓ(w) for all w ∈ 𝔖_{λ}d } .

Then, as is well–known, 𝒟_{λ} is a complete set of right coset representatives for 𝔖_{λ} in 𝔖_{n}. Moreover, if d ∈ 𝒟_{λ} then d is the unique element of minimal length in the coset 𝔖_{λ}d; see, for example, [14, Prop. 2.1.1]. It is not hard to see that T 𝒟_{λ} is a complete set of minimal length coset representatives for G_{−λ} = 𝔖_{λ} in G_{r,n}. We want to generalize this observation to all reflection subgroups.

Recall that if μ = (μ_{1}, …, μ_{k}) ∈ Λ_{n}^{±} then μ^{+} = (|μ_{1}|, …, |μ_{k}|) is a composition of n. Consulting the definitions, G_{μ} ∩ 𝔖_{n} = 𝔖_{μ^{+}}. Similarly, define

T_{μ} = G_{μ} ∩ T = ⟨t_{i} | t_{i} ∈ G_{μ}⟩
= ⟨t_{i} | \overline{μ}_{j−1} < i ≤ \overline{μ}_{j} for some j with μ_{j} > 0⟩.

Then, T_{μ} ≅ (ℤ/rℤ)^{|μ|^{+}}.

With this notation, (2.2) gives the following description of G_{μ} as a semidirect product of T_{μ} and 𝔖_{μ^{+}}.

4.1. Lemma. Suppose that μ ∈ Λ_{n}^{±}. Then G_{μ} = T_{μ} ⋊ 𝔖_{μ^{+}}.

Since T ≅ (ℤ/rℤ)^{n} is an abelian group, every subgroup of T is a normal subgroup of T. In particular, if G_{μ} is a reflection subgroup of G_{r,n} then T_{μ} is normal in T and T/T_{μ} ≅ (ℤ/rℤ)^{|μ|^{−}} ≅ T_{−μ}. Further, T_{μ}T_{−μ} = T = T_{−μ}T_{μ}, for all μ ∈ Λ_{n}^{±}.

Mimicking the definition of 𝒟_{μ^{+}} we have:

4.2. Definition. Suppose that μ ∈ Λ_{n}^{±}. Set

ℰ_{μ} = { e ∈ G_{r,n} : ℓ(e) ≤ ℓ(g) for all g ∈ G_{μ}e } .

We can now prove the main result of this section which shows that ℰ_{μ} is a (distinguished) set of coset representatives for G_{μ} in G_{r,n}.

4.3. Theorem. Suppose that μ ∈ Λ_{n}^{±}. Then ℰ_{μ} = T_{−μ} × 𝒟_{μ^{+}} and ℰ_{μ} is a complete set of right coset representatives for G_{μ} in G_{r,n}.

Proof. We first show that T_{−μ} × 𝒟_{μ^{+}} is a complete set of coset representatives for G_{μ} in G_{r,n}. Suppose that t^{α}w ∈ G_{r,n}, where α ∈ ℤ_{r}^{n} and w ∈ 𝔖_{n}. Define β = (β_{1}, …, β_{n}) ∈ ℤ_{r}^{n} by

β_{i} = \begin{cases}α_{i}, & if t_{i} ∉ G_{μ} ⟺ t_{i} ∈ T_{−μ}, \\ 0, & if t_{i} ∈ G_{μ} ⟺ t_{i} ∉ T_{−μ}. \end{cases}

Then, by definition, t^{β} ∈ T_{−μ}. Moreover, G_{μ}t^{α}w = G_{μ}t^{β}w and ℓ(t^{α}w) ≥ ℓ(t^{β}w), with equality if and only if α = β.

Write w = vd, where v ∈ 𝔖_{μ^{+}} and d ∈ 𝒟_{μ^{+}}. Let γ = βv = (β_{1^{v}}, …, β_{n^{v}}). Then t^{β}v = vt^{γ}, by (2.1), so that t^{γ} = v^{−1}t^{β}v ∈ T_{−μ} since 𝔖_{μ} centralizes T_{−μ}. Consequently, G_{μ}t^{α}w = G_{μ}t^{γ}d, where t^{γ} ∈ T_{−μ} and d ∈ 𝒟_{μ^{+}}. However, by Lemma 4.1,

[G_{r,n} : G_{μ}] = [T : T_{−μ}] · [𝔖_{n} : 𝔖_{μ^{+}}] = #(T_{−μ} × 𝒟_{μ^{+}}).

Therefore, T_{−μ} × 𝒟_{μ^{+}} is a complete set of right coset representatives for G_{μ} in G_{r,n}.

It remains to prove that ℰ_{μ} = T_{−μ} × 𝒟_{μ^{+}}. Suppose that, as above, we have G_{μ}t^{α}w = G_{μ}t^{γ}d, for α ∈ ℤ_{r}^{n}, w ∈ 𝔖_{n}, t^{γ} ∈ T_{−μ} and d ∈ 𝒟_{μ^{+}}. The argument of the first paragraph shows that ℓ(t^{γ}d) ≤ ℓ(t^{α}w) with equality if and only if t^{α} ∈ T_{−μ} and w ∈ 𝒟_{μ^{+}}. That is, if and only if α = γ and w = d. Hence, ℰ_{μ} = T_{−μ} × 𝒟_{μ^{+}} as claimed. □

Theorem 4.3 shows that every coset of a reflection subgroup contains a unique element of minimal Π–length. We call ℰ_{μ} the set of distinguished coset representatives for G_{μ} in G_{r,n}. 4.4. Example Suppose that r ≥ 2 and consider G_{r,2} = (ℤ/rℤ) ≀ 𝔖_{2}. Then Π = {t_{1}, t_{2}, s_{1}} and G_{r,2} has six reflection subgroups. The following table describes these groups and the corresponding sets of distinguished right coset representatives.

\begin{matrix}μ & G_{μ} & Π_{μ} & ℰ_{μ} \\ (−1, −1) & 1 & ∅ & T × 𝔖_{2} \\ (1, −1) & \{ t_{1}^{k} : 0 ≤ k < r \} & \{t_{1}\} & ⟨t_{2}⟩ × 𝔖_{2} \\ (−1, 1) & \{ t_{2}^{k} : 0 ≤ k < r \} & \{t_{2}\} & ⟨t_{1}⟩ × 𝔖_{2} \\ (1, 1) & T & \{t_{1}, t_{2}\} & 𝔖_{2} \\ (−2) & 𝔖_{2} & \{s_{1}\} & T \\ (2) & T ⋊ 𝔖_{2} & \{t_{1}, t_{2}, s_{1}\} & 1 \end{matrix}

For each reflection subgroup we have given the factorization of ℰ_{μ} from Theorem 4.3. Observe that the reflection subgroups do not depend in a crucial way on r and that |ℰ_{μ}| = |G_{r,n}|/|G_{μ}| is a polynomial in r, for μ ∈ Λ_{n}^{±} (and r ≥ 2). ♢

We now give combinatorial interpretations of the set of distinguished coset representatives ℰ_{μ}, for μ ∈ Λ_{n}^{±}, which is similar to the description of 𝒟_{μ^{+}} in terms of row standard tableaux (see [24, Prop. 3.3]).

Fix a composition λ ∈ Λ_{n}. The diagram of λ is the set

[λ] = { (i, j) ∈ ℕ^{2} : 1 ≤ j ≤ λ_{i} and 1 ≤ i ≤ ℓ(λ) } .

Here ℓ(λ) is the number of non–zero parts of λ. We think of [λ] as being an array of boxes in the plane.

Now suppose that μ ∈ Λ_{n}^{±}. A μ–tableau is a map 𝔱: [μ^{+}] ⟶ n_{ζ}. We identify a μ– tableau with a diagram for μ^{+} which is labeled by elements of n_{ζ}. If 𝔱 is a μ–tableau let |𝔱| be the tableau obtained by taking the complex modulus of the entries in 𝔱; that is, |𝔱|(x) = |𝔱(x)|, for all x ∈ [μ^{+}]. 4.5. Example Let μ = (2, −3, 1, −1). Then four μ–tableaux are:

1 2 3 4 5 6 7  _{1 2 ; ^{3ζ 4ζ^{2} 5ζ^{3}} ; 6 ; 7ζ^{4}}  _{3 6 ; ^{2ζ 5ζ^{2} 7ζ^{3}} ; 1 ; 4ζ}  and  _{7 6ζ ; ^{2ζ 5ζ^{2} 3ζ^{3}} ; 1 ; 4ζ}.

♢

As remarked at the end of section 2 we can think of G_{r,n} as the group of permutations of n_{ζ} such that (mζ^{i})^{g} = m^{g}ζ^{i}, for all mζ^{i} ∈ n_{ζ} and all g ∈ G_{r,n}. Consequently, G_{r,n} acts on the set of μ–tableaux by composition of maps. Thus, if 𝔱 is a μ–tableau and g ∈ G_{r,n} then 𝔱^{g} is the tableau with 𝔱^{g}(x) = 𝔱(x)^{g}, for x ∈ [μ^{+}].

Let 𝔱^{μ} be the μ–tableau which has the numbers 1, …, n entered in order, from left to right and then top to bottom, along the rows of [μ^{+}]. The first μ–tableau in Example 4.5 is 𝔱^{μ} when μ = (−2, 3, −1, 1).

So far none of the combinatorial definitions above distinguish between compositions and signed compositions. We now single out a set of μ–tableaux that are in bijection with ℰ_{μ}. First, define a total order ⪯ on n_{ζ} by declaring that aζ^{i} ⪯ bζ^{j} if a < b, or a = b and i > j. Then

ζ^{m−1} ⪯ ζ^{m−2} ⪯ ⋯ ⪯ ζ ⪯ 1 ⪯ 2ζ^{m−1} ⪯ ⋯ ⪯ 2 ⪯ ⋯ ⪯ nζ^{m−1} ⪯ ⋯ ⪯ n.

4.6. Definition. Suppose that μ ∈ Λ_{n}^{±}. A μ–tableau 𝔱 is row standard if it satisfies the following three conditions:

a) The set of entries in the tableau |𝔱| is {1, …, n}.

b) The entries in row i of 𝔱 belong to {1, …, n} whenever μ_{i} > 0. c) In each row the entries of 𝔱 appear, from left to right, in increasing order with respect

to ⪯.

For example, the first three of the (2, −3, 1, −1)–tableaux in Example 4.5 are row standard. The last tableau in this example is not row standard because it fails conditions (b) and (c).

The action of G_{r,n} on the set of μ–tableau which satisfy condition (a) of Definition 4.6 gives a realization of the regular representation of G_{r,n}. Consequently, the map g ↦ 𝔱^{μ}g, for g ∈ G_{r,n}, is a bijection from G_{r,n} to the set of these μ–tableaux. If 𝔱 is such a μ–tableau let d_{𝔱} be the unique element of G_{r,n} such that 𝔱 = 𝔱^{μ}d_{𝔱}.

4.7. Proposition. Suppose that μ ∈ Λ_{n}^{±}. Then

ℰ_{μ} = { d_{𝔱} : 𝔱 is a row standard μ–tableau } .

Proof. By definition, the orbit 𝔱^{μ}G_{μ} = { 𝔱^{μ}g : g ∈ G_{μ} } of 𝔱^{μ} under G_{μ} consists of all those tableaux which can be obtained by permuting the entries of each row of 𝔱^{μ} and multiplying the entries in row i by a power of ζ when μ_{i} > 0. Consequently, 𝔱^{μ} is the unique row standard μ–tableaux in 𝔱^{μ}G_{μ}, so that each right coset of G_{μ} in G_{r,n} contains a unique element e such that 𝔱^{μ}e is row standard. Now, ℰ_{μ} = T_{−μ}𝒟_{μ^{+}} by Theorem 4.3 and T_{−μ} acts on 𝔱^{μ} by multiplying the entries in row i by different powers of ζ when μ_{i} < 0. If d ∈ 𝒟_{μ^{+}} then it is well–known that the entries in 𝔱^{μ}d increase from left to right along each row; see, for example, [24, Prop. 3.3]. Hence, 𝔱^{μ}e is row standard whenever e ∈ ℰ_{μ}. This completes the proof. □

In the case of the symmetric groups the set of distinguished coset representatives can be described combinatorially in terms of ‘descents’. Explicitly, if w ∈ 𝔖_{n} then its descent set is

Des(w) = { s ∈ S : ℓ(sw) < ℓ(w) } = { s_{i} : 1 ≤ i < n and i^{w} > (i + 1)^{w} } .

If μ is a composition of n then the connection between distinguished coset representatives and descents is that

(4.8)  𝒟_{μ} = { d ∈ 𝔖_{n} : Des(d) ⊆ S − Π_{−μ} } .

There is an analogous description of ℰ_{μ}, for μ ∈ Λ_{n}^{±}. If α ∈ ℤ_{r}^{n} define the colour of t^{α} to be the set Col(t^{α}) = { t_{i} ∈ T : α_{i} > 0 } . Then using Theorem 4.3 it is easy to see that if μ ∈ Λ_{n}^{±} then

ℰ_{μ} = { t^{α}w ∈ G_{r,n} : Col(α) ∪ Des(w) ⊆ Π − Π_{−μ} } .

We remark that it is easy to rephrase this last statement combinatorially in terms of words in G_{r,n}.

4.9. Remark. It is easy to check that ℰ_{μ}^{−1} = 𝒟_{μ}^{−1} × T_{−μ} = 𝒟_{μ}^{−1} × T_{−μ} is a complete set of left coset representatives for G_{μ} in G_{r,n}. Moreover, e ∈ ℰ_{μ}^{−1} if and only if ℓ(e) ≤ ℓ(g) for all g ∈ eG_{μ}, so every left coset of G_{μ} in G_{r,n} contains a unique element of minimal Π–length.

4.10. Remark. Mak [20] has shown that every coset of a reflection subgroup contains a unique element of minimal length with respect to the length function ℓ_{0} defined in Remark 2.4. Mak’s set of coset representatives is different from ℰ_{μ}. Nonetheless, it does admit a factorization which is similar to the factorization of ℰ_{μ} given in Theorem 4.3. To describe this if μ = (μ_{1}, …, μ_{k}) ∈ Λ_{n}^{±} then set

ℰ′_{μ} = ∏_{\begin{matrix}k≥j≥1 \\ μ_{j}<0 \end{matrix}} ∏_{\overline{μ}_{j}≥i>\overline{μ}_{j−1}} [\{1\} ∪ \{ s_{\overline{μ}_{j−1}+1}s_{\overline{μ}_{j−1}+2} … s_{i−1}t_{i}^{k} : 1 ≤ k < r \} ]×𝒟_{μ},

where the product is taken in order from left to right in terms of decreasing values of i. One can show that ℰ′_{μ} is Mak’s set of coset representatives for G_{μ} in G_{r,n}. As we will never need this result we leave the proof to the reader.

5. Double Coset representatives

Our next aim is to describe the double cosets of reflection subgroups. In order to do this we first recall some well–known facts about the symmetric group 𝔖_{n}. Suppose that μ and ν are compositions of n. Then 𝔖_{μ} and 𝔖_{ν} are Young, or parabolic, subgroups of 𝔖_{n}. Set 𝒟_{μν} = 𝒟_{μ} ∩ 𝒟_{ν}^{−1}. Then 𝒟_{μν} is a complete set of (𝔖_{μ}, 𝔖_{ν})–double coset representatives in 𝔖_{n}; see, for example, [24, Prop. 4.4]. Moreover, if d ∈ 𝒟_{μν} then d^{−1}𝔖_{μ}d ∩ 𝔖_{ν} is a Young subgroup of 𝔖_{n}; see, for example, [24, Lemma 4.3]. Define μd∩ν to be the unique composition of n such that 𝔖_{μd∩ν} = d^{−1}𝔖_{μ}d ∩ 𝔖_{ν}. We remark that the composition μd ∩ ν can be determined by comparing the row stabilizers of the tableaux 𝔱^{μ}d and 𝔱^{ν}.

5.1. Lemma. Suppose that μ, ν ∈ Λ_{n}^{±} and d ∈ 𝒟_{μ^{+}}_{ν^{+}}. Then d^{−1}G_{μ}d ∩ G_{ν} is a reflection subgroup of G_{r,n}.

Proof. The group G_{ν} consists of those elements of G_{r,n} which act on 𝔱^{ν} by first multiplying each entry of row l by possibly different powers of ζ, if ν_{l} > 0, and then permuting the entries in each row of the resulting tableaux. Similarly, the group d^{−1}G_{μ}d consists of those elements of G_{r,n} which act on the row standard tableau 𝔱^{μ}d by multiplying each entry of row k by different powers of ζ, if μ_{k} > 0, and then permuting the entries in each row. Consequently, the subgroup d^{−1}G_{μ}d ∩ G_{ν} is generated by the elements {s_{i}, t_{j}}, where i runs over those integers for which i and i+1 are in the same row of 𝔱^{ν} and in the same row of 𝔱^{μ}d, and j ∈ n is in row l of 𝔱^{ν} with ν_{l} > 0 and j is in row k of 𝔱^{μ}d with μ_{k} > 0 (cf. the proof of [24, Lemma 4.3]). Therefore, d^{−1}G_{μ}d ∩ G_{ν} = G_{σ}, where σ is the unique signed composition such that σ^{+} = μ^{+}d ∩ ν^{+} and σ_{i} > 0 if and only if ν_{j} > 0 and μ_{k} > 0, where \overline{σ}_{i} appears in row j of 𝔱^{ν} and row k of 𝔱^{μ}d. □

Suppose that d ∈ 𝒟_{μ^{+}}_{ν^{+}}, for μ, ν ∈ Λ_{n}^{±}. Then d^{−1} ∈ 𝒟_{ν^{+}}_{μ^{+}}, since 𝒟_{ν^{+}}_{μ^{+}} = 𝒟_{μ^{+}ν^{+}}^{−1}. Therefore, G_{μ} ∩ dG_{ν}d^{−1} is also a reflection subgroup of G_{r,n}.

5.2. Definition. Suppose that μ, ν ∈ Λ_{n}^{±} and d ∈ 𝒟_{μ^{+}}_{ν^{+}}. Then μd ∩ ν is the signed composition of n such that G_{μd∩ν} = d^{−1}G_{μ}d ∩ G_{ν} and μ ∩ dν is the signed composition such that G_{μ∩dν} = G_{μ} ∩ dG_{ν}d^{−1}.

Note that the proof of Lemma 5.1 gives a recipe for computing μd ∩ ν. Note also that μd ∩ ν = d^{−1}μ ∩ ν, for d ∈ 𝒟_{μ^{+}}_{ν^{+}} and μ, ν ∈ Λ_{n}^{±}.

We now describe a set of (G_{μ}, G_{ν})–double coset representatives. We do this by generalizing the description of the double cosets of the Young subgroups of the symmetric group in terms of row semistandard tableaux.

5.3. Definition. Suppose that μ ∈ Λ_{n}^{±}. A μ–tableau T : [μ^{+}] ⟶ n_{ζ} is row semistandard if

a) The entries in row i of T belong to {1, …, n} whenever μ_{i} > 0.

b) The entries in each row of T appear in weakly increasing order, from left to right, with respect to ⪯.

There is a map from the set of row semistandard tableau to the set of row standard tableaux. To define this first observe that a row semistandard μ–tableau T determines a unique total order <_{T} on [μ^{+}] where x <_{T} x′, for x, x′ ∈ [μ^{+}], if

a) |T(x)| < |T(x′)|, or

b) |T(x)| = |T(x′)| and x is in an earlier row of [μ^{+}] than x′, or

c) |T(x)| = |T(x′)| and x and x′ are in the same row and x is to the left of x′.

Let x_{1} <_{T} ⋯ <_{T} x_{n} be the nodes in [μ^{+}]. Then the μ–tableau T^{∗} is defined by the requirements that |T^{∗}(x_{i})| = i and arg T^{∗}(x_{i}) = arg T(x_{i}), for 1 ≤ i ≤ n. By construction, T^{∗} is a row standard μ–tableaux. Moreover, it is easy to see that the map T ↦ T^{∗} is injective.

5.4. Definition. Suppose that μ, ν ∈ Λ_{n}^{±} and let T be a μ–tableau. Then T has type ν if a) |ν_{j}| = # { x ∈ [μ^{+}] : |T(x)| = j }, for j ≥ 1.

b) If ν_{j} > 0 then ν_{j} = # { x ∈ [μ^{+}] : T(x) = j }. Let T_{ζ}(μ, ν) = { T : [μ^{+}]⟶n_{ζ} : T is row semistandard μ–tableau of type ν }. If μ and ν are compositions let T (μ, ν) = { T : [μ^{+}]⟶n : T is row semistandard μ–tableau of type ν }.

See Example 5.9 below for these definitions in action.

5.5 ([24, Prop. 4.4]). Suppose that μ, ν ∈ Λ_{n}. Then

𝒟_{μν} = { d_{T^{∗}} : T ∈ T (μ, ν) }

is a complete set of (𝔖_{μ}, 𝔖_{ν}) double coset representatives in 𝔖_{n}. Moreover, if d ∈ 𝒟_{μν} then ℓ(d) ≤ ℓ(w), for all w ∈ 𝔖_{μ}d𝔖_{ν}, with equality if and only if w = d.

If 𝔱 is a row standard tableau let ν(𝔱)′ be the tableau obtained by replacing each entry mζ^{a} in 𝔱 with kζ^{a′} if m appears in row k of 𝔱^{ν}, where a′ = 0 if ν_{k} > 0 and a′ = a otherwise. Now define ν(𝔱) to be the row semistandard tableau obtained by reordering the entries in each row of ν(𝔱)′ so that they are in increasing order. Then ν(𝔱) is a row semistandard tableau of type ν.

For example, let ν = (2, −2, 1) and 𝔱 = \begin{matrix}3 & 4ζ^{a} & 5ζ^{b} \\ 1 & 2 \end{matrix} , where 0 ≤ a, b < r. Then, by

definition, ν(𝔱)′ = \begin{matrix}2 & 2ζ^{a} & 3 \\ 1 & 1 \end{matrix} and ν(𝔱) = \begin{matrix}2ζ^{a} & 2 & 3 \\ 1 & 1 \end{matrix} .

5.6. Proposition. Suppose that μ and ν are signed compositions of n and let

ℰ_{μν} = { d_{T^{∗}} : T ∈ T_{ζ}(μ, ν) } .

Then ℰ_{μν} is a complete set of (G_{μ}, G_{ν}) double coset representatives in G_{r,n}. Moreover, if e ∈ ℰ_{μν} then ℓ(e) ≤ ℓ(g), for all g ∈ G_{μ}eG_{ν}.

Proof. By Proposition 4.7 the right cosets of G_{μ} in G_{r,n} are naturally indexed by the row standard μ–tableaux. Hence, the (G_{μ}, G_{ν})–double cosets are indexed by the G_{ν}–orbits of the row standard μ–tableaux. Using the definitions it is easy to see that two μ–tableaux 𝔰 and 𝔱 belong to the same G_{ν}–orbit if and only if ν(𝔰) = ν(𝔱). Moreover, if 𝔱 is row standard then ν(𝔱) is row semistandard. Finally, if T is a row semistandard μ–tableau of type ν then T^{∗} is a row standard μ–tableau such that T = ν(T^{∗}). Hence, ℰ_{μν} is a complete set of (G_{μ}, G_{ν})–double coset representatives in G_{r,n}.

To complete the proof we need to show that if T ∈ T_{ζ}(μ, ν) then d_{T^{∗}} is an element of minimal length in the double coset G_{μ}d_{T^{∗}}G_{ν}. For convenience, let d = d_{|T^{∗}|}. Then, d ∈ 𝒟_{μ^{+}}_{ν^{+}} by (5.5). Now, by the last paragraph G_{μ}d_{T^{∗}}G_{ν} = ⋃_{𝔱} G_{μ}d_{𝔱}, where 𝔱 runs over the row standard μ–tableau 𝔱 such that ν(𝔱) = T. By definition, d_{T^{∗}} = t_{1}^{α_{1}} … t_{n}^{α_{n}}d, where if x ∈ [μ^{+}] then T^{∗}(x) = ζ^{α_{i}}i^{d} if and only if T(x) = ζ^{α_{i}}k and i^{d} is in row k of 𝔱^{ν}. Now suppose that 𝔱 is any row standard μ–tableaux such that ν(𝔱) = T. Then, using (5.5) again, d_{𝔱} = t_{1}^{β_{1}} … t_{n}^{β_{n}}d_{|𝔱|} = t_{1}^{β_{1}} … t_{n}^{β_{n}}du, for some u ∈ 𝔖_{ν} and where β_{i} = α_{i^{w}}, for some w ∈ 𝔖_{n} (since ν(𝔱) = T). Therefore,

ℓ(d_{𝔱}) = β_{1} + ⋯ + β_{n} + ℓ(du) = α_{1} + ⋯ + α_{n} + ℓ(du)
≥ α_{1} + ⋯ + α_{n} + ℓ(d) = ℓ(d_{T^{∗}}),

with equality if and only if u = 1. By Theorem 4.3, d_{𝔱} is the unique element of minimal length in the coset G_{μ}d_{𝔱}, for each such 𝔱. Therefore, ℓ(d_{T^{∗}}) ≤ ℓ(g) for all g ∈ G_{μ}d_{T^{∗}}G_{ν} as claimed. □

Note that we are not claiming that each double coset of two reflection subgroups of G_{r,n} contains a unique element of minimal length. Indeed, the proof of Proposition 5.6 shows that if T is a row semistandard μ–tableau of type ν then the double coset G_{μ}d_{T^{∗}}G_{ν} contains more than one element of minimal length if and only if there exist integers b, c, not both zero, such that mζ^{b} and mζ^{c} appear in the same row of T, for some m ∈ n. For future comparison we make this statement explicit.

If d ∈ 𝒟_{μ^{+}}_{ν^{+}} let T_{d} ∈ T (μ^{+}, ν^{+}) be the unique row semistandard tableau such that d = d_{T_{d}^{∗}} as in (5.5). If X ⊆ G_{r,n} let X^{−1} = { g : g^{−1} ∈ X }.

5.7. Lemma. Suppose that μ, ν ∈ Λ_{n}^{±}. Then

ℰ_{μν} = ∐_{d∈𝒟_{_{μ^{+}}_{ν^{+}}}}{ t_{1}^{α_{1}} … t_{n}^{α_{n}} ∈ T_{−μ∩d(−ν)} | α_{i} ≤ α_{j} whenever i^{d} and j^{d} are in the same row of T_{d}^{∗} and the same row of 𝔱^{ν}} d

Moreover,

ℰ_{μ} ∩ ℰ_{ν}^{−1} = ∐_{d∈𝒟_{_{μ^{+}}_{ν^{+}}}} T_{−μ∩d(−ν)}d = { e ∈ G_{r,n} : ℓ(e) ≤ ℓ(g) for all g ∈ G_{μ}eG_{ν} }

is the set of elements in G_{r,n} which are of minimal length in their (G_{μ}, G_{ν})–double coset.

Proof. Observe that 𝒟_{μ^{+}}_{ν^{+}} = 𝒟_{(−μ)^{+}(−ν)^{+}}. Therefore, if d ∈ 𝒟_{μ^{+}}_{ν^{+}} then the signed composition −μ ∩ d(−ν) in the statement of the Corollary makes sense by Definition 5.2. (Note, however, that the two signed compositions −μ∩d(−ν) and −(μ∩dν) are not equal in general.)

By Proposition 5.6, we have ℰ_{μν} = { d_{T^{∗}} : T ∈ T_{ζ}(μ, ν) }. Fix a row semistandard μ–tableau T of type ν. Then, as in the proof of Proposition 5.6, d_{T^{∗}} = t_{1}^{α_{1}} … t_{n}^{α_{n}}d, where d = d_{|T^{∗}|} ∈ 𝒟_{μ^{+}}_{ν^{+}} and, for all x ∈ [μ^{+}] if T^{∗}(x) = ζ^{α_{i}}i^{d} then T(x) = ζ^{α_{i}}k where i^{d} is in row k of 𝔱^{ν}. In particular, α_{i} = 0 if t_{i} ∈ T_{μ} or if t_{i^{d}} ∈ T_{ν}. Therefore, α_{i} > 0 only if t_{i} ∈ T_{−μ} ∩ dT_{−ν}d^{−1} = T_{−μ∩d(−ν)}. If t_{i} ∈ T_{−μ∩d(−ν)} then the integer α_{i} can take any value in {0, …, r − 1} provided that this is compatible with T being row semistandard. That is, we require that α_{i} ≤ α_{j} whenever i^{d} and j^{d} are in the same row of T^{∗} and in the same row of 𝔱^{ν}. This gives the decomposition of ℰ_{μν} in the statement of Lemma.

For the final claim, suppose that d ∈ 𝒟_{μ^{+}}_{ν^{+}} and let T = ν(𝔱^{μ}d). By the last paragraph, if t ∈ T then ν(𝔱^{μ}td_{T^{∗}}) = T if and only if t ∈ T_{−μ∩d(−ν)}. By the last paragraph again, if t ∈ T_{−μ∩d(−ν)}d then td is an element of minimal length in the double coset G_{μ}tdG_{ν}. That ℰ_{μ} ∩ ℰ_{ν}^{−1} = ∐_{d} T_{−μ∩d(−ν)}d is now follows from the definition of row semistandard tableaux. □

5.8. Corollary. Suppose that μ, ν ∈ Λ_{n}^{±} and d ∈ 𝒟_{μ^{+}}_{ν^{+}}. Then G_{r,n} contains |T_{−μ∩d(−ν)}| elements of the form t^{α}d which are of minimal length in their (G_{μ}, G_{ν})–double coset, for some α ∈ ℤ_{r}^{n}. Moreover, if T = ν(𝔱^{μ}d) then |T_{−μ∩d(−ν)}| = r^{wt(T)}, where wt(T) is the number of pairs (i, j) such that j appears in row i of T and μ_{i} < 0 and ν_{j} < 0.

Proof. That |T_{−μ∩d(−nu)}| counts number of elements of the form t^{α}d which are of minimal length in their (G_{μ}, G_{ν})–double coset is immediate from Lemma 5.7. The second claim follows from the observation that the tableaux { 𝔱^{μ}td : t ∈ T_{−μ∩d(−ν)} } differ only in that any of the numbers appearing in row i of 𝔱^{μ}d and row j of 𝔱^{ν} can be multiplied by arbitrary powers of ζ whenever μ_{i} < 0 and ν_{j} < 0. □

5.9. Example Suppose that r ≥ 2 and n = 5 and let μ = (3, −2) and ν = (−2, −2, 1). Then the set of row semistandard μ–tableaux T of type ν, together with the corresponding row standard tableau T^{∗} and the coset representatives d_{T^{∗}} ∈ ℰ_{μν} = ℰ_{(3,−2)(−2,−2,1)}, is as follows (we set d = d_{|T^{∗}|}).

\begin{matrix}T & T^{∗} & d_{T^{∗}} & |T_{−μ∩d(−ν)}| & μ ∩ dν \\ \begin{matrix}1 & 1 & 2 \\ 2ζ^{a} & 3 \end{matrix} & \begin{matrix}1 & 2 & 3 \\ 4ζ^{a} & 5 \end{matrix} & t_{4}^{a} & r & (−2, −1^{3}) \\ \begin{matrix}1 & 1 & 3 \\ 2ζ^{b} & 2ζ^{c} \end{matrix} & \begin{matrix}1 & 2 & 5 \\ 3ζ^{b} & 4ζ^{c} \end{matrix} & t_{4}^{b}t_{5}^{c}s_{3}s_{4} & r^{2} & (−2, 1, −2) \\ \begin{matrix}1 & 2 & 2 \\ 1ζ^{a} & 3 \end{matrix} & \begin{matrix}1 & 3 & 4 \\ 2ζ^{a} & 5 \end{matrix} & t_{4}^{a}s_{3}s_{2} & r & (−1, −2, −1^{2}) \\ \begin{matrix}1 & 2 & 3 \\ 1ζ^{a} & 2ζ^{b} \end{matrix} & \begin{matrix}1 & 3 & 5 \\ 2ζ^{a} & 4ζ^{b} \end{matrix} & t_{4}^{a}t_{5}^{b}s_{3}s_{2}s_{4} & r^{2} & (−1^{2}, 1, −1^{2}) \\ \begin{matrix}2 & 2 & 3 \\ 1ζ^{b} & 1ζ^{c} \end{matrix} & \begin{matrix}3 & 4 & 5 \\ 1ζ^{b} & 2ζ^{c} \end{matrix} & t_{4}^{b}t_{5}^{c}s_{3}s_{2}s_{4}s_{3}s_{1}s_{2} & r^{2} & (−2, 1, −2) \end{matrix}

where 0 ≤ a, b, c < r and b ≤ c. We use exponentials in the signed compositions to indicate consecutive repeated parts. Therefore, there are 2r^{2} + 3r (G_{μ}, G_{ν})–double cosets in G_{r,n}. When checking the entries in this table observe that the signed composition μ ∩ dν = μ ∩ νd^{−1} can be computed by intersecting G_{μ} with the ‘row stabilizer’ of 𝔱^{ν}d^{−1} as in the proof of Lemma 5.1. Note that |T_{−μ∩d(−ν)}| can be computed without finding −μ ∩ d(−ν) by using Corollary 5.8. ♢

5.10. Remark. If μ and ν are compositions of n then 𝒟_{μν} = 𝒟_{μ} ∩ 𝒟_{ν}^{−1} is a complete set of minimal length (𝔖_{μ}, 𝔖_{ν})–double coset representatives in 𝔖_{n} by (5.5). In contrast, it is not hard to show that ℰ_{μν} ⊆ ℰ_{μ} ∩ ℰ_{ν}^{−1} with ℰ_{μ} ∩ ℰ_{ν}^{−1} being strictly bigger than ℰ_{μν} in general. For example, if we take μ = (3, −2) and ν = (−2^{2}, 1) then |ℰ_{μ}∩ℰ_{ν}^{−1}| = 3r^{2}+2r, whereas |ℰ_{μν}| = 2r^{2} + 3r by Example 5.9. So ℰ_{μν} ⊊ ℰ_{μ} ∩ ℰ_{ν}^{−1} since r > 1.

6. The cyclotomic Solomon algebra

Suppose that R is a commutative ring (with one) and let RG_{r,n} be the group ring of G_{r,n} over R. In this section we use the distinguished coset representatives of the reflection subgroups of G_{r,n} to define an analogue of Solomon’s descent algebra for the complex reflection group G_{r,n}.

Recall that for each reflection subgroup G_{μ} of G_{r,n} we have a distinguished set ℰ_{μ} of right coset representatives, for μ ∈ Λ_{n}^{±}. Define

E_{μ} = ∑_{e∈ℰ_{μ}} e ∈ RG_{r,n}.

The main aim of this paper is to understand the subalgebra of RG_{r,n} which is generated by these elements.

6.1. Definition. Suppose that r > 1. The cyclotomic Solomon algebra

Sol(G_{r,n}) = Sol_{R}(G_{r,n})

is the subalgebra of RG_{r,n} generated by { E_{μ} : μ ∈ Λ_{n}^{±} }.

It is not yet clear that Sol(G_{r,n}) is a subalgebra of RG_{r,n}; this will be established in Theorem 6.8 below. We begin by taking advantage of the factorization of ℰ_{μ} given by Theorem 4.3. To do this, for i = 1, …, n and λ ∈ Λ_{n} define

F_{i} = ∑_{k=0}^{r−1} t_{i}^{k} and D_{λ} = ∑_{d∈𝒟_{λ}} d,

Then F_{i} and D_{λ} are both elements of RG_{r,n}.

6.2. Lemma. Suppose that 1 ≤ i, j ≤ n and that w ∈ 𝔖_{n}. Then a) F_{i}F_{j} = F_{j}F_{i} and F_{i}^{2} = rF_{i}.

b) F_{i}w = wF_{i^{w}}.

Proof. As T is an abelian group part (a) is true and part (b) is immediate from the definitions and (2.1). □

Hence, if 1 ≤ i ≤ n then F_{i} is a multiple of an idempotent if the characteristic of R does not divide r and, otherwise, it is a nilpotent element of RG_{r,n}.

Suppose that μ ∈ Λ_{n}^{±}. In order to factorize E_{μ} set

F_{−μ} = ∏_{t_{i}∈T_{−μ}} F_{i} = ∏_{i:μ_{i}<0} F_{\overline{μ}_{i−1}+1} ⋯ F_{\overline{μ}_{i}}.

Then, by Lemma 6.2(a), (F_{−μ})^{2} = r^{|μ|^{−}}F_{−μ}.

By Lemma 6.2, 𝔖_{n} acts on {F_{1}, …, F_{n}} by conjugation. If w ∈ 𝔖_{n} and i ∈ n then we set F_{i}^{w} = w^{−1}F_{i}w = F_{i^{w}}. Similarly, if μ ∈ Λ_{n}^{±} let

F_{−μ}^{w} = ∏_{t_{i}∈T_{−μ}} F_{i}^{w}.

Then F_{−μ}w = wF_{−μ}^{w}, for all w ∈ 𝔖_{n}, by Lemma 6.2(b).

6.3. Lemma. Suppose that μ ∈ Λ_{n}^{±} is a signed composition of n. Then:

a) E_{μ} = F_{−μ}D_{μ^{+}}.

b) If w ∈ 𝔖_{μ^{+}} then F_{−μ}^{w} = F_{−μ}, so that F_{−μ}w = wF_{−μ}.

Proof. Part (a) is an immediate consequence of the factorization ℰ_{μ} = T_{−μ} × 𝒟_{μ^{+}} of ℰ_{μ} given by Theorem 4.3. For part (b), use Lemma 6.2(b) and the fact that the elements of the two subgroups 𝔖_{μ} and T_{−μ} commute. □

Definition 6.1 is motivated by Solomon’s [27] definition of the descent algebra of a finite Coxeter group. As an important special case, the Solomon descent algebra Sol(𝔖_{n}) of 𝔖_{n} is the subalgebra of R𝔖_{n} generated by { D_{λ} : λ ∈ Λ_{n} }. The next result, due to Solomon, shows that { D_{λ} : λ ∈ Λ_{n} } is basis of Sol(𝔖_{n}).

6.4 (Solomon [27, Theorem 1]).

a) The set { D_{μ} : μ ∈ Λ_{n} } is linearly independent in Sol(𝔖_{n}).

b) Suppose that μ and ν are composition of n. Then

D_{μ}D_{ν} = ∑_{d∈𝒟_{μν}} D_{μ∩dν}.

By the remarks before Lemma 5.1, part (b) is equivalent to the following formula:

D_{μ}D_{ν} = ∑_{σ∈Λ_{n}} d_{μνσ}D_{σ},

where d_{μνσ} = # { d ∈ 𝒟_{μν} : 𝔖_{σ} = 𝔖_{μ} ∩ d^{−1}𝔖_{ν}d }. In fact, Solomon proved an analogous result for an arbitrary finite Coxeter group W, where the Young subgroups 𝔖_{μ} are replaced with the parabolic subgroups of W and D_{μ} by the sum of the ‘distinguished’ (left) coset representatives which are of minimal length in their coset.

As we now recall, part (a) of Solomon’s theorem is easy to prove. Recall that S = {s_{1}, …, s_{n−1}}. For each composition μ ∈ Λ_{n} let S_{ν} = Π_{−ν}, so that S_{ν} ⊆ S. Now define Y_{μ} ∈ R𝔖_{n} ⊂ RG_{r,n} by

Y_{μ} = ∑_{\begin{matrix}w∈𝔖_{n} \\ Des(w)=S_{μ} \end{matrix}} w.

By (6.4) the descent sets partition 𝔖_{n}, so the set { Y_{μ} : μ ∈ Λ_{n} } is linearly independent in R𝔖_{n}. By (6.4) again, we can write

D_{μ} = ∑_{\begin{matrix}ν∈Λ_{n} \\ S_{ν}⊆S−S_{μ} \end{matrix}} Y_{ν}.

Hence, { D_{μ} : μ ∈ Λ_{n} } is a linearly independent subset of R𝔖_{n}, as claimed. We build upon this idea to prove that the E_{μ}’s are linearly independent.

6.5. Proposition. The set { E_{μ} : μ ∈ Λ_{n}^{±} } is linearly independent in Sol(G_{r,n}).

Proof. Suppose that there exist scalars a_{μ} ∈ R such that

∑_{μ∈Λ_{n}^{±}} a_{μ}E_{μ} = 0.

By Lemma 6.3, E_{μ} = F_{−μ}D_{μ^{+}}. Therefore, the last displayed equation becomes

0 = ∑_{μ∈Λ_{n}^{±}} a_{μ}F_{−μ}D_{μ^{+}} = ∑_{μ∈Λ_{n}^{±}} a_{μ}F_{−μ} ∑_{\begin{matrix}ν∈Λ_{n} \\ S_{ν}⊆S−S_{μ^{+}} \end{matrix}} Y_{ν}

= ∑_{ν∈Λ_{n}} ( ∑_{\begin{matrix}μ∈Λ_{n}^{±} \\ S_{μ^{+}} ⊆S−S_{ν} \end{matrix}} a_{μ}F_{−μ} ) Y_{ν}

Now, RG_{r,n} = ⨁_{t∈T} tR𝔖_{n}, as an R–module, and { Y_{ν} : ν ∈ Λ_{n} } is a linearly independent subset of R𝔖_{n}. Therefore, for any composition ν ∈ Λ_{n} we must have

(6.6)  0 = ∑_{\begin{matrix}μ∈Λ_{n}^{±} \\ S_{μ^{+}} ⊆S−S_{ν} \end{matrix}} a_{μ}F_{−μ}.

We use this equation to argue by induction on ν to show that a_{μ} = 0 for all μ ∈ Λ_{n}^{±}.

First suppose that ν = (n). Then S_{ν} = S and the summation in (6.6) becomes a sum over those signed compositions μ with S_{μ^{+}} = ∅. Hence, μ^{+} = (1^{n}) and (6.6) becomes

0 = ∑_{\begin{matrix}μ∈Λ_{n}^{±} \\ μ^{+}=(1^{n}) \end{matrix}} a_{μ}F_{−μ} = ∑_{\begin{matrix}μ∈Λ_{n}^{±} \\ Π_{μ}⊆\{t_{1},...,t_{n}\} \end{matrix}} a_{μ}F_{−μ}.

Each monomial t_{i_{1}} … t_{i_{k}}, where 1 ≤ i_{1} < ⋯ < i_{k} ≤ n, occurs in a unique F_{−μ} when μ^{+} = (1^{n}). Hence, a_{μ} = 0 for all μ ∈ Λ_{n}^{±} with μ^{+} = (1^{n}), as claimed.

Now suppose that ν ≠ (n). By induction we may assume that a_{μ} = 0 whenever S_{μ^{+}} ⊊ S − S_{ν}. Therefore, by (6.6) we have

0 = ∑_{\begin{matrix}μ∈Λ_{n}^{±} \\ S_{μ^{+}} =S−S_{ν} \end{matrix}} a_{μ}F_{−μ} = ∑_{\begin{matrix}μ∈Λ_{n}^{±} \\ Π_{μ}−(S−S_{ν})⊆\{t_{1},...,t_{n}\} \end{matrix}} a_{μ}F_{−μ}.

So, by exactly the same argument as before, a_{μ} = 0 whenever μ^{+} = ν. Hence, a_{μ} = 0, for all μ ∈ Λ_{n}^{±}, and { E_{μ} : μ ∈ Λ_{n}^{±} } is linearly independent as required. □

The next result that we need amounts to a proof of part (b) of Solomon’s theorem (6.4). Once again, we state the result only for the symmetric group even though it is valid for an arbitrary finite Coxeter group. All of the results quoted in (6.7) follow easily from the fact that 𝔖_{μd∩ν} = d^{−1}𝔖_{μ}d ∩ 𝔖_{ν}, for d ∈ 𝒟_{μν}.

If μ, ν ∈ Λ_{n} and 𝔖_{ν} ⊆ 𝔖_{μ} then we write ν ⊆ μ and set 𝒟_{ν}^{μ} = 𝒟_{ν} ∩ 𝔖_{μ}. It is easy to check that 𝒟_{ν}^{μ} is a complete set of coset representatives for 𝔖_{ν} in 𝔖_{μ}.

6.7 (Bergeron, Bergeron, Howlett and Taylor [5, Lemmas 2.2 and 2.4]). Suppose that μ and ν are compositions of n. Then

a) If σ ⊆ ν then 𝒟_{σ} = 𝒟_{σ}^{ν}𝒟_{ν}.

b) 𝒟_{μ} = ∐_{d∈𝒟_{μν}} d𝒟_{μd∩ν}^{ν}.

c) If d ∈ 𝒟_{μ} and μd is a composition of n (that is, d^{−1}𝔖_{μ}d = 𝔖_{σ} for some σ ∈ Λ_{n}), then 𝒟_{μ} = d𝒟_{μd}.

We can now establish one of the main results of this paper.

6.8. Theorem. Suppose that r > 1 and that μ and ν are signed compositions of n. Then

E_{μ}E_{ν} = ∑_{d∈𝒟_{_{μ^{+}}_{ν^{+}}}} |T_{−μ∩d(−ν)}| E_{μ∩dν}.

Proof. We use most of the results in this section to compute E_{μ}E_{ν}:

E_{μ}E_{ν} = F_{−μ}D_{μ^{+}}F_{−ν}D_{ν^{+}},  by Lemma 6.3(a),
= ∑_{d∈𝒟_{_{μ^{+}}_{ν^{+}}}} F_{−μ}dD_{μ^{+}d∩ν^{+}}^{ν^{+}}F_{−ν}D_{ν^{+}},  by Lemma 6.7(b),

= ∑_{d∈𝒟_{_{μ^{+}}_{ν^{+}}}} F_{−μ}dF_{−ν}D_{μ^{+}d∩ν^{+}}^{ν^{+}}D_{ν^{+}},  by Lemma 6.3(b),

= ∑_{d∈𝒟_{_{μ^{+}}_{ν^{+}}}} F_{−μ}F_{−ν}^{d^{−1}}dD_{μ^{+}d∩ν^{+}},  by Lemma 6.7(b),

= ∑_{d∈𝒟_{_{μ^{+}}_{ν^{+}}}} F_{−μ}F_{−ν}^{d^{−1}}D_{μ^{+}∩dν^{+}},  by Lemma 6.7(c).

Fix d ∈ 𝒟_{μ^{+}}_{ν^{+}} and consider F_{−μ}F_{−ν}^{d^{−1}}. Now F_{i}^{2} = rF_{i} = |T_{i}|F_{i}, for 1 ≤ i ≤ n. So,

F_{−μ}F_{−ν}^{d^{−1}} = |T_{−μ} ∩ dT_{−ν}d^{−1}| ∏_{t_{i}∈T_{−μ}∩dT_{−ν}d^{−1}} F_{i}.

First, T_{−μ} ∩ dT_{−ν}d^{−1} = T_{−μ∩d(−ν)} since d ∈ 𝒟_{μ^{+}}_{ν^{+}} = 𝒟_{(−μ)^{+}(−ν)^{+}}. Next, the subgroup of T generated by T_{−μ} and dT_{−ν}d^{−1} is T_{−(μ∩dν)} since t_{i} ∈ T_{−(μ∩dν)} = T/T_{μ∩dν} if and only if t_{i} ∉ T_{μ} and t_{i} ∉ dT_{ν}d^{−1}. Therefore, F_{−μ}F_{−ν}^{d^{−1}} = |T_{−μ∩d(−ν)}|F_{−(μ∩dν)}. Hence, using Lemma 6.3 once more,

E_{μ}E_{ν} = ∑_{d∈𝒟_{_{μ^{+}}_{ν^{+}}}} |T_{−μ∩d(−ν)}| F_{−(μ∩dν)}D_{(μ∩dν)^{+}} = ∑_{d∈𝒟_{_{μ^{+}}_{ν^{+}}}} |T_{−μ∩d(−ν)}| E_{μ∩dν},

as required. □

Corollary 5.8gives a combinatorial interpretation of the structure constants |T_{−μ∩d(−ν)}| of Sol(G_{r,n}). This shows that Theorem 6.8 is a direct generalization of (6.4)(b). A second combinatorial interpretation of the structure constants |T_{−μ∩d(−ν)}| is given in Proposition 10.3 below.

Combining Theorem 6.8 and Proposition 6.5 we obtain the following.

6.9. Corollary. Suppose that r > 1. The cyclotomic Solomon algebra Sol(G_{r,n}) is a subalgebra of RG_{r,n} which is free as an R–module of rank 2 · 3^{n−1}.

6.10. Example Suppose that r > 1. Then, by Example 5.9, we have

E_{(3,−2)}E_{(−2^{2},1)} = 2r^{2}E_{(−2,1,−2)} + rE_{(−2,−1^{3})} + rE_{(−1,−2,−1^{2})} + r^{2}E_{(−1^{2},1,−1^{2})}.

See Example 10.4 for a second way of computing this product using Proposition 10.3.

Notice that by (6.4) and Theorem 6.8 we can recover the multiplication in Sol(𝔖_{n}) by setting r = 1 and identifying μ and μ^{+}, for all μ ∈ Λ_{n}^{±}, so that

D_{(3,2)}D_{(2^{2},1)} = 2D_{(2,1,2)} + D_{(2,1^{3})} + D_{(1,2,1^{2})} + D_{(1^{5})}.
♢

7. The generic cyclotomic Solomon algebra

By Theorem 6.8, if r > 1 then the structure constants of the algebra Sol(G_{r,n}) are polynomials in r. Consequently, these algebras admit a simultaneous deformation.

Recall that if r ≥ 2 and μ, ν ∈ Λ_{n}^{±} then |T_{−μ∩d(−ν)}| = r^{wt(T_{d})} by Corollary 5.8.

Let x be an indeterminate over ℤ and suppose that μ, ν, σ ∈ Λ_{n}^{±}. Define polynomials d_{μνσ}(x) ∈ ℕ[x] by

d_{μνσ}(x) = ∑_{\begin{matrix}d∈𝒟_{_{μ^{+}}_{ν^{+}}} \\ σ=μ∩dν \end{matrix}} x^{wt(T_{d})}.

We abuse notation and consider d_{μνσ}(x) to be a polynomial over R. For q ∈ R we let d_{μνσ}(q) be the evaluation of this polynomial at q. Then, by Theorem 6.8,

E_{μ}E_{ν} = ∑_{σ∈Λ_{n}^{±}} d_{μνσ}(r)E_{σ}.

7.1. Definition. Suppose that n ≥ 1 and that R is a commutative ring. The cyclotomic Solomon algebra with parameter q ∈ R is the R–algebra Sol_{q}(n) = Sol_{R,q}(n) with generating set { E_{μ} : μ ∈ Λ_{n}^{±} } and relations

E_{μ}E_{ν} = ∑_{σ∈Λ_{n}^{±}} d_{μνσ}(q)E_{σ},

for μ, ν ∈ Λ_{n}^{±}. The generic cyclotomic Solomon algebra is the ℤ[x]–algebra Sol_{x}(n), where x is an indeterminate over ℤ.

We are abusing notation slightly in Definition 7.1 because from here onwards E_{μ} is a generator of Sol_{q}(n) and not necessarily the element defined in the previous section. This abuse is justified by the following result.

7.2. Corollary. Suppose that q = r · 1_{R}, where r > 1. Then Sol_{q}(n) and Sol(G_{r,n}) are canonically isomorphic R–algebras where the isomorphism Sol_{q}(n) → Sol(G_{r,n}) is given by E_{μ} ↦ E_{μ}, for μ ∈ Λ_{n}^{±}.

Proof. By Theorem 6.8 there is a natural surjection Sol_{q}(n) ⟶ Sol(G_{r,n}). By Corollary 6.9 this map is an isomorphism. □

The explicit description of the algebra Sol(G_{r,n}) as a subalgebra of the group algebra RG_{r,n} makes the algebra Sol(G_{r,n}) slightly easier to work with than the more general algebras Sol_{q}(n). For example, we know that E_{μ} = F_{−μ}D_{μ^{+}} in RG_{r,n} but we have no such factorization in general. As we will soon see, however, almost all of the properties of the algebras Sol(G_{r,n}) hold for the algebras Sol_{q}(n).

7.3. Proposition. Suppose that n ≥ 1 and that q ∈ R. Then

a) Sol_{q}(n) is free as an R–module with basis { E_{μ} : μ ∈ Λ_{n}^{±} }. In particular, Sol_{q}(n) has rank 2 · 3^{n−1}.

b) Sol_{q}(n) ≅ Sol_{x}(n) ⊗_{ℤ[x]} R, where R is considered as a ℤ[x]–module by letting x act on R as multiplication by q (and 1 ∈ ℤ acts as multiplication by 1_{R}).

c) Sol_{q}(n) is a unital associative R–algebra with multiplicative identity E_{(n)}.

Proof. First consider the generic Solomon algebra over ℤ[x]. Suppose that

∑_{μ∈Λ_{n}^{±}} f_{μ}(x)E_{μ} = 0,

for some f_{μ}(x) ∈ R[x]. Then f_{μ}(r) = 0, for r = 2, 3, 4 … and all μ ∈ Λ_{n}^{±}, by Corollary 7.2 and Proposition 6.5. As non–zero polynomials have only finitely many roots, we conclude that f_{μ}(x) = 0, for all μ ∈ Λ_{n}^{±}. Consequently, Sol_{x}(n) is free as a ℤ[x]–module with basis { E_{μ} : μ ∈ Λ_{n}^{±} }.

Now fix q ∈ R and consider R as a ℤ[x]–module by letting x act on R as multiplication by q (and 1 ∈ ℤ act as multiplication by 1_{R}). Then the R–algebra Sol_{x}(n) ⊗_{ℤ[x]} R is free as an R–module with basis { E_{μ} ⊗ 1 : μ ∈ Λ_{n}^{±} } and it satisfies the relations of Sol_{q}(n). As Sol_{q}(n) is spanned by the elements { E_{μ} : μ ∈ Λ_{n}^{±} } ⊆ Sol_{q}(n) it follows that Sol_{q}(n) ≅ Sol_{x}(n) ⊗_{ℤ[x]} R. This proves (a) and (b).

To prove (c) it is now enough to prove the corresponding statements for the generic Solomon algebra Sol_{x}(n). We first show that E_{(n)} is the identity element of Sol_{x}(n). This is equivalent to the polynomial identities

d_{μ(−n)α}(x) = δ_{μα} = d_{α(−n)μ}(x),

for all μ, α ∈ Λ_{n}^{±}. All of these identities follow directly from the definitions because T_{(−n)∩d(−ν)} = 1 = T_{−μ∩d′(−n)}, for all μ, ν ∈ Λ_{n}^{±}, d ∈ 𝒟_{(n)μ^{+}} and d′ ∈ 𝒟_{ν^{+}(n)}. Similarly, the associativity of Sol_{x}(n) is equivalent to the polynomial identities

∑_{α,β∈Λ_{n}^{±}} d_{μνα}(x)d_{ασβ}(x) = ∑_{α,β∈Λ_{n}^{±}} d_{μαβ}(x)d_{νσα}(x),

for all μ, ν, σ ∈ Λ_{n}^{±}. As in the first paragraph of the proof, by Corollary 7.2these identities hold when x = 2, 3, … since the algebras Sol(G_{r,n}) are associative for r ≥ 2. As these identities hold for infinitely many values of x, they lift to the required polynomial identities.

□

Part (b) of the Proposition justifies our calling the ℤ[x]–algebra Sol_{x}(n) the generic cyclotomic Solomon algebra.

As we next describe, the algebras Sol_{q}(n) have many interesting subalgebras.

7.4. Lemma. Suppose that d_{μνσ}(q) ≠ 0, for μ, ν, σ ∈ Λ_{n}^{±}. Then Π_{σ} = Π_{μ} ∩ dΠ_{ν}d^{−1}, for some d ∈ 𝒟_{μ^{+}}_{ν^{+}}

Proof. By definition, the polynomial d_{μνσ}(x) is non–zero only if G_{σ} = G_{μ} ∩ dG_{ν}d^{−1} for some d ∈ 𝒟_{μ^{+}}_{ν^{+}}. Consequently, if d_{μνσ}(q) ≠ 0 then Π_{σ} = Π_{μ} ∩ dΠ_{ν}d^{−1}, for some d ∈ 𝒟_{μ^{+}}_{ν^{+}}. □

Notice, in particular, that this implies that the poset structure on Λ_{n}^{±} given by defining μ ⪯ ν whenever Π_{ν} ⊆ Π_{μ} is compatible with the ideal structure of Sol_{q}(n).

7.5. Proposition. Suppose that n ≥ 1 and that q ∈ R. Then Sol_{q}(n) has a filtration by two–sided ideals

Sol_{q}(n) = 𝒮_{0} ⊃ ⋯ ⊃ 𝒮_{n} ⊃ 0

where 𝒮_{i} is the R–submodule of Sol_{q}(n) with basis { E_{μ} : μ ∈ Λ_{n}^{±} such that |μ|^{+} ≥ i }, for i = 0, …, n.

Proof. By Lemma 7.4, d_{μνσ}(q) ≠ 0 only if Π_{σ} = Π_{μ} ∩ dΠ_{ν}d^{−1}, for some d ∈ 𝒟_{μ^{+}}_{ν^{+}}. Consulting the definitions, |μ|^{+} = |Π_{μ} ∩ T |. Therefore, d_{μνσ}(q) ≠ 0 only if |σ|^{+} ≤ min{|μ|^{+}, |ν|^{+}}. Hence, 𝒮_{i} is a two–sided ideal of Sol(G_{r,n}), for 0 ≤ i ≤ n, and the Proposition follows. □

7.6. Proposition. Suppose that n ≥ 1 and that q ∈ R. Let

Sol_{q}^{+}(n) = ∑_{μ∈Λ_{n}} RE_{μ}
Sol_{q}^{±}(n) = ∑_{±μ∈Λ_{n}} RE_{μ}
Sol_{q}^{1}(n) = ∑_{\begin{matrix}μ∈Λ_{n}^{±} \\ μ_{i}>0 for i>1 \end{matrix}} RE_{μ}.

Then Sol_{q}^{+}(n), Sol_{q}^{±}(n) and Sol_{q}^{1}(n) are all subalgebras of Sol_{q}(n). Moreover, Sol_{q}^{+}(n) is naturally isomorphic to Sol(𝔖_{n}) via the R–linear map E_{μ} ↦ D_{μ}, for μ ∈ Λ_{n}.

Proof. All of these results can be proved directly using the definition of the polynomials d_{μνσ}(x), for μ, ν, σ ∈ Λ_{n}^{±}. Note that Sol_{q}^{+}(n) = 𝒮_{n} in the notation of Proposition 7.5, so in this case the result is already known. The isomorphism Sol_{q}^{+}(n) ≅ Sol(𝔖_{n}) is trivial because if μ ∈ Λ_{n} then T_{−μ} = 1, so that E_{μ} = D_{μ} by Lemma 6.3. □

8. The representation theory of Sol_{q}(n)

In this section we construct all of the irreducible representations of the algebras Sol_{q}(n) over an arbitrary field. Even though Sol_{q}(n) is, in general, not commutative, it turns out that every irreducible Sol_{q}(n)–module is one dimensional — so that Sol_{q}(n) is a basic algebra for all n and q. As an application of these results we give a basis for the radical of Sol_{q}(n) when R is an arbitrary field.

Let ∼ be the equivalence relation on the set of signed compositions where two signed compositions are ∼–equivalent if one can be obtained by reordering the parts of the other. More explicitly, if λ = (λ_{1}, …, λ_{k}) and μ = (μ_{1}, …, μ_{l}) then λ ∼ μ if and only if k = l and λ_{i} = μ_{i^{v}}, for some v ∈ 𝔖_{k}.

8.1. Lemma. Suppose that λ, μ ∈ Λ_{n}^{±}. Then the following are equivalent:

a) λ ∼ μ;

b) G_{λ} = w^{−1}G_{μ}w, for some w ∈ 𝔖_{n};

c) G_{λ} = g^{−1}G_{μ}g, for some g ∈ G_{r,n}.

Proof. We leave the proof for the reader. □

8.2. Lemma. Suppose that λ, μ ∈ Λ_{n}^{±}. Then

a) If μ ≁ λ then d_{μλμ}(q) ≠ 0 only if |Π_{λ}| > |Π_{μ}|.

b) If μ ∼ λ then d_{μαμ}(q) = d_{λαλ}(q), for all α ∈ Λ_{n}^{±}.

Proof. By Lemma 7.4 d_{μλμ}(q) ≠ 0 only if Π_{μ} = Π_{μ} ∩ dΠ_{λ}d^{−1}, for some d ∈ 𝒟_{μ^{+}λ^{+}}. Hence, part (a) follows since λ ≁ μ.

Consulting the definition of the polynomials d_{μνσ}(x), to prove (b) it is enough to show that if r ≥ 2 then in the group G_{r,n} we have

(†)  ∑_{\begin{matrix}d∈𝒟_{λ^{+}α^{+}} \\ λ=λ∩dα \end{matrix}} |T_{−λ∩d(−α)}| = ∑_{\begin{matrix}d∈𝒟_{μ^{+}α^{+}} \\ μ=μ∩dα \end{matrix}} |T_{−μ∩d(−α)}|.

We prove this by showing that the ‘obvious’ bijection 𝒟_{λ^{+}α^{+}} ⟶ 𝒟_{μ^{+}α^{+}} preserves each of the summands in this equation.

First note that by Lemma 8.1we can find an element w ∈ 𝔖_{n} such that G_{λ} = w^{−1}G_{μ}w since λ ∼ μ. That is, T_{λ}𝔖_{λ^{+}} = w^{−1}T_{μ}w · w^{−1}𝔖_{μ^{+}}w, so that T_{λ} = w^{−1}T_{μ}w and 𝔖_{λ^{+}} = w^{−1}𝔖_{μ^{+}}w. Consequently, the map 𝔖_{λ^{+}}\𝔖_{n}/𝔖_{α^{+}} → 𝔖_{μ^{+}}\𝔖_{n}/𝔖_{α^{+}} given by C ↦ wC defines a bijection since if d ∈ 𝒟_{λ^{+}α^{+}} then 𝔖_{λ^{+}}d𝔖_{α^{+}} = w^{−1}𝔖_{μ^{+}}wd𝔖_{α^{+}}. Let d ↦ d′ be map from 𝒟_{λ^{+}α^{+}} to 𝒟_{μ^{+}α^{+}} determined by 𝔖_{μ^{+}}wd𝔖_{α^{+}} = 𝔖_{μ^{+}}d′𝔖_{α^{+}}.

Now fix d ∈ 𝒟_{λ^{+}α^{+}} such that λ = λ ∩ dα. Then

T_{−λ∩d(−α)} = T_{−λ} ∩ dT_{−α}d^{−1}
= w^{−1}T_{−μ}w ∩ dT_{−α}d^{−1}
= w^{−1}(T_{−μ} ∩ wdT_{−α}(wd)^{−1})w.

Write wd = ud′v, for u ∈ 𝔖_{μ^{+}} and v ∈ 𝔖_{α^{+}}. Then we have

T_{−λ∩d(−α)} = w^{−1}(T_{−μ} ∩ (ud′v)T_{−α}v^{−1}(d′)^{−1}u^{−1})w,
= w^{−1}u(T_{−μ} ∩ d′T_{−α}(d′)^{−1})u^{−1}w,
= w^{−1}u(T_{−μ} ∩ T_{d′(−α)})u^{−1}w,

where the second equality follows because 𝔖_{μ} normalizes T_{−μ} and the last equality follows because 𝔖_{α} normalizes T_{−α}. Hence, we have shown that |T_{−λ∩d(−α)}| = |T_{−μ∩d′(−α)}|, for all d ∈ 𝒟_{λ^{+}α^{+}}. This establishes (†), so the Lemma is proved. □

8.3. Theorem. Suppose that R is a field, q ∈ R and n ≥ 0.

a) If λ ∈ Λ_{n}^{±} then Sol_{q}(n) has a unique one dimensional representation I(λ) upon which E_{α} acts as multiplication by d_{λαλ}(q), for α ∈ Λ_{n}^{±}.

b) Every irreducible representation of Sol_{q}(n) is isomorphic to I(λ), for some λ ∈ Λ_{n}^{±}.

c) If λ ∼ μ then I(λ) ≅ I(μ).

Proof. Choose a total order ≥ on Λ_{n}^{±} such that |Π_{λ}| ≥ |Π_{μ}| whenever λ > μ, for λ, μ ∈ Λ_{n}^{±}. Let 𝒮_{λ} be the R–submodule of Sol_{q}(n) with basis { E_{μ} : λ ≥ μ ∈ Λ_{n}^{±} } and let 𝒮′_{λ} be the R–submodule with basis { E_{μ} : λ > μ ∈ Λ_{n}^{±} }. Then 𝒮_{λ} and 𝒮′_{λ} are both right Sol_{q}(n)–modules by Lemma 7.4. Hence the quotient module I(λ) = 𝒮_{λ}/𝒮′_{λ} = R(E_{λ} + 𝒮′_{λ}) is one dimensional Sol_{q}(n)–module. By definition, if α ∈ Λ_{n}^{±} then E_{α} acts on I(λ) as multiplication by d_{λαλ}(q). Hence, I(λ) is the one dimensional Sol_{q}(n)–module described in part (a).

Now suppose that Λ_{n}^{±} = {λ_{1} > λ_{1} > ⋯ > λ_{N}}, where N = 2 · 3^{n−1} = dim Sol_{q}(n). Then

Sol_{q}(n) = 𝒮_{λ_{1}} ⊃ 𝒮_{λ_{2}} ⊃ ⋯ ⊃ 𝒮_{λ_{N}} ⊃ 0

is a filtration of Sol_{q}(n) by two–sided ideals with quotients 𝒮_{λ_{i}}/𝒮_{λ_{i+1}} ≅ I(λ_{i}), since 𝒮_{λ_{i+1}} = 𝒮′_{λ_{i}}. As every irreducible Sol_{q}(n)–module arises as a composition factor of Sol_{q}(n) part (b) now follows.

Finally, if λ ∼ μ then I(λ) ≅ I(μ) by Lemma 8.2(b). Hence, part (c) holds. □

8.4. Corollary. Every field is a splitting field for Sol_{q}(n).

Proof. Suppose that D is an irreducible Sol_{q}(n)–module. Then D is one dimensional by the Proposition, and hence absolutely irreducible. □

If A is an algebra over a field then let Rad A be its radical. Thus, Rad A is the unique maximal nilpotent ideal of A and A is semisimple if and only if Rad A = 0. Recall that a ∈ A is nilpotent if a^{k} = 0, for some k > 0, whereas an ideal I of A is nilpotent if I^{k} = 0 for some k > 0.

8.5. Corollary. Suppose that R is a field. Then Rad Sol_{q}(n) is the set of nilpotent elements in Sol_{q}(n).

Proof. By definition every element of Rad Sol_{q}(n) is nilpotent. To prove the converse let M be the number of irreducible Sol_{q}(n)–modules. By Theorem 8.3 every irreducible Sol_{q}(n)–module is one dimensional. Therefore, Sol_{q}(n)/ Rad Sol_{q}(n) ≅ R^{M} by the Wedderburn Theorem. In particular, Sol_{q}(n)/ Rad Sol_{q}(n) contains no nilpotent elements, so the result follows. □

8.6. Corollary. Suppose that R is a field and q ∈ R. Then Sol_{q}(n) is semisimple if and only if n = 1 and q ≠ 0.

Proof. If n ≥ 2 then Sol_{q}(n) is not semisimple because there exist distinct signed compositions λ, ν ∈ Λ_{n}^{±} such that λ ∼ μ. Therefore, E_{λ} − E_{μ} ∈ Rad Sol_{q}(n), so that Rad Sol_{q}(n) ≠ 0. If n = 1 then a quick calculation verifies that I(1) ≅ I(−1) if and only if q = 0 which implies the result. □

Each ∼–equivalence class of Λ_{n}^{±} contains a unique signed composition μ = (μ_{1}, …, μ_{k}) such that μ_{1} ≥ ⋯ ≥ μ_{k}. If μ ∈ Λ_{n}^{±} and μ_{1} ≥ ⋯ ≥ μ_{k} then we call μ a signed partition of n. Let Λ_{n}^{⊕} be the set of all signed partitions of n. By the remarks above, the 𝔖_{n}–conjugacy classes of reflection subgroups of G_{r,n} are indexed by the signed partitions of n. We note that Λ_{n}^{⊕} is naturally in bijection with the set of bipartitions of n, however, for us the signed partitions are more natural because we have already defined a reflection subgroup G_{λ} for each λ ∈ Λ_{n}^{⊕}.

8.7. Theorem. Suppose that R is a field of characteristic zero and that q ≠ 0. Then

{ I(λ) : λ ∈ Λ_{n}^{⊕} }

is a complete set of pairwise non–isomorphic irreducible Sol_{q}(n)–modules.

Proof. As the ∼–equivalence classes of Λ_{n}^{±} are indexed by the signed partitions of n, { I(λ) : λ ∈ Λ_{n}^{⊕} } is a complete set of irreducible Sol_{q}(n)–modules by parts (b) and (c) of Theorem 8.3. It remains then to show that if λ, μ ∈ Λ_{n}^{⊕} then I(λ) ≅ I̸(μ) if λ ≠ μ. Now, R is a field of characteristic zero and q ≠ 0, so d_{λνλ}(q) ≠ 0 if and only if d_{λνλ}(x) ≠ 0, for λ, ν ∈ Λ_{n}^{±}. However, d_{λλλ}(x) ∈ 1 + xℕ[x] since 1 ∈ 𝒟_{λ^{+}λ^{+}} and Π_{λ} = Π_{λ} ∩ 1 · Π_{λ} · 1^{−1}. Therefore, d_{λλλ}(q) ≠ 0 and so, using Lemma 8.2(a) again, if λ ≠ μ then I(λ) ≅ I̸(μ). □

8.8. Corollary. Suppose that R is a field of characteristic zero and q ≠ 0. Then

{ E_{λ} − E_{μ} : λ ∈ Λ_{n}^{⊕}, μ ∈ Λ_{n}^{±}, λ ∼ μ and λ ≠ μ }

is a basis of Rad Sol_{q}(n). Consequently, dim Sol_{q}(n)/ Rad Sol_{q}(n) = |Λ_{n}^{⊕}|.

Proof. Suppose that λ ∼ μ where λ ∈ Λ_{n}^{⊕}, μ ∈ Λ_{n}^{±} and λ ≠ μ. Then, by Theorem 8.7 and Lemma 8.2, E_{λ} − E_{μ} acts as multiplication by zero on every irreducible Sol_{q}(n)– module. Therefore, E_{λ} − E_{μ} belongs to Rad Sol_{q}(n) whenever λ ≅ μ. Consequently, dim Sol_{q}(n)/ Rad Sol_{q}(n) ≤ |Λ_{n}^{⊕}|. However, dim Sol_{q}(n)/ Rad Sol_{q}(n) = |Λ_{n}^{⊕}| by Theorem 8.7, so the result follows. □

Suppose that R is a field of characteristic zero and that q ≠ 0. Define the character table of Sol_{q}(n) to be the matrix

C_{q}(n) = (d_{λμλ}(q))_{λ,μ∈Λ_{n}^{⊕}}.

Then C_{q}(n) is the character table of Sol_{q}(n)/ Rad Sol_{q}(n), by Theorem 8.7, so it completely determines the maximal semisimple quotient of Sol_{q}(n). The character table C_{q}(n) is explicitly known for all q ≠ 0 and all n ≥ 1 since the polynomials d_{λμσ}(x) are explicitly known for all λ, μ, σ ∈ Λ_{n}^{±} by Corollary 5.8.

8.9. Example Suppose that R is a field of characteristic zero and that q = 2 = n. Then Sol_{2}(2) ≅ Sol(G_{2,2}) and the character table C_{2}(2) of Sol_{2}(2) is the following matrix.

(2) (1^{2}) (1, −1) (−2) (−1^{2})
(2)  1
(1^{2})  1  2
(1, −1) 1  2  2
(−2)  1  .  .  4
(−1^{2}) 1  2  4  4  8

As all of the diagonal entries of C_{q}(2) are powers of 2 it follows that if R is any field of characteristic different from 2 then { I(λ) : λ ∈ Λ_{n}^{⊕} } is a complete set of pairwise non– isomorphic irreducible Sol_{q}(2)–modules. If R is a field of characteristic 2 then I(2) is the only irreducible Sol_{q}(2)–module. This is in agreement with Theorem 8.11 below.

By comparing the character table of Sol(G_{2,2}) with the character table of the group G_{2,2} (the Coxeter group of type B_{2}) it is easy to see that there cannot be a ring homomorphism from Sol(G_{2,2}) into the character ring of G_{2,2}. This is in marked contrast with the Solomon algebras of Coxeter groups for which such a homomorphism always exists. ♢

8.10. Remark. As discussed in Remark 4.10, Mak has shown that the cosets of the reflection subgroups of G_{r,n} have a unique element of minimal length with respect to the Bremke–Malle length function ℓ_{0} (see Remark 2.4). For each μ ∈ Λ_{n}^{±} let ℰ′_{μ} be Mak’s set of distinguished coset representatives for G_{μ} and let E′_{μ} = ∑_{e∈ℰ′_{μ}} e ∈ RG_{r,n}. Define

Σ′(G_{r,n}) = ∑_{μ∈Λ_{n}^{±}} RE′_{μ}.

If r > 2 then Σ′(G_{r,n}) is not, in general, a subalgebra of RG_{r,n}. The smallest counter example occurs when r = n = 3.

Now suppose that r = 2. Then G_{2,n} is a Coxeter group of type B_{n} and Bonnafé and Hohlweg [9] have shown that Σ′(G_{2,n}) is a subalgebra of RG_{2,n} and, moreover, that Σ′(G_{2,n}) is isomorphic to the Mantaci-Reutenauer algebra [23]. Now, the algebras Sol(G_{2,n}) and Σ′(G_{2,n}) are both free of rank 2 · 3^{n−1}, so it is natural to ask whether these algebras are isomorphic. In fact, Sol(G_{2,n}) ≅ Σ̸′(G_{2,n}) if n > 1. This can be proved by induction on n starting from the following observation. Bonnafé and Hohlweg have shown in [9, Table V] that the following matrix is the character table of the semisimple quotient of Σ′(G_{2,2}).

(2) (1^{2}) (1, −1) (−2) (−1^{2})
(2)  1
(1^{2})  1  2
(1, −1) 1  2  2
(−2)  1  .  .  2
(−1^{2}) 1  2  4  4  8

Observe that the ((−2), (−2))–entry in this character table is different to the corresponding entry in the character table of Sol(G_{2,2}) given in Example 8.9. Therefore, Sol(G_{2,2}) and Σ′(G_{2,2}) are not isomorphic algebras because they have non–isomorphic maximal semisimple quotients.

We close this section by classifying the irreducible Sol_{q}(n)–modules over an arbitrary field. This classification is a direct generalization of the corresponding results for the descent algebra of the symmetric groups [2] – although our proofs are necessarily different because there is no homomorphism from Sol_{q}(n) into the character ring of G_{r,n}.

For λ ∈ Λ_{n} let N_{𝔖_{n}}(𝔖_{λ}) = { w ∈ 𝔖_{n} : 𝔖_{λ} = w^{−1}𝔖_{λ}w } be the normalizer of 𝔖_{λ} in 𝔖_{n}.

8.11. Theorem. Suppose that R is field, q ∈ R and λ ∈ Λ_{n}^{⊕}. Then the following are equivalent:

a) d_{λλλ}(q) = 0;

b) q^{|λ|^{−}}[N_{𝔖_{n}}(𝔖_{λ^{+}}) : 𝔖_{λ^{+}}] = 0 in R;

c) E_{λ} ∈ Rad Sol_{q}(n);

d) E_{λ} is nilpotent; and,

e) I(λ) ≅ I(μ), for some μ ∈ Λ_{n}^{⊕} with |Π_{μ}| > |Π_{λ}|.

Proof. By definition,

d_{λλλ}(q) = ∑_{\begin{matrix}d∈𝒟_{λ^{+}λ^{+}} \\ λ=λ∩dλ \end{matrix}} |T_{−λ∩d(−λ)}| = ∑_{\begin{matrix}d∈𝒟_{λ^{+}λ^{+}} \\ λ=λ∩dλ \end{matrix}} q^{|λ|^{−}} = q^{|λ|^{−}}[N_{𝔖_{n}}(𝔖_{λ^{+}}) : 𝔖_{λ^{+}}],

since |T_{−λ}| = q^{|λ|^{−}} and T_{−λ∩d(−λ)} = T_{−λ} if λ = λ ∩ dλ. Hence, (a) and (b) are equivalent. Further, (c) and (d) are equivalent by Corollary 8.5.

To complete the proof it is enough to show that (a) ⟹ (c) ⟹ (e) ⟹ (a). In order to do this let Sol_{q}(n) = 𝒮_{λ_{1}} ⊃ 𝒮_{λ_{2}} ⊃ ⋯ ⊃ 𝒮_{λ_{N}} ⊃ 0 be the filtration of Sol_{q}(n) by two sided ideals which was constructed in the proof of Theorem 8.3 using a total order > on Λ_{n}^{±}. Recall that |Π_{μ}| ≥ |Π_{ν}| whenever μ > ν, for μ, ν ∈ Λ_{n}^{±}. Then 𝒮_{λ_{i}} is a subalgebra of Sol_{q}(n) which is also a quotient of Sol_{q}(n) since 𝒮_{λ_{i}} ≅ Sol_{q}(n)/𝒮_{λ_{i+1}}, for 1 ≤ i ≤ N. Therefore, by Theorem 8.3, every irreducible 𝒮_{λ_{i}}–module is isomorphic to I(μ) for some μ ∈ Λ_{n}^{⊕} with μ ≥ λ_{i}, for 1 ≤ i ≤ N. In particular, every irreducible 𝒮_{λ}–module is isomorphic to I(μ) for some μ ≥ λ.

We can now return to the proof of the Theorem.

First, suppose (a) holds, so that d_{λλλ}(q) = 0. By definition, if μ ∈ Λ_{n}^{⊕} then E_{λ} acts on I(μ) as multiplication by d_{μλμ}(q). By Lemma 8.2(a), if μ > λ then E_{λ} acts on I(μ) as multiplication by 0, whereas E_{λ} acts on I(λ) as multiplication by 0 since d_{λλλ}(q) = 0. Therefore, E_{λ} ∈ Rad 𝒮_{λ} and (c) holds because Rad 𝒮_{λ} ⊆ Rad Sol_{q}(n).

Next, suppose that (c) holds. Then E_{λ} belongs to the radical of 𝒮_{λ}. Now, 𝒮_{λ} ⊂ 𝒮_{λ_{l−1}} so, as vector spaces, Rad 𝒮_{λ} = RE_{λ} + Rad 𝒮_{λ_{l−1}}. On the other hand, dim 𝒮_{λ} = dim 𝒮_{λ_{l−1}} + 1, so it follows that the algebras 𝒮_{λ} and 𝒮_{λ_{l−1}} have the same number of irreducible modules. Hence, I(λ) ≅ I(μ) for some signed partition μ > λ. That is, (e) holds.

Finally, assume that (e) holds. Then I(λ) ≅ I(μ), for some signed partition μ > λ. Therefore, E_{λ} acts on these modules as multiplication by d_{λλλ}(q) = d_{μλμ}(q). Consequently, d_{λλλ}(q) = 0 by Lemma 8.2, so (a) holds.

This completes the proof of the Theorem. □

In the following Corollaries note that the integer d_{λλλ}(q) = q^{|λ|^{−}}[N_{𝔖_{n}}(𝔖_{λ^{+}}) : 𝔖_{λ^{+}}] is explicitly known by Theorem 8.11 (and Corollary 5.8).

8.12. Corollary. Suppose that R is a field and q ∈ R. Then

{ I(λ) : λ ∈ Λ_{n}^{⊕} and d_{λλλ}(q) ≠ 0 }

is a complete set of pairwise non–isomorphic irreducible Sol_{q}(n)–modules.

Proof. This follows from Theorem 8.11 and Theorem 8.3. □

Similarly, combining the Theorem 8.11with Corollary 8.5and Corollary 8.8, we obtain the general description of the radical of Sol_{q}(n) when R is a field.

8.13. Corollary. Suppose that R is a field and q ∈ R. Then

{ E_{λ} − E_{μ} : λ ∈ Λ_{n}^{⊕}, μ ∈ Λ_{n}^{±}, λ ∼ μ and λ ≠ μ } ⋃ { E_{λ} : λ ∈ Λ_{n}^{⊕} and d_{λλλ}(q) = 0 }

is a basis of Rad Sol_{q}(x).

Finally, we can use Theorem 8.11 to describe the radical and irreducible modules for each of the subalgebras of Sol_{q}(n) described in Proposition 7.6. For brevity we state only the following result.

8.14. Corollary. Suppose that R is a field, n ≥ 1 and q ∈ R. Let A be one of the subalgebras Sol_{q}^{+}(n), Sol_{q}^{±}(n), Sol_{q}^{1}(n) of Sol_{q}(n). Then Rad A = A ∩ Rad Sol_{q}(n).

9. The Hopf algebra of cyclotomic Solomon algebras

In this section we fix r > 1 and show that the direct sum of cyclotomic Solomon algebras ⨁_{n≥0} Sol(G_{r,n}) is a concatenation Hopf algebra, where G_{r,0} = {1_{G_{r,0}}} is the trivial group. Further, this Hopf algebra is a Hopf subalgebra of the Hopf algebra of colored permutations introduced by Baumann and Hohlweg [3].

Most of the results in this section hold over an arbitrary integral domain, however, the main results of this section (Theorem 9.7 and Corollary 9.8), hold only in characteristic zero. Consequently, for this section we fix a field 𝕜 of characteristic zero and we work only over this field. Thus, all tensor products are over 𝕜, all modules are 𝕜-vector spaces and all algebras are 𝕜–algebras. In particular, the cyclotomic Solomon algebras Sol(G_{r,n}) = Sol_{𝕜}(G_{r,n}) are 𝕜–algebras.

We first recall some general facts about bialgebras and Hopf algebras.

A 𝕜–coalgebra is a triple (A, δ, ε) consisting of a 𝕜–vector space A together with two linear maps δ: A⟶A ⊗ A (comultiplication) and ε: A⟶𝕜 (the counit) such that

(δ ⊗ id_{A}) ∘ δ = (id_{A} ⊗δ) ∘ δ and (ε ⊗ id_{A}) ∘ δ = (id_{A} ⊗ε) ∘ δ,

where id_{A} is the identity map on A.

A 𝕜–bialgebra is a coalgebra (A, δ, ε) such that A is a 𝕜–algebra and the structure maps δ: A ⟶ A ⊗ A and ε: A ⟶ 𝕜 are algebra homomorphisms. A Hopf algebra is a quadruple (A, δ, ε, S) where (A, δ, ε) is a bialgebra and S: A ⟶ A (the antipode) is a linear map such that μ(S ⊗ id_{A})δ = ηε = μ(1 ⊗ S)δ. Here μ: A ⊗ A⟶A : (a, b) ↦ ab is the multiplication map and η: 𝕜⟶A; 1 ↦ 1_{A} is the unit map for the algebra A.

Finally, a graded bialgebra is a triple (A, δ, ε) where A = ⨁_{n∈ℕ} A_{n} is ℕ–graded bialgebra and the maps δ and ε are graded (degree zero) vector space homomorphisms. A graded Hopf algebra is a graded bialgebra which is equipped with an antipode which is a graded vector space homomorphism of degree zero. A graded bialgebra, or a graded Hopf algebra, A = ⨁_{n≥0} A_{n} is connected if A_{0} = 𝕜.

Following Baumann and Hohlweg [3], we next define the (graded connected) Hopf algebra of coloured permutations. This will require some preparation. As a graded vector space this Hopf algebra is the direct sum of the group algebras of groups G_{r,n}:

𝒢 (r) := ⨁_{n≥0} 𝕜G_{r,n}.

We need some more notation before we can describe the Hopf algebra structure on 𝒢 (r).

First, suppose that m and n are non–negative integers. Then G_{r,m} × G_{r,n} is naturally isomorphic to the reflection subgroup G_{(m,n)} of G_{r,m+n}. By identifying G_{r,m} × G_{r,n} and G_{(m,n)} we have an embedding G_{r,m} × G_{r,n} ↪ G_{r,m+n}. Explicitly, this embedding sends the generators {s_{0}, …, s_{m−1}} of G_{r,m} to {s_{0}, …, s_{m−1}} in G_{r,m+n} and the generators {s_{0}, …, s_{n−1}} of G_{r,n} to {t_{m+1}, s_{m+1}, …, s_{m+n−1}}, respectively.

By Proposition 4.7 there is a natural bijection between the set ℰ_{(m,n)} = 𝒟_{(m,n)} of right coset representatives of G_{(m,n)} in G_{r,n} and the set of row standard (m, n)–tableau. The product ∗ on the Hopf algebra 𝒢 (r) is the bilinear map determined by

u ∗ v = ∑_{e∈ℰ_{(m,n)}}(u × v)e = (u × v)E_{(m,n)},

for u ∈ G_{r,m}, v ∈ G_{r,n} and where u × v is multiplication inside G_{r,m+n} (the internal product on 𝒢 (r)). The product ∗ on 𝒢 (r) is called the shuffle product, or the external product, on 𝒢 (r) because, by Proposition 4.7, ℰ_{(m,n)} is in bijection with the ways of shuffling the two sets {1, …, m} and {m + 1, …, m + n} together. It is easy to check that E_{(0)} = 1_{G_{r,0}} ∈ Sol(G_{r,0}) is the unit for the shuffle product.

To define the coproduct on 𝒢 (r) observe that for m = 0, …, n any element g ∈ G_{r,n} can be written uniquely in the form g = e_{m}^{−1}(g_{(m)} × g_{(n)}), where g_{(m)} ∈ G_{r,m}, g_{(n)} ∈ G_{r,n} and e_{m} ∈ ℰ_{(m,n)}. Using this notation, the coproduct Δ on 𝒢 (r) is the linear map determined by

Δ(g) = ∑_{m=0}^{n} g_{(m)} ⊗ g_{(n)},

for g ∈ G_{r,n}. 9.1. Example In order to better distinguish between the elements G_{r,n} for different values of n recall from the end of section 2 that there is a natural bijection between G_{r,n} and the set of words G_{r,n} = \{ \underline{ω} = ω_{1} … ω_{n} : ω_{i} ∈ n_{ζ} and \{|ω_{1}|, …, |ω_{n}|\} = n \} . To give an example of the shuffle product and the coproduct on 𝒢 (r) we identify G_{r,n} and G_{r,n} using this bijection.

Suppose that 0 ≤ a, b, c, d < r. Then, using the identification above,

1ζ^{a} 2ζ^{b} ∗ 2ζ^{c} 1ζ^{d} = 1ζ^{a}2ζ^{b}4ζ^{c}3ζ^{d} + 1ζ^{a}3ζ^{b}4ζ^{c}2ζ^{d} + 1ζ^{a}4ζ^{b}3ζ^{c}2ζ^{d}
+ 2ζ^{a}3ζ^{b}4ζ^{c}1ζ^{d} + 2ζ^{a}4ζ^{b}3ζ^{c}1ζ^{d} + 3ζ^{a}4ζ^{b}2ζ^{c}1ζ^{d}

and

Δ(2ζ^{a}3ζ^{b}1ζ^{c}4ζ^{d}) = ∅ ⊗ 2ζ^{a}3ζ^{b}1ζ^{c}4ζ^{d} + 1ζ^{c} ⊗ 1ζ^{a}2ζ^{b}3ζ^{d} + 2ζ^{a}1ζ^{c} ⊗ 1ζ^{b}2ζ^{d}
+ 2ζ^{a}3ζ^{b}1ζ^{c} ⊗ 1ζ^{d} + 2ζ^{a}3ζ^{b}1ζ^{c}4ζ^{d} ⊗ ∅,

where ∅ is the empty word in G_{r,0}. ♢

As remarked above, E_{(0)} = 1_{G_{r,0}} is the multiplicative unit for the shuffle product. The counit of 𝒢 (r) is the linear map ε: 𝒢 (r)⟶𝕜 defined by

ε(w) = \begin{cases}1 & if w = E_{(0)} ∈ G_{r,0} \\ 0 & otherwise. \end{cases}

9.2. Theorem (Baumann and Hohlweg [3, Theorem 1]). The triple (𝒢 (r), Δ, ε) is a graded connected bialgebra.

In fact, (𝒢 (r), Δ, ε) is a Hopf algebra at least when 𝕜 is a field because every connected ℕ-graded 𝕜-bialgebra is a Hopf algebra; see [28, Ex. 1, page 238].

We remind the reader that r > 1 is fixed throughout this section.

9.3. Definition. The cyclotomic Hopf algebra is the graded vector space

Sol(r) = ⨁_{n≥0} Sol(G_{r,n}).

The cyclotomic Hopf algebra is naturally graded with Sol(r)_{n} = Sol(G_{r,n}) and, as a vector space, Sol(r)_{n} is finite dimensional with basis { E_{μ} : μ ∈ Λ_{n}^{±} }. For convenience, we set E_{n} = E_{(n)}, for n ∈ ℤ.

Our next aim is to show that Sol(r) is a Hopf subalgebra of 𝒢 (r). We begin with a Lemma which generalizes 6.7(a).

9.4. Lemma. Suppose that α, β ∈ Λ_{n}^{±} with G_{α} ⊆ G_{β}. Then ℰ_{α}^{β} = ℰ_{α} ∩ G_{β} is a complete set of minimal length right coset representatives for G_{α} in G_{β} and ℰ_{α} = ℰ_{α}^{β}ℰ_{β}.

Proof. It is clear that ℰ_{α}^{β} is a complete set of right coset representatives for G_{α} in G_{β}. Moreover, by definition, if e ∈ ℰ_{α}^{β} then e is the unique element of minimal length in the coset G_{α}e. To prove the second statement observe that

G_{r,n} = ∐_{d∈ℰ_{β}} G_{β}d = ∐_{d∈ℰ_{β}} ( ∐_{e∈ℰ_{α}^{β}} G_{α}e)d.

So, ℰ_{α}^{β}ℰ_{β} is a complete set of coset representatives for G_{α} in G_{r,n}. Therefore, ℰ_{α} = ℰ_{α}^{β}ℰ_{β} since the elements of both sides are of minimal length in their respective cosets. □

9.5. Proposition. Suppose that μ ∈ Λ_{m}^{±} and ν ∈ Λ_{n}^{±}. Then

E_{μ} ∗ E_{ν} = E_{μ⊔ν} ∈ Sol(G_{r,n+m})

where μ ⊔ ν = (μ_{1}, …, μ_{l}, ν_{1}, …, ν_{k}) is the concatenation of two signed permutations.

Proof. By definition, E_{μ} ∗E_{ν} = (E_{μ} ×E_{ν})E_{(m,n)} where, as above, we interpret E_{μ} ×E_{ν} as an element of 𝕜G_{(m,n)} ⊆ 𝕜G_{r,n}. Therefore, it is enough to prove that ℰ_{μ⊔ν} = ℰ_{μ×ν}^{(m,n)}ℰ_{(m,n)}. However, this follows immediately from the previous Lemma because G_{μ⊔ν} = G_{μ} × G_{ν} ⊆ G_{(m,n)}. □

Notice that the Proposition says that Sol(r) is a subalgebra of 𝒢 (r) and that, as an algebra, Sol(r) is freely generated by the elements { E_{n} : n ∈ ℤ }.

9.6. Proposition. Suppose that n is a positive integer. Then

a) Δ(E_{n}) = ∑_{m=0}^{n} E_{m} ⊗ E_{n−m};

b) Δ(E_{−n}) = ∑_{m=0}^{n} E_{−m} ⊗ E_{m−n}.

Proof. Part (a) follows directly from the definitions. This result is well known because E_{n} = 1_{G_{r,n}} is the identity element of 𝕜G_{r,n}, so we omit the details.

For part (b), observe that E_{−n} = F_{(n)} = ∑_{t∈T} t. Therefore,

Δ(E_{−n}) = ∑_{α=(α_{1},...,α_{n})∈ℤ_{n}^{r}} Δ(t_{1}^{α_{1}} … t_{n}^{α_{n}})

= ∑_{α=(α_{1},...,α_{n})∈ℤ_{n}^{r}} ∑_{m=0}^{n} t_{1}^{α_{1}} … t_{m}^{α_{m}} ⊗ t_{1}^{α_{m+1}} … t_{n−m}^{α_{m}}

= ∑_{m=0}^{n} ∑_{\begin{matrix}β∈ℤ_{m}^{r} \\ γ∈ℤ_{n−m}^{r} \end{matrix}} t_{1}^{β_{1}} … t_{m}^{β_{m}} ⊗ t_{1}^{γ_{1}} … t_{n−m}^{γ_{n−m}}

= ∑_{m=0}^{n} E_{−m} ⊗ E_{m−n},

as required. □

We henceforth adopt the unusual convention that ∑_{m=a}^{b} f(m) = ∑_{m=b}^{a} f(m) if b < a. This allows us to write the Proposition 9.6 more compactly as Δ(E_{n}) = ∑_{m=0}^{μ_{k}} E_{m} ⊗ E_{μ_{k}−m}, for n ∈ ℤ.

As the coproduct is an algebra homomorphism 𝒢 (r) → 𝒢 (r) ⊗ 𝒢 (r) it follows from the last two Propositions that Sol(r) is a sub-bialgebra of 𝒢 (r).

Let P be a set of non-commuting indeterminates over 𝕜. The concatenation Hopf algebra on P is the free associative 𝕜-algebra 𝕜⟨P⟩ on P with counit ε, where ε(f(P)) = f(0) is the constant term of f(P) ∈ 𝕜⟨P⟩, coproduct δ(p) = p ⊗ 1 + 1 ⊗ p for any p ∈ P, and antipode S determined by S(p_{1} … p_{k}) = (−1)^{k}p_{k} … p_{1}, for p_{1}, …, p_{k} ∈ P. Any function deg : P ⟶ ℕ extends to a degree function on the monomials in 𝕜⟨P⟩ by setting deg(p_{1} … p_{k}) = deg(p_{1}) + ⋯ + deg(p_{k}). In this way, 𝕜⟨P⟩ = ⨁_{n≥0} 𝕜⟨P⟩_{n} becomes a graded connected Hopf algebra, where 𝕜⟨P⟩_{n} is the space of homogeneous polynomials p_{1} … p_{k} in P with deg(p_{1} … p_{k}) = n.

We can now prove the main result of this section. Up until now we have not used the assumption that 𝕜 is a field of characteristic zero. This assumption is necessary, however, for the proof of the following Theorem.

9.7. Theorem. Suppose that 𝕜 is a field of characteristic zero. Then (Sol(r), Δ, ε) is isomorphic to the graded connected concatenation Hopf algebra 𝕜⟨P⟩ on a set of noncommuting indeterminates P = { P_{n} : n ∈ ℤ \ {0} } where deg P_{±n} = n, for n > 0.

Proof. Our argument is modeled on the proof of [22, Theorem 2.1].

Let x be a formal variable and consider the algebra Sol(r)⟦x⟧ of formal power series in x over Sol(r), where x commutes with Sol(r). For each positive integer n define elements P_{±n} ∈ Sol(r)⟦x⟧ using the generating series

∑_{n>0} P_{n}x^{n} = log(1 + E_{1}x + E_{2}x^{2} + ⋯)

and

∑_{n>0} P_{−n}x^{n} = log(1 + E_{−1}x + E_{−2}x^{2} + ⋯).

A straightforward calculation using Proposition 9.5 and the Taylor series expansion of log(1 + t) shows that

P_{n} = ∑_{α∈Λ_{n}} \frac{(−1)^{ℓ(α)−1}}{ℓ(α)} E_{α} and P_{−n} = ∑_{−α∈Λ_{n}} \frac{(−1)^{ℓ(α)−1}}{ℓ(α)} E_{α}.

(Recall that ℓ(α) is the number of non-zero parts in α.) Therefore, P_{n}, P_{−n} ∈ Sol(G_{r,n}) are homogeneous of degree n; in particular, P_{±n} ∈ Sol(r), for all n > 0. Consequently, the elements { P_{±n} : n > 0 } generate a subalgebra of Sol(r).

Similarly, since ∑_{n≥0} E_{±n}x^{n} = exp(∑_{n>0} P_{±n}x^{n}), another completely formal calculation using the Taylor series expansion of exp(x) and Proposition 9.5 shows that if n > 0 then

E_{n} = ∑_{α∈Λ_{n}} \frac{1}{ℓ(α)!}P_{α} and E_{−n} = ∑_{−α∈Λ_{n}} \frac{1}{ℓ(α)!}P_{α},

where we set P_{α} = P_{α_{1}} ∗ ⋯ ∗ P_{α_{k}}, for α = (α_{1}, …, α_{k}) ∈ ±Λ_{n}. Therefore, by the last paragraph, the set P = { P_{n} : n ∈ ℤ \ {0} } freely generates Sol(r) as an algebra. That is, Sol(r) = ⟨P_{±n} | n > 0⟩ as an algebra.

We claim that Δ(P_{n}) = P_{n} ⊗ 1 + 1 ⊗ P_{n}, for n ∈ ℤ \ {0}. This will complete the proof because it shows that these elements generate a concatenation Hopf algebra 𝕜⟨P⟩ inside Sol(r). Starting from the definition of P_{±n} we have that

∑_{n>0} Δ(P_{±n}x^{n}) = Δ( ∑_{n≥0} log(∑_{n≥0} E_{±n}x^{n})) = log ( ∑_{n≥0} Δ(E_{±n})x^{n}),

where the last equality follows by the linearity of Taylor expansions since Δ is an algebra homomorphism. Using Proposition 9.6 to expand the right hand side of the last equation, exactly as in the proof of [22, (2.9)], shows that Δ(P_{±n}) = P_{n} ⊗ 1 + 1 ⊗ P_{n}. This proves our claim and so completes the proof. □

9.8. Corollary. Suppose that r > 1. Then the graded vector space Sol(r) equipped with the product ∗, coproduct Δ, unit E_{0} and counit ε, is a graded connected Hopf subalgebra of 𝒢 (r).

10. A second bialgebra structure on Sol(r)

In this section we show that the cyclotomic Hopf algebra Sol(r) has a second bialgebra structure with the same coproduct Δ as in section 9, but where the product is inherited from multiplication in the groups G_{r,n}, for r, n ≥ 0. More precisely, the internal product is the unique bilinear map ·: 𝒢 (r)⟶𝒢 (r) such that if w ∈ G_{r,m} and v ∈ G_{r,n} then

w · v = \begin{cases}wv, & if n = m, \\ 0, & otherwise. \end{cases}

We frequently abuse notation and write xy = x · y, for x, y ∈ 𝒢 (r). It is straightforward to check that (Sol(r), Δ, ε, ·) is a bialgebra.

As each of the group algebras 𝕜G_{r,n} are associative algebras it follows that (𝒢 (r), ·) is an associative algebra. Note, however, that (𝒢 (r), ·) does not have a multiplicative unit, so we cannot expect to obtain a second Hopf algebra structure on Sol(r) in this way. Note also that the internal product · does not respect the grading on 𝒢 (r) = ⨁_{n} 𝕜G_{r,n}.

By Theorem 6.8, Sol(r) is a subalgebra of the algebra (𝒢 (r), ·). We will show that ( Sol(r), ·, Δ) is a sub-bialgebra of 𝒢 (r). To prove this we need only show that Δ is an algebra homomorphism with respect to the internal product. The argument that we give generalizes that used by Malvenuto [21, Remark 5.15] to prove the analogous statement for the descent algebra of the symmetric group. We start with some new definitions.

A pseudo signed composition of n is an element c = (c_{1}, c_{2}, …, c_{k}) ∈ ℤ^{k}, for some k > 0, such that |c| = |c_{1}| + |c_{2}| + ⋯ |c_{k}| = n. A pseudo composition is an element of ℕ^{k}, for some k > 0. The difference between (signed) compositions and pseudo (signed) compositions is that pseudo (signed) compositions can contain zeros. If c is a pseudo signed composition let \overline{c} be the signed composition obtained by omitting the zeros from c. For example, if c = (−2, 0, 3, 0, 1) then \overline{c} = (−2, 3, 1).

If c ∈ ℤ^{k} is a pseudo signed composition then set E_{c} = E\overline{c}. If c, c′ ∈ ℤ^{k} are two pseudo signed composition of the same length then c + c′ ∈ ℤ^{k}, where addition is defined componentwise. We extend the operation of concatenation to pseudo signed compositions in the obvious way so that if c ∈ ℤ^{k} and c′ ∈ ℤ^{l} then c ⊔ c′ ∈ ℤ^{k+l}.

Two integers c and c′ are sign equivalent, and we write c ∼_{sgn} c′, if c and c′ are both non–negative, or both non-positive. Similarly, two (pseudo) signed compositions c = (c_{1}, …, c_{k}) and c′ = (c′_{1}, …, c′_{k}) are sign equivalent if c_{i} ∼_{sgn} c′_{i}, for i = 1, …, k. Again, we write c ∼_{sgn} c′.

10.1. Proposition. Suppose that μ ∈ Λ_{n}^{±} and that ℓ(μ) = k. Then

Δ(E_{μ}) = ∑_{\begin{matrix}c′ ∼_{sgn} c′′∈ℤ^{k} \\ μ=c′+c′′ \end{matrix}} E_{c′} ⊗ E_{c′′}.

Proof. We argue by induction on k. As Δ(E_{0}) = E_{0} ⊗ E_{0} the case k = 0 is clear. So we may assume that k > 0. Let ν = (μ_{1}, …, μ_{k−1}) so that μ = ν ⊔ (μ_{k}). Then, by Proposition 9.5 and Proposition 9.6,

Δ(E_{μ}) = Δ(E_{ν} ∗ E_{μ_{k}}) = Δ(E_{ν}) ∗ Δ(E_{μ_{k}})

= ( ∑_{\begin{matrix}c′ ∼_{sgn} c′′∈ℤ^{k−1} \\ ν=c′+c′′ \end{matrix}} E_{c′} ⊗ E_{c′′}) ∗ ( ∑_{m=0}^{μ_{k}} E_{m} ⊗ E_{μ_{k}−m}),

by induction on k. (If μ_{k} < 0 then recall our unusual convention for summations from after Proposition 9.6.) Therefore, using Proposition 9.5 for the second equality,

Δ(E_{μ}) = ∑_{\begin{matrix}c′ ∼_{sgn} c′′∈ℤ^{k−1} \\ ν=c′+c′′ \end{matrix}} ∑_{m=0}^{μ_{k}} E_{c′} ∗ E_{m} ⊗ E_{c′′} ∗ E_{μ_{k}−m}

= ∑_{\begin{matrix}c′ ∼_{sgn} c′′∈ℤ^{k−1} \\ ν=c′+c′′ \end{matrix}} ∑_{m=0}^{μ_{k}} E_{c′⊔(m)} ⊗ E_{c′′⊔(μ_{k}−m)}

= ∑_{\begin{matrix}c′ ∼_{sgn} c′′∈ℤ^{k} \\ μ=c′+c′′ \end{matrix}} E_{c′} ⊗ E_{c′′}

as required. □

Let k, l > 0 be positive integers and let Mat_{kl}(ℤ) be the set of k × l integer matrices. If M ∈ Mat_{kl}(ℤ) let row(M) = (r_{1}, …, r_{k}) be the pseudo composition where r_{i} is the sum of the absolute entries in row i of M, for 1 ≤ i ≤ k. Similarly, let col(M) = (c_{1}, …, c_{k}) be the pseudo composition where c_{j} is the sum of the absolute values of the entries in column j of M. Finally, if M ∈ Mat_{kl}(ℤ) let comp(M) be the signed composition obtained by listing the non–zero entries in M in order, from left to right and then top to bottom; thus, if M = (m_{ij}) then comp(M) = \overline{(m_{11}, …, m_{1l}, m_{21}, …, m_{k1}, …, m_{kl})}.

If c = (c_{1}, …, c_{k}) is a pseudo signed composition then define c^{+} = (|c_{1}|, …, |c_{k}|). In the next definition we are most interested in the case when μ and ν are signed compositions. We include pseudo signed compositions in the definition because they are needed in the proof of Theorem 10.5 below.

10.2. Definition. Suppose that μ = (μ_{1}, …, μ_{k}) and ν = (ν_{1}, …, ν_{l}) are pseudo signed compositions of n. Let

N_{μν} = { M = (m_{ij}) ∈ Mat_{kl}(ℤ) | row(M) = μ^{+}, and col(M) = ν^{+}, m_{ij} ≤ 0 if μ_{i} < 0 or if ν_{j} < 0, and m_{ij} ≥ 0 if μ_{i} > and ν_{j} > 0}.

Suppose now that M = (m_{ij}) ∈ N_{μν}. The weight of M is the non-negative integer

wt(M) = − ∑_{\begin{matrix}i:μ_{i}<0 \\ j:ν_{j}<0 \end{matrix}} m_{ij},

where in the sum 1 ≤ i ≤ k and 1 ≤ j ≤ l (note that m_{ij} ≤ 0 for all such i, j). If μ and ν are signed compositions let T_{M} be the unique row semistandard tableau in T (μ, ν) such that j appears |m_{ij}| times in row i of T, for 1 ≤ i ≤ k and 1 ≤ j ≤ l.

Note that if μ and ν are compositions and M = (m_{ij}) ∈ N_{μν} then wt(M) = 0 and m_{ij} ≥ 0, for 1 ≤ i ≤ ℓ(μ) and 1 ≤ j ≤ ℓ(ν).

10.3. Proposition. Suppose that μ and ν are signed compositions of n. Then

E_{μ}E_{ν} = ∑_{M∈N_{μν}} r^{wt(M)}E_{comp(M)}

Proof. By Theorem 6.8, E_{μ}E_{ν} = ∑_{d∈𝒟_{_{μ^{+}}_{ν^{+}}}} |T_{−μ∩d(−ν)}| E_{μ∩dν}. Therefore, to prove the Proposition it is enough to show that there exists a bijection N_{μν} → 𝒟_{μ^{+}}_{ν^{+}}; M ↦ d_{M} such that comp(M) = μ ∩ d_{M}ν and r^{wt(M)} = |T_{−μ∩d_{M} (−ν)}|.

First, observe that the map N_{μν} → T (μ, ν); M ↦ T_{M} is a bijection because its inverse is the map which sends the tableau T ∈ T (μ, ν) to M_{T} = (m_{ij}), where |m_{ij}| is the number of times that j appears in row i of T, and where the sign of m_{ij} is determined by the constraints on N_{μν}. Next, by (5.5), the map T (μ, ν) → 𝒟_{μ^{+}}_{ν^{+}}; T ↦ d_{T^{∗}} is a bijection. Hence, the map

N_{μν} → 𝒟_{μ^{+}}_{ν^{+}}; M ↦ d_{M} = d_{T_{M}^{∗}}

is a bijection.

Fix M ∈ N_{μν}. Then wt(M) = wt(T_{M}) in the notation of Corollary 5.8, so that r^{wt(M)} = |T_{−μ∩d_{M} (−ν)}|. Hence, it remains to prove that comp(M) = μ ∩ d_{M}ν. The permutation d_{M} is determined by the row semistandard tableau T which, by the last paragraph, also determines M = (m_{ij}). If m_{ij} ≠ 0 then |m_{ij}| is equal to the number of times that j appears in row i of T. Writing G_{μ} = G_{μ_{1}} ×⋯×G_{μ_{k}} and G_{ν} = G_{ν_{1}} ×⋯×G_{ν_{l}}, and abusing notation slightly, we see that m_{ij} computes the intersection of G_{μ_{i}} with d_{M}G_{ν_{j}}d_{M}^{−1}; more precisely,

G_{μ_{i}} ∩ d_{M}G_{ν_{j}}d_{M}^{−1} ≅ \begin{cases}G(r, 1, m_{ij}), & if m_{ij} > 0, \\ 𝔖_{−m_{ij}}, & if m_{ij} < 0. \end{cases}

Comparing this with the recipe given in the proof of Lemma 5.1 for computing μ ∩ d_{M}ν we see that comp(M) = μ ∩ d_{M}ν, as required. □

Garsia–Remmel [13, Prop. 1.1] (see also [12, §4]), proved the analogue of this result for the Solomon algebras of the symmetric groups. This is equivalent to special case of Proposition 10.3 when μ and ν are both compositions of n. If μ, ν ∈ Λ_{n} then the bijection N_{μν} ⟶^{≃} 𝒟_{μν} is well–known; see, for example, [19, Theorem 1.3.10]. 10.4. Example As in Example 5.9, suppose that μ = (3, −2) and ν = (−2^{2}, 1). The following table lists all of the elements of N_{μν}, together with the associated signed composition and row semistandard μ–tableau of type ν and the weight of the matrix.

\begin{matrix}M & comp(M) & T_{M} & wt(M) \\ ( \begin{matrix}−2 & −1 & 0 \\ 0 & −1 & −1 \end{matrix} ) & (−2, −1^{3}) & 1 1 2 2 3 & 1 \\ ( \begin{matrix}−2 & 0 & 1 \\ 0 & −2 & 0 \end{matrix} ) & (−2, 1, −2) & 1 1 3 2 2 & 2 \\ ( \begin{matrix}−1 & −2 & 0 \\ −1 & 0 & −1 \end{matrix} ) & (−1, −2, −1^{2}) & 1 2 2 1 3 & 1 \\ ( \begin{matrix}−1 & −1 & 1 \\ −1 & −1 & 0 \end{matrix} ) & (−1^{2}, 1, −1^{2}) & 1 2 3 1 2 & 2 \\ ( \begin{matrix}0 & −2 & 1 \\ −2 & 0 & 0 \end{matrix} ) & (−2, 1, −2) & 2 2 3 1 1 & 2 \end{matrix}

The reader might like to compare this table with the one given in Example 5.9. Combining the information above with Proposition 10.3 shows that

E_{(3,−2)}E_{(−2^{2},1)} = 2r^{2}E_{(−2,1,−2)} + rE_{(−2,−1^{3})} + rE_{(−1,−2,−1^{2})} + r^{2}E_{(−1^{2},1,−1^{2})}.

This calculation agrees with Example 6.10, as it must. ♢

Suppose that M′ = (m′_{ij}), M′′ = (m′′_{ij}) ∈ Mat_{kl}(ℤ), for some k, l > 0. Then M′ and M′′ are signed equivalent, and we write M′ ∼_{sgn} M′′, if m′_{ij} ∼_{sgn} m′′_{ij}, for 1 ≤ i ≤ k and 1 ≤ j ≤ l.

We can now prove the main result of this section.

10.5. Theorem. Suppose that r > 1. Then Sol(r) equipped with product ·, coproduct Δ and counit ε, is a bialgebra.

Proof. As remarked at the beginning of this section, it remains to show that the coproduct Δ : Sol(r) ⟶ Sol(r) ⊗ Sol(r) is an algebra homomorphism with respect to the internal product ·. By linearity it is enough to show that

Δ(E_{μ}E_{ν}) = Δ(E_{μ})Δ(E_{ν}),

for all signed compositions μ and ν. Further, we may assume that |μ| = |ν| since otherwise both sides of this equation are zero. Let k = ℓ(μ) and l = ℓ(ν) and for M ∈ N_{μν} let ℓ(M) = ℓ(comp(M)). Then, by Proposition 10.3 and Proposition 10.1,

Δ(E_{μ}E_{ν}) = ∑_{M∈N_{μν}} r^{wt(M)}Δ(E_{comp(M)})

= ∑_{M∈N_{μν}} ∑_{c′ ∼_{sgn} c′′∈ℤ^{ℓ(M)} comp(M)=c′+c′′} r^{wt(M)}E_{c′} ⊗ E_{c′′},

For the moment, fix a matrix M ∈ N_{μν} and c′, c′′ ∈ ℤ^{ℓ(M)} such that c′ ∼_{sgn} c′′ and comp(M) = c′ + c′′. Since c′ ∼_{sgn} c′′ there exist unique matrices M′ = (m′_{ij}), M′′ = (m′′_{ij}) ∈ Mat_{kl}(ℤ) such that M = M′ + M′′, M ∼_{sgn} M′ ∼_{sgn} M′′, comp(M′) = \overline{c′} and comp(M′′) = \overline{c′′}. Note that wt(M) = wt(M′) + wt(M′′) since M′ ∼_{sgn} M′′. Therefore, the last equation becomes

Δ(E_{μ}E_{ν}) = ∑_{M∈N_{μν}} ∑_{\begin{matrix}M′ ∼_{sgn} M′′ \\ M′+M′′=M \end{matrix}} r^{wt(M′)}E_{comp(M′)} ⊗ r^{wt(M′′)}E_{comp(M′′)}

For each pair M′ and M′′ in the second sum let μ′ = row(M′) and μ′′ = row(M′′). Then μ′ and μ′′ are pseudo signed compositions such that μ = μ′ + μ′′ and μ′ ∼_{sgn} μ′′. Similarly, ν′ = col(M′) and ν′′ = col(M′′) are pseudo signed compositions such that ν = ν′ + ν′′ and ν′ ∼_{sgn} ν′′. By signed equivalence, M′ ∈ N_{μ′ν′} and M′′ ∈ N_{μ′′ν′′}. Moreover, M′ and M′′ run through N_{μ′ν′} and N_{μ′′ν′′}, respectively, for all possible μ′, μ′′, ν′ and ν′′, as M runs through N_{μν}. Observe that if M′ ∈ N_{μ′ν′} and M′′ ∈ N_{μ′′ν′′}, for μ′, μ′′, ν′ and ν′′ as above, then M′ ∼_{sgn} M′′ since μ′ ∼_{sgn} μ′′ and ν′ ∼_{sgn} ν′′. Therefore, we can reverse the order of summation in the last displayed equation to obtain

Δ(E_{μ}E_{ν}) = ∑_{\begin{matrix}μ′ ∼_{sgn} μ′′ \\ μ=μ′+μ′′ \\ ν′ ∼_{sgn} ν′′ \\ ν=ν′+ν′′ \end{matrix}} ∑_{\begin{matrix}M′∈N_{μ′ν′} \\ M′′∈N_{μ′′ν′′} \end{matrix}} r^{wt(M′)}E_{comp(M′)} ⊗ r^{wt(M′′)}E_{comp(M′′)}

= ∑_{\begin{matrix}μ′ ∼_{sgn} μ′′ \\ μ=μ′+μ′′ \\ ν′ ∼_{sgn} ν′′ \\ ν=ν′+ν′′ \end{matrix}}( ∑_{M′∈N_{μ′ν′}}r^{wt(M′)}E_{comp(M′)})⊗( ∑_{M′′∈N_{μ′′ν′′}}r^{wt(M′′)}E_{comp(M′′)})

= ( ∑_{\begin{matrix}μ′ ∼_{sgn} μ′′ \\ μ=μ′+μ′′ \end{matrix}} E_{μ′} ⊗ E_{μ′′})( ∑_{\begin{matrix}ν′ ∼_{sgn} ν′′ \\ ν=ν′+ν′′ \end{matrix}} E_{ν′} ⊗ E_{ν′′})

= Δ(E_{μ})Δ(E_{ν}),

where the last two equalities follow by Proposition 10.3 and Proposition 10.1 respectively. This completes the proof. □

Acknowledgments

Many of the results in this paper were inspired by extensive computer calculations using programs written using Gap [26]. We thank N. Bergeron and M. Aguiar for useful conversations.

References

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[Page 1.]

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School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia.

E-mail address: a.mathas@usyd.edu.au

Department of Mathematics, Dartmouth College, Hanover, NH 03755–3551, USA.

E-mail address: rosa.c.orellana@dartmouth.edu