SEMINORMAL FORMS AND GRAM DETERMINANTS FOR CELLULAR ALGEBRAS

Andrew Mathas

School of Mathematics and Statistics F07, University of Sydney, Sydney NSW 2006, Australia

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

Abstract. This paper develops an abstract framework for constructing ‘seminormal forms’ for cellular algebras. That is, given a cellular R-algebra A which is equipped with a family of JM-elements we give a general technique for constructing orthogonal bases for A, and for all of its irreducible representations, when the JM-elements separate A. The seminormal forms for A are defined over the field of fractions of R. Significantly, we show that the Gram determinant of each irreducible A-module is equal to a product of certain structure constants coming from the seminormal basis of A. In the non-separated case we use our seminormal forms to give an explicit basis for a block decomposition of A.

1. Introduction

The purpose of this paper is to give an axiomatic way to construct “seminormal forms” and to compute Gram determinants for the irreducible representations of semisimple cellular algebras. By this we mean that, starting from a given cellular basis {a_{st}^{λ}} for a cellular algebra A, we give a new cellular basis {f_{st}^{λ}} for the algebra which is orthogonal with respect to a natural bilinear form on the algebra. This construction also gives a “seminormal basis” for each of the cell modules of the algebra. We show that the Gram determinant of the cell modules (the irreducible A–modules) can be computed in terms of the structure constants of the new cellular basis of A. Combining these results gives a recipe for computing the Gram determinants of the irreducible A–modules.

Of course, we cannot carry out this construction for an arbitrary cellular algebra A. Rather, we assume that the cellular algebra comes equipped with a family of “Jucys– Murphy” elements. These are elements of A which act on the cellular basis of A via upper triangular matrices. We will see that, over a field, the existence of such a basis {f_{st}^{λ}} forces A to be (split) semisimple (and, conversely, every split semisimple algebra has a family of JM–elements). The cellular algebras which have JM–elements include the group algebras of the symmetric groups, any split semisimple algebra, the Hecke algebras of type A, the q–Schur algebras, the (degenerate) Ariki–Koike algebras, the cyclotomic q–Schur Algebras, the Brauer algebras and the BMW algebras.

At first sight, our construction appears to be useful only in the semisimple case. However, in the last section of this paper we apply these ideas in the non–semisimple case to construct a third cellular basis {g_{st}^{λ}} of A. We show that this basis gives an explicit decomposition of A into a direct sum of smaller cellular subalgebras. In general, these subalgebras need not be indecomposable, however, it turns out that these subalgebras are indecomposable in many of the cases we know about. As an application, we give explicit bases for the block decomposition of the group algebras of the symmetric groups, the Hecke algebras of type A, the Ariki–Koike algebras with q ≠ 1, the degenerate Ariki– Koike algebras and the (cyclotomic) q–Schur algebras.

There are many other accounts of seminormal forms in the literature; see, for example, [1, 8, 13, 20]. The main difference between this paper and previous work is that, starting from a cellular basis for an algebra we construct seminormal forms for the entire algebra, rather than just the irreducible modules. The main new results that we obtain are explicit formulae for the Gram determinants of the cell modules in the separated case, and a basis for a block decomposition of the algebra in the non–separated case. These seminormal forms that we construct have the advantage that they are automatically defined over the field of fractions of the base ring; this is new for the Brauer and BMW algebras.

Finally, we remark that cellular algebras provide the right framework for studying seminormal forms because it turns out that an algebra has a family of separating JM–elements if and only if it is split semisimple (see Example 2.13), and every split semisimple algebra is cellular. In the appendix to this paper, Marcos Soriano, gives an alternative matrix theoretic approach to some of the results in this paper which, ultimately, rests on the Cayley–Hamilton theorem.

2. Cellular algebras and JM–elements

We begin by recalling Graham and Lehrer’s [5] definition of a cellular algebra. Let R be commutative ring with 1 and let A be a unital R–algebra and let K be the field of fractions of R.

2.1. Definition (Graham and Lehrer). A cell datum for A is a triple (Λ, T, C) where Λ = (Λ, >) is a finite poset, T (λ) is a finite set for each λ ∈ Λ, and

C: ∐_{λ∈Λ} T (λ) × T (λ)⟶A; (s, t) ↦ a_{st}^{λ}

is an injective map (of sets) such that:

a) { a_{st}^{λ} | λ ∈ Λ, s, t ∈ T (λ) } is an R–free basis of A;

b) For any x ∈ A and t ∈ T (λ) there exist scalars r_{tvx} ∈ R such that, for any

s ∈ T (λ),

a_{st}^{λ}x ≡ ∑_{v∈T (λ)} r_{tvx}a_{sv}^{λ} (mod A^{λ}),

where A^{λ} is the R–submodule of A with basis { a_{yz}^{μ} | μ > λ and y, z ∈ T (μ) }.

c) The R–linear map determined by ∗: A ⟶ A; a_{st}^{λ} = a_{ts}^{λ}, for all λ ∈ Λ and s, t ∈

T (λ), is an anti–isomorphism of A. If a cell datum exists for A then we say that A is a cellular algebra.

Henceforth, we fix a cellular algebra A with cell datum (Λ, T, C) as above. We will also assume that T (λ) is a poset with ordering ▷_{λ}, for each λ ∈ Λ. For convenience we set T (Λ) = ∐_{λ∈Λ} T (λ). We consider T (Λ) as a poset with the ordering s ▷ t if either (1) s, t ∈ T (λ), for some λ ∈ Λ, and s ▷_{λ} t, or (2) s ∈ T (λ), t ∈ T (μ) and λ > μ. We write s ⊵ t if s = t or s ▷ t. If s ⊵ t we say that s dominates t.

Note that, by assumption A, is a free R–module of finite rank T (Λ).

Let A_{K} = A ⊗_{R} K. As A is free as an R–module, A_{K} is a cellular algebra with cellular basis { a_{st}^{λ} ⊗ 1_{K} | λ ∈ Λ and s, t ∈ T (λ) }. We consider A as a subalgebra of A_{K} and, abusing notation, we also consider a_{st}^{λ} to be elements of A_{K}.

We recall some of the general theory of cellular algebras. First, applying the ∗ involution to part (b) of Definition 2.1 we see that if y ∈ A and s ∈ T (λ) then there exist scalars r′_{suy} ∈ R such that, for all t ∈ T (λ),

(2.2)  ya_{st}^{λ} ≡ ∑_{u∈T (λ)} r′_{suy}a_{ut}^{λ} (mod A^{λ}).

Consequently, A^{λ} is a two–sided ideal of A, for any λ ∈ Λ.

Next, for each λ ∈ Λ define the cell module C(λ) to be the free R–module with basis { a_{t}^{λ} | t ∈ T (λ) } and with A–action given by

a_{t}^{λ}x = ∑_{v∈T (λ)} r_{tvx}a_{v}^{λ},

where r_{tvx} is the same scalar which appears in Definition 2.1. As r_{tvx} is independent of s this gives a well–defined A–module structure on C(λ). The map ⟨ , ⟩_{λ}: C(λ)×C(λ)⟶R which is determined by

(2.3)  ⟨a_{t}^{λ}, a_{u}^{λ}⟩_{λ}a_{sv}^{λ} ≡ a_{st}^{λ}a_{uv}^{λ} (mod A^{λ}),

for s, t, u, v ∈ T (λ), defines a symmetric bilinear form on C(λ). This form is associative in the sense that ⟨ax, b⟩_{λ} = ⟨a, bx^{∗}⟩_{λ}, for all a, b ∈ C(λ) and all x ∈ A. From the definitions, for any s ∈ T (λ) the cell module C(λ) is naturally isomorphic to the A– module spanned by { a_{st}^{λ} + A^{λ} | t ∈ T (λ) }. The isomorphism is the obvious one which sends a_{t}^{λ} ↦ a_{st}^{λ} + A^{λ}, for t ∈ T (λ).

For λ ∈ Λ we define rad C(λ) = { x ∈ C(λ) | ⟨x, y⟩_{λ} = 0 for all y ∈ C(λ) }. As the bilinear form on C(λ) is associative it follows that rad C(λ) is an A–submodule of C(λ). Graham and Lehrer [5, Theorem 3.4] show that the A_{K}–module D(λ) = C(λ)/ rad C(λ) is absolutely irreducible and that { D(λ) ≠ 0 | λ ∈ Λ } is a complete set of pairwise non– isomorphic irreducible A_{K}–modules.

The proofs of all of these results follow easily from Definition 2.1. For the full details see [5, §2–3] or [14, Chapt. 2].

In this paper we are interested only in those cellular algebras which come equipped with the following elements.

2.4. Definition. A family of JM–elements for A is a set {L_{1}, …, L_{M}} of commuting elements of A together with a set of scalars, { c_{t}(i) ∈ R | t ∈ T (Λ) and 1 ≤ i ≤ M }, such that for i = 1, …, M we have L_{i}^{∗} = L_{i} and, for all λ ∈ Λ and s, t ∈ T (λ),

a_{st}^{λ}L_{i} ≡ c_{t}(i)a_{st}^{λ} + ∑_{v▷t} r_{tv}a_{sv}^{λ} (mod A^{λ}),

for some r_{tv} ∈ R (which depend on i). We call c_{t}(i) the content of t at i.

Implicitly, the JM–elements depend on the choice of cellular basis for A.

Notice that we also have the following left hand analogue of the formula in (2.4):

(2.5)  L_{i}a_{st}^{λ} ≡ c_{s}(i)a_{st}^{λ} + ∑_{u▷s} r′_{su}a_{ut}^{λ} (mod A^{λ}),

for some r′_{su} ∈ R.

2.6. Let ℒ_{K} be the subalgebra of A_{K} which is generated by {L_{1}, …, L_{M}}. By definition, ℒ_{K} is a commutative subalgebra of A_{K}. It is easy to see that each t ∈ T (Λ) gives rise to a one dimensional representation K_{t} of ℒ_{K} on which L_{i} acts as multiplication by c_{t}(i), for 1 ≤ i ≤ M. In fact, since ℒ_{K} is a subalgebra of A_{K}, and A_{K} has a filtration by cell modules, it follows that { K_{t} | t ∈ T (Λ) } is a complete set of irreducible ℒ_{K}–modules.

These observations give a way of detecting when D(λ) ≠ 0 (cf. [5, Prop. 5.9(i)]).

2.7. Proposition. Let A_{K} be a cellular algebra with a family of JM–elements and fix λ ∈ Λ, and s ∈ T (λ). Suppose that whenever t ∈ T (Λ) and s ▷ t then c_{t}(i) ≠ c_{s}(i), for some i with 1 ≤ i ≤ M. Then D(λ) ≠ 0.

Proof. By Definition 2.4, for any μ ∈ Λ the ℒ_{K}–module composition factors of C(μ) are precisely the modules { K_{s} | s ∈ T (μ) }. Observe that if u, v ∈ T (Λ) then K_{u} ≅ K_{v} as ℒ_{K}–modules if and only if c_{u}(i) = c_{v}(i), for 1 ≤ i ≤ M. Therefore, our assumptions imply that K_{t} is not an ℒ_{K}–module composition factor of any cell module C(μ) whenever λ > μ. Consequently, K_{t} is not an ℒ_{K}–module composition factor of D(μ) whenever λ > μ. However, by [5, Prop. 3.6], D(μ) is a composition factor of C(λ) only if λ ≥ μ. Therefore, a_{t}^{λ} ∉ rad C(λ) and, consequently, D(λ) ≠ 0 as claimed. □

Motivated by Proposition 2.7, we break our study of cellular algebras with JM–elements into two cases depending upon whether or not the condition in Proposition 2.7 is satisfied.

2.8. Definition (Separation condition). Suppose that A is a cellular algebra with JM– elements {L_{1}, …, L_{M}}. The JM–elements separate T (λ) (over R) if whenever s, t ∈ T (Λ) and s ▷ t then c_{s}(i) ≠ c_{t}(i), for some i with 1 ≤ i ≤ M.

In essence, the separation condition says that the contents c_{t}(i) distinguish between the elements of T (Λ). Using the argument of Proposition 2.7 we see that the separation condition forces A_{K} to be semisimple.

2.9. Corollary. Suppose that A_{K} is a cellular algebra with a family of JM–elements which separate T (Λ). Then A_{K} is (split) semisimple.

Proof. By the general theory of cellular algebras [5, Theorem 3.8], A_{K} is (split) semisimple if and only if C(λ) = D(λ) for all λ ∈ Λ. By the argument of Proposition 2.7, the separation condition implies that if t ∈ T (λ) then K_{t} does not occur as an ℒ_{K}–module composition factor of D(μ) for any μ > λ. By [5, Prop. 3.6], D(μ) is a composition factor of C(λ) only if λ ≥ μ, so the cell module C(λ) = D(λ) is irreducible. Hence, A_{K} is semisimple as claimed. □

In Example 2.13 below we show that every split semisimple algebra is a cellular algebra with a family of JM–elements which separate T (Λ).

2.10. Remark. Corollary 2.9 says that if a cellular algebra A has a family of JM–elements which separate T (Λ) then A_{K} is split semisimple. Conversely, we show in Example 2.13 below that every split semisimple algebra has a family of JM–elements which separate T (λ). However, if A is semisimple and A has a family of JM–elements then it is not true that the JM–elements must separate A; the problem is that an algebra can have different families of JM–elements. As described in Example 2.18 below, the Brauer and BMW algebras both have families of JM–elements. Combined with work of Enyang [4, Examples 7.1 and 10.1] this shows that there exist BMW and Brauer algebras which are semisimple and have JM–elements which do not separate T (Λ).

2.11. Remark. Following ideas of Grojnowski [12, (11.9)] and (2.6) we can use the algebra ℒ_{K} to define formal characters of A_{K}–modules as follows. Let { K_{t} | t ∈ L(Λ) } be a complete set of non–isomorphic irreducible ℒ_{K}–modules, where L(Λ) ⊆ T (Λ). If M is any A_{K}–module let [M : K_{t}] be the decomposition multiplicity of the irreducible ℒ_{K}– module K_{t} in M. Define the formal character of M to be

ch M = ∑_{t∈L(Λ)}[M : K_{t}] e^{t},

which is element of the free ℤ–module with basis { e^{t} | t ∈ L(Λ) }.

We close this introductory section by giving examples of cellular R–algebras which have a family of JM–elements. Rather than starting with the simplest example we start with the motivating example of the symmetric group. The latter examples are either less well–known or new. 2.12 Example (Symmetric groups) The first example of a family of JM–elements was given by Jucys [11] and, independently, by Murphy [16]. (In fact, these elements first appear in the work of Young [22].) Let A = R𝔖_{n} be the group ring of the symmetric group of degree n. Define

L_{i} = (1, i) + (2, i) + ⋯ + (i − 1, i), for i = 2, …, n.

Murphy [16] showed that these elements commute and he studied the action of these elements on the seminormal basis of the Specht modules. The seminormal basis of the Specht modules can be extended to a seminormal basis of R𝔖_{n}, so Murphy’s work shows that the group algebra of the symmetric group fits into our general framework. We do not give further details because a better approach to the symmetric groups in given by the special case q = 1 of Example 2.15 below which concerns the Hecke algebra of type A. ♢ 2.13 Example (Semisimple algebras) By Corollary 2.9 every cellular algebra over a field which has a family of JM–elements which separate T (Λ) is split semisimple. In fact, the converse is also true.

Suppose that A_{K} is a split semisimple algebra. Then the Wedderburn basis of matrix units in the simple components of A_{K} is a cellular basis of A_{K}. We claim that A_{K} has a family of JM–elements. To see this it is enough to consider the case when A_{K} = Mat_{n}(K) is the algebra of n × n matrices over K. Let e_{ij} be the elementary matrix with has a 1 in row i and column j and zeros elsewhere. Then it is easy to check that {e_{ij}} is a cellular basis for A_{K} (with Λ = {1}, say, and T (λ) = {1, …, n})). Let L_{i} = e_{ii}, for 1 ≤ i ≤ n. Then {L_{1}, …, L_{n}} is a family of JM–elements for A_{K} which separate T (Λ).

By the last paragraph, any split semisimple algebra A_{K} has a family of JM–elements {L_{1}, …, L_{M}} which separate T (Λ), where M = d_{1} + ⋯ + d_{r} and d_{1}, …, d_{r} are the dimensions of the irreducible A_{K}–modules. The examples below show that we can often find a much smaller set of JM–elements. In particular, this shows that the number M of JM–elements for an algebra is not an invariant of A! Nevertheless, in the separated case we will show that the JM–elements are always linear combinations of the diagonal elementary matrices coming from the different Wedderburn components of the algebra. Further, the subalgebra of A_{K} generated by a family of JM–elements is a maximal abelian subalgebra of A_{K}. ♢

If A_{K} is a cellular algebra and explicit formulae for the Wedderburn basis of A_{K} are known then we do not need this paper to understand the representations of A_{K}. One of the points of this paper is that if we have a cellular basis for an R–algebra A together with a family of JM–elements then we can construct a Wedderburn basis for A_{K}. 2.14 Example (A toy example) Let A = R[X]/(X − c_{1}) … (X − c_{n}), where X is an indeterminate over R and c_{1}, …, c_{n} ∈ R. Let x be the image of X in A under the canonical projection R[X] ⟶ A. Set a_{i} := a_{ii}^{i} = ∏_{j=1}^{i−1}(x − c_{j}), for i = 1, …, n + 1. Then A is a cellular algebra with Λ = {1, …, n}, T (i) = {i}, for 1 ≤ i ≤ n, and with cellular basis {a_{11}^{1}, …, a_{nn}^{n}}. Further, x is a JM–element for A because

a_{i}x = (x − c_{1}) … (x − c_{i−1})x = c_{i}a_{i} + a_{i+1},

for i = 1, …, n. Thus, c_{i}(x) = c_{i}, for all i. The ‘family’ of JM–elements {x} separates T (Λ) if and only if c_{1}, …, c_{n} are pairwise distinct. ♢ 2.15 Example (Hecke algebras of type A) Fix an integer n > 1 and an invertible element q ∈ R. Let ℋ = ℋ_{R,q}(𝔖_{n}) be the Hecke algebra of type A. In particular, if q = 1 then ℋ_{R,q}(𝔖_{n}) ≅ R𝔖_{n}. In general, ℋ is free as an R–module with basis { T_{w} | w ∈ 𝔖_{n} } and with multiplication determined by

T_{(i,i+1)}T_{w} = \begin{cases}T_{(i,i+1)w}, & if i^{w} > (i + 1)^{w}, \\ qT_{(i,i+1)w} + (q − 1)T_{w}, & otherwise. \end{cases}

Recall that a partition of n is a weakly decreasing sequence of positive integers which sum to n. Let Λ be the set of partitions of n ordered by dominance [14, 3.5]. If λ = (λ_{1}, …, λ_{k}) is a partition let [λ] = { (r, c) | 1 ≤ c ≤ λ_{r}, r ≤ k } be the diagram of λ. A standard λ–tableau is a map t : [λ]⟶{1, …, n} such that t is monotonic increasing in both coordinates (i.e. rows and columns).

Given λ ∈ Λ let T (λ) be the set of standard λ–tableau, ordered by dominance (the Bruhat order; see [14, Theorem 3.8]). Murphy [18] has shown that ℋ has a cellular basis of the form { m_{st}^{λ} | λ ∈ Λ and s, t ∈ T (λ) }.

Set L_{1} = 0 and define

L_{i} = ∑_{j=1}^{i−1} q^{j−i}T_{(i,j)}, for 2 ≤ i ≤ n.

It is a straightforward, albeit tedious, exercise to check that these elements commute; see, for example, [14, Prop. 3.26]. The cellular algebra ∗ involution of ℋ is the linear extension of the map which sends T_{w} to T_{w^{−1}}, for w ∈ 𝔖_{n}. So L_{i}^{∗} = L_{i}, for all i.

For any integer k let [k]_{q} = 1 + q + ⋯ + q^{k−1} if k ≥ 0 and set [k]_{q} = −q^{−k}[−k]_{q} if k < 0. Let t be a standard tableau and suppose that i appears in row r and column c of t, where 1 ≤ i ≤ n. The q–content of i in t is c_{t}(i) = [c−r]_{q}. Then, by [14, Theorem 3.32],

m_{st}^{λ}L_{i} = c_{t}(i)m_{st}^{λ} + more dominant terms.

Hence, {L_{1}, …, L_{n}} is a family of JM–elements for ℋ . Moreover, if [1]_{q}[2]_{q} … [n]_{q} ≠ 0 then a straightforward induction shows that the JM–elements separate T (Λ); see [14, Lemma 3.34]. ♢ 2.16 Example (Ariki–Koike algebras) Fix integers n, m ≥ 1, an invertible element q ∈ R and an m–tuple u = (u_{1} …, u_{m}) ∈ R^{m}. The Ariki–Koike algebra ℋ_{R,q,u} is a deformation of the group algebra of the complex reflection group of type G(m, 1, n); that is, the group (ℤ/mℤ)≀𝔖_{n}. The Ariki–Koike algebras are generated by elements T_{0}, T_{1}, …, T_{n−1} subject to the relations (T_{0} − u_{1}) … (T_{0} − u_{m}) = 0, (T_{i} − q)(T_{i} + 1) = 0 for 1 ≤ i < n, together with the braid relations of type B.

Let Λ be the set of m–multipartitions of n; that is, the set of m–tuples of partitions which sum to n. Then Λ is a poset ordered by dominance. If λ ∈ Λ then a standard λ– tableau is an m–tuple of standard tableau t = (t^{(1)}, …, t^{(m)}) which, collectively, contain the numbers 1, …, n and where t^{(s)} has shape λ^{(s)}. Let T (λ) be the set of standard λ– tableaux ordered by dominance [3, (3.11)]. It is shown in [3] that the Ariki–Koike algebra has a cellular basis of the form { m_{st}^{λ} | λ ∈ Λ and s, t ∈ T (λ) }.

For i = 1, …, n set L_{i} = q^{1−i}T_{i−1} … T_{1}T_{i}T_{1} … T_{i−1}. These elements commute, are invariant under the ∗ involution of ℋ_{R,q,u} and

m_{st}^{λ}L_{i} = c_{t}(i)m_{st}^{λ} + more dominant terms,

where c_{t}(i) = u_{s}q^{c−r} if i appears in row r and column c of t^{(s)}. All of these facts are proved in [10, §3]. Hence, {L_{1}, …, L_{n}} is a family of JM–elements for ℋ_{R,q,u}. In this case, if [1]_{q} … [n]_{q} ∏_{1≤i<j≤m} ∏_{|d|<n}(q^{d}u_{i} − u_{j}) ≠ 0 and q ≠ 1 then the JM–elements separate T (Λ) by [10, Lemma 3.12].

There is an analogous family of JM–elements for the degenerate Ariki–Koike algebras. See [2, §6] for details. ♢ 2.17 Example (Schur algebras) Let Λ be the set of partitions of n, ordered by dominance, and for μ ∈ Λ let 𝔖_{μ} be the corresponding Young subgroup of 𝔖_{n} and set m_{μ} = ∑_{w∈𝔖_{μ}} T_{w} ∈ ℋ . Then the q–Schur algebra is the endomorphism algebra

S_{R,q}(n) = End_{ℋ} ( ⨁_{μ∈Λ} m_{μ}ℋ ).

For λ ∈ Λ let T (λ) be the set of semistandard λ–tableaux, and let T_{μ}(λ) ⊆ T (λ) be the set of semistandard λ–tableaux of type μ; see [14, §4.1]. The main result of [3] says that S_{R,q}(n) has a cellular basis { φ_{ST}^{λ} | λ ∈ Λ and S, T ∈ T (λ) } where the homomorphism φ_{ST}^{λ} is given by left multiplication by a sum of Murphy basis elements m_{st}^{λ} ∈ ℋ which depend on S and T.

Let μ = (μ_{1}, …, μ_{k}) be a partition in Λ. For i = 1, …, k let L_{i}^{μ} be the endomorphism of m_{μ}ℋ which is given by

L_{i}^{μ}(m_{μ}h) = ∑_{j=μ_{1}+···+μ_{j−1}+1}^{μ_{1}+···+μ_{j}} L_{j}m_{μ}h,

for all h ∈ ℋ . Here, L_{1}, …, L_{n} are the JM–elements of the Hecke algebra ℋ . We can consider L_{i}^{μ} to be an element of S_{R,q}(n). Using properties of the JM–elements of ℋ it is easy to check that the L_{i}^{μ} commute, that they are ∗–invariant and by [9, Theorem 3.16] that

φ_{ST}^{λ}L_{i}^{μ} = \begin{cases}c_{T}(i)φ_{ST}^{λ} + more dominant terms terms, & if T ∈ T_{μ}(λ), \\ 0, & otherwise. \end{cases}

Here c_{T}(i) is the sum of the q–contents of the nodes in T labelled by i [14, §5.1]. Hence { L_{i}^{μ} | μ ∈ Λ } is a family of JM–elements for S_{R,q}(n). If [1]_{q} … [n]_{q} ≠ 0 then the JM– elements separate T (Λ); see [14, Lemma 5.4].

More generally, the q–Schur algebras S_{R,q}(n, r) of type A and the cyclotomic q–Schur algebras both have a family of JM–elements; see [9, 10] for details. ♢ 2.18 Example (Birman–Murakami–Wenzl algebras) Let r and q be invertible indeterminates over R and let n ≥ 1 an integer. Let ℬ_{n}(q, r) be the Birman–Murakami–Wenzl algebra, or BMW algebra. The BMW algebra is generated by elements T_{1}, …, T_{n−1} which satisfy the relations (T_{i} − q)(T_{i} + q^{−1})(T_{i} − r^{−1}) = 0, the braid relations of type A, and the relations E_{i}T_{i±1}^{±1}E_{i} = r^{±1}E_{i} and E_{i}T_{i} = T_{i}E_{i} = r^{−1}E_{i}, where E_{i} = 1 − \frac{T_{i}−T_{i}^{−1}}{q−q^{−1}} ; see [4, 13].

The BMW algebra ℬ_{n}(q, r) is a deformation of the Brauer algebra. Indeed, both the Brauer and BMW algebras have a natural diagram basis indexed by the set of n–Brauer diagrams; that is, graphs with vertex set \{1, …, n, \overline{1}, …, \overline{n}\} such that each vertex lies on a unique edge. For more details see [7].

Let λ be a partition of n − 2k, where 0 ≤ k ≤ ⌊\frac{n}{2}⌋. An n–updown λ–tableau t is an n–tuple t = (t_{1}, …, t_{n}) of partitions such that t_{1} = (1), t_{n} = λ and |t_{i}| = |t_{i−1}| ± 1, for 2 ≤ i ≤ n. (Here |t_{i}| is the sum of the parts of the partition t_{i}.)

Let Λ be the set of partitions of n − 2k, for 0 ≤ k ≤ ⌊\frac{n}{2}⌋ ordered again by dominance. For λ ∈ Λ let T (λ) be the set of n–updown tableaux. Enyang [4, Theorem 4.8 and §5] has given an algorithm for constructing a cellular basis of ℬ_{n}(q, r) of the form { m_{st}^{λ} | λ ∈ Λ and s, t ∈ T (λ) }. Enyang actually constructs a basis for each cell module of ℬ_{n}(q, r) which is “compatible” with restriction, however, his arguments give a new cellular basis {m_{st}^{λ}} for ℬ_{n}(q, r) which is indexed by pairs of n–updown λ–tableaux for λ ∈ Λ.

Following [13, Cor. 1.6] set L_{1} = 1 and define L_{i+1} = T_{i}L_{i}T_{i}, for i = 2, …, n. These elements are invariant under the ∗ involution of ℬ_{n}(q, r) and Enyang [4, §6] has shown that L_{1}…. , L_{n} commute and that

m_{st}^{λ}L_{i} = c_{t}(i)m_{st}^{λ} + more dominant terms,

where c_{t}(i) = q^{2(c−r)} if [t_{i}] = [t_{i−1}] ∪ {(r, c)} and c_{t}(i) = r^{−1}q^{2(r−c)} if [t_{i}] = [t_{i−1}] \ {(r, c)}. Hence, L_{1}, …, L_{n} is a family of JM–elements for ℬ_{n}(q, r). When R = ℤ[r^{±1}, q^{±1}] the JM–elements separate T (Λ).

The BMW algebras include the Brauer algebras essentially as a special case. Indeed, it follows from Enyang’s work [4, §8–9] that the Brauer algebras have a family of JM– elements which separate T (Λ).

Rui and Si [21] have recently computed the Gram determinants of for the irreducible modules of the Brauer algebras in the semisimple case. ♢

It should be possible to find JM–elements for other cellular algebras such as the partition algebras and the cyclotomic Nazarov–Wenzl algebras [2].

3. The separated case

Throughout this section we assume that A is a cellular algebra with a family of JM– elements which separate T (Λ) over R. By Corollary 2.9 this implies that A_{K} is a split semisimple algebra.

For i = 1, …, M let 𝒞 (i) = { c_{t}(i) | t ∈ T (Λ) }. Thus, 𝒞 (i) is the set of possible contents that the elements of T (Λ) can take at i.

We can now make the key definition of this paper.

3.1. Definition. Suppose that s, t ∈ T (λ), for some λ ∈ Λ and define

F_{t} = ∏_{i=1}^{M} ∏_{\begin{matrix}c∈𝒞 (i) \\ c≠c_{t}(i) \end{matrix}} \frac{L_{i} − c}{c_{t}(i) − c}.

Thus, F_{t} ∈ A_{K}. Define f_{st}^{λ} = F_{s}a_{st}^{λ}F_{t} ∈ A_{K}.

3.2. Remark. Rather than working over K we could instead work over a ring R′ in which the elements { c_{s}(i) − c_{t}(i) | s ≠ t ∈ T (Λ) and 1 ≤ i ≤ M } are invertible. All of the results below, except those concerned with the irreducibe A_{K}–modules or with the semisimplicity of A_{K}, are valid over R′. However, there seems to be no real advantage to working over R′ in this section. In section 4 we work over a similar ring when studying the nonseparated case.

We extend the dominance order ▷ on T (Λ) to ∐_{λ∈Λ} T (λ) × T (λ) by declaring that (s, t) ▷ (u, v) if s ⊵ u, t ⊵ v and (s, t) ≠ (u, v).

We now begin to apply our definitions. The first step is easy.

3.3. Lemma. Assume that A has a family of JM–elements which separate T (Λ). a) Suppose that s, t ∈ T (λ). Then there exist scalars b_{uv} ∈ K such that

f_{st}^{λ} = a_{st}^{λ} + ∑_{\begin{matrix}u,v∈T (μ),μ∈Λ \\ (u,v)▷(s,t) \end{matrix}} b_{uv}a_{uv}^{μ}.

b) { f_{st}^{λ} | s, t ∈ T (λ) for someλ ∈ Λ } is a basis of A_{K}.

c) Suppose that s, t ∈ T (λ). Then and (f_{st}^{λ})^{∗} = f_{ts}^{λ}.

Proof. By the definition of the JM–elements (2.4), for any i and any c ∈ 𝒞 (i) with c ≠ c_{t}(i) we have

a_{st}^{λ} \frac{L_{i} − c}{c_{t}(i) − c} ≡ a_{st}^{λ} + ∑_{v▷t} b_{v}a_{sv}^{λ} (mod A_{K}^{λ}).

By (2.5) this is still true if we act on a_{st}^{λ} with L_{i} from the left. These two facts imply part (a). Note that part (a) says that the transition matrix between the two bases {a_{st}^{λ}} and {f_{st}^{λ}} of A_{K} is unitriangular (when the rows and columns are suitably ordered). Hence, (b) follows. Part (c) follows because, by definition, (a_{st}^{λ})^{∗} = a_{ts}^{λ} and L_{i}^{∗} = L_{i}, so that F_{t}^{∗} = F_{t} and (f_{st}^{λ})^{∗} = F_{t}a_{ts}^{λ}F_{s} = f_{ts}^{λ}. □

Given s, t ∈ T (Λ) let δ_{st} be the Kronecker delta; that is, δ_{st} = 1 if s = t and δ_{st} = 0, otherwise.

3.4. Proposition. Suppose that s, t ∈ T (λ), for some λ ∈ Λ, that u ∈ T (Λ) and fix i with 1 ≤ i ≤ M. Then

a) f_{st}^{λ}L_{i} = c_{t}(i)f_{st}^{λ},

b) f_{st}^{λ}F_{u} = δ_{tu}f_{su}^{λ},

c) L_{i}f_{st}^{λ} = c_{s}(i)f_{st}^{λ},
d) F_{u}f_{st}^{λ} = δ_{us}f_{ut}^{λ}.

Proof. Notice that statements (a) and (c) are equivalent by applying the ∗ involution. Similarly, (b) and (d) are equivalent. Thus, it is enough to show that (a) and (b) hold. Rather than proving this directly we take a slight detour.

Let N = |T (Λ)| and fix v = v_{1} ∈ T (μ) with v ▷ t. We claim that a_{uv}^{μ}F_{t}^{N} = 0, for all u ∈ T (μ). By the separation condition (2.8), there exists an integer j_{1} with c_{t}(j_{1}) ≠ c_{v}(j_{1}). Therefore, by (2.4), a_{uv}^{μ}(L_{j_{1}} −c_{v}(j_{1})) is a linear combination of terms a_{wx}^{ν}, where x ▷ v ▷ t. However, (L_{j_{1}} − c_{v}(j_{1})) is a factor of F_{t}, so a_{uv}^{μ}F_{t} is a linear combination of terms of the form a_{wx}^{ν} where x ▷ v ▷ t. Let v_{2} ∈ T (μ_{2}) be minimal such that a_{u_{2}v_{2}}^{μ_{2}} appears with non–zero coefficient in a_{uv}^{μ}F_{t}, for some u_{2} ∈ T (μ_{2}). Then v_{2} ▷ v_{1} ▷ t, so there exists an integer j_{2} such that c_{t}(j_{2}) ≠ c_{v_{2}}(j_{2}). Consequently, (L_{j_{2}} − c_{v_{2}}(j_{2})) is a factor of F_{t}, so a_{uv}^{μ}F_{t}^{2} is a linear combination of terms of the form a_{wx}^{ν}, where x ▷ v_{2} ▷ v_{1} ▷ t. Continuing in this way proves the claim.

For any s, t ∈ T (λ) let f′_{st} = F_{s}^{N}a_{st}^{λ}F_{t}^{N}. Fix j with 1 ≤ j ≤ M. Then, because the JM–elements commute,

f′_{st}L_{j} = F_{s}^{N}a_{st}^{λ}F_{t}^{N}L_{j} = F_{s}^{N}a_{st}^{λ}L_{j}F_{t}^{N} = F_{s}^{N}(c_{t}(i)a_{st}^{λ} + x)F_{t}^{N},

where x is a linear combination of terms of the form a_{uv}^{μ} with v ▷ t and u, v ∈ T (μ) for some μ ∈ Λ. However, by the last paragraph xF_{t}^{N} = 0, so this implies that f′_{st}L_{j} = c_{t}(j)f′_{st}. Consequently, every factor of F_{t} fixes f′_{st}, so f′_{st} = f′_{st}F_{t}. Moreover, if u ≠ t then we can find j such that c_{t}(j) ≠ c_{u}(j) by the separation condition, so that f′_{st}F_{u} = 0 since (L_{j} − c_{u}(j)) is a factor of F_{u}. As F_{u}f′_{st} = (f′_{ts}F_{u})^{∗}, we have shown that

(3.5)  F_{u}f′_{st}F_{v} = δ_{us}δ_{tv}f′_{st},

for any u, v ∈ T (Λ).

We are now almost done. By the argument of Lemma 3.3(a) we know that

f′_{st} = a_{st}^{λ} + ∑_{\begin{matrix}u,v∈T (μ) \\ (u,v)▷(s,t) \end{matrix}} s_{uv}a_{uv}^{μ},

for some s_{uv} ∈ K. Inverting this equation we can write

a_{st}^{λ} = f′_{st} + ∑_{\begin{matrix}u,v∈T (μ) \\ (u,v)▷(s,t) \end{matrix}} s′_{uv}f′_{uv},

for some s′_{uv} ∈ K. Therefore,

f_{st}^{λ} = F_{s}a_{st}^{λ}F_{t} = F_{s}(f′_{st} + ∑_{\begin{matrix}u,v∈T (μ) \\ (u,v)▷(s,t) \end{matrix}} s′_{uv}f′_{uv})F_{t} = F_{s}f′_{st}F_{t} = f′_{st},

where the last two equalities follow from (3.5). That is, f_{st}^{λ} = f′_{st}. We now have that

f_{st}^{λ}L_{i} = f′_{st}L_{i} = c_{t}(i)f′_{st} = c_{t}(i)f_{st}^{λ},

proving (a). Finally, if u ∈ T (Λ) then

f_{st}^{λ}F_{u} = f′_{st}F_{u} = δ_{tu}f′_{st} = δ_{tu}f_{st}^{λ},

proving (b). (In fact, (b) also follows from (a) and the separation condition.) □

3.6. Remark. The proof of the Proposition 3.4 is the only place where we explicitly invoke the separation condition. Of course, all of the results which follow rely on this key result.

3.7. Theorem. Suppose that the JM–elements separate T (Λ) over R. Let s, t ∈ T (λ) and u, v ∈ T (μ), for some λ, μ ∈ Λ. Then there exist scalars { γ_{t} ∈ K | t ∈ T (Λ) } such that

f_{st}^{λ}f_{uv}^{μ} = \begin{cases}γ_{t}f_{sv}^{λ}, & ifλ = μ and t = u, \\ 0, & otherwise. \end{cases}

In particular, γ_{t} depends only on t ∈ T (Λ) and { f_{st}^{λ} | s, t ∈ T (λ) and λ ∈ Λ } is a cellular basis of A_{K}.

Proof. Using the definitions, f_{st}^{λ}f_{uv}^{μ} = f_{st}^{λ}F_{u}a_{uv}^{μ}F_{v}. So f_{st}^{λ}f_{uv}^{μ} ≠ 0 only if u = t by Proposition 3.4(b).

Now suppose that u = t (so that μ = λ). Using Lemma 3.3, we can write f_{st}^{λ}f_{tv}^{λ} = ∑_{w,x} r_{wx}f_{wx}^{μ}, where r_{wx} ∈ R and the sum is over pairs w, x ∈ T (μ), for some μ ∈ Λ. Hence, by parts (b) and (d) of Proposition 3.4

(3.8)  f_{st}^{λ}f_{tv}^{λ} = F_{s}f_{st}^{λ}f_{tv}^{λ}F_{v} = ∑_{\begin{matrix}μ∈Λ \\ w,x∈T (μ) \end{matrix}} r_{wx}F_{s}f_{wx}^{μ}F_{v} = r_{sv}f_{sv}^{λ}.

Thus, it remains to show that scalar r_{sv} is independent of s, v ∈ T (λ). Using Lemma 3.3 to compute directly, there exist scalars b_{wx}, c_{yz}, r_{wx} ∈ K such that

f_{st}^{λ}f_{tv}^{λ} ≡ (a_{st}^{λ} + ∑_{\begin{matrix}w,x∈T (λ) \\ (w,x)▷(s,t) \end{matrix}} b_{wx}a_{wx}^{λ})(a_{tv}^{λ} + ∑_{\begin{matrix}y,z∈T (λ) \\ (y,z)▷(t,v) \end{matrix}} c_{yz}a_{yz}^{λ}) mod A_{K}^{λ}

≡ (⟨a_{t}^{λ}, a_{t}^{λ}⟩_{λ} + ∑_{\begin{matrix}u∈T (λ) \\ u▷t \end{matrix}}(b_{su} + c_{uv})⟨a_{u}^{λ}, a_{t}^{λ}⟩_{λ} + ∑_{\begin{matrix}x,y∈T (λ) \\ x,y▷t \end{matrix}} b_{sx}c_{yv}⟨a_{x}^{λ}, a_{y}^{λ}⟩_{λ})a_{sv}^{λ}

+ ∑_{\begin{matrix}w,x∈T (λ) \\ (w,x)▷(s,v) \end{matrix}} r_{wx}a_{wx}^{λ} (mod A_{K}^{λ}).

The inner products in the last equation come from applying (2.3). (For typographical convenience we also use the fact that the form is symmetric in the sum over u.) That is, there exists a scalar γ ∈ A, which does not depend on s or on v, such that f_{st}^{λ}f_{tv}^{λ} = γa_{sv}^{λ} plus a linear combination of more dominant terms. By Lemma 3.3(b) and (3.8), the coefficient of f_{sv}^{λ} in f_{st}^{λ}f_{tv}^{λ} is equal to the coefficient of a_{sv}^{λ} in f_{st}^{λ}f_{tv}^{λ}, so this completes the proof. □

We call { f_{st}^{λ} | s, t ∈ T (λ) and λ ∈ Λ } the seminormal basis of A. This terminology is justified by Remark 3.13 below.

3.9. Corollary. Suppose that A_{K} is a cellular algebra with a family of JM–elements which separate T (Λ). Then γ_{t} ≠ 0, for all t ∈ T (Λ).

Proof. Suppose by way of contradiction that γ_{t} = 0, for some t ∈ T (λ) and λ ∈ Λ. Then, by Theorem 3.7, f_{tt}^{λ}f_{uv}^{μ} = 0 = f_{uv}^{μ}f_{tt}^{λ}, for all u, v ∈ T (μ), μ ∈ Λ. Therefore, Kf_{tt}^{λ} is a one dimensional nilpotent ideal of A_{K}, so A_{K} is not semisimple. This contradicts Corollary 2.9, so we must have γ_{t} ≠ 0 for all t ∈ T (Λ). □

Next, we use the basis {f_{st}^{λ}} to identify the cell modules of A as submodules of A.

3.10. Corollary. Suppose that λ ∈ Λ and fix s, t ∈ T (λ). Then

C(λ) ≅ f_{st}^{λ}A_{K} = Span { f_{sv}^{λ} | v ∈ T (λ) } .

Proof. As f_{uv}^{μ} = F_{u}a_{uv}^{μ}F_{v}, for u, v ∈ T (μ), the cell modules for the cellular bases {a_{uv}^{λ}} and {f_{uv}^{λ}} of A_{K} coincide. Therefore, C(λ) is isomorphic to the A_{K}–module C(λ)′ which is spanned by the elements { f_{su}^{λ} + A_{K}^{λ} | u ∈ T (λ) }.

On other hand, if u, v ∈ T (μ), for μ ∈ Λ, then f_{st}^{λ}f_{uv}^{μ} = δ_{tu}γ_{t}f_{sv}^{λ} by Theorem 3.7. Now γ_{t} ≠ 0, by Corollary 3.9, so { f_{sv}^{λ} | v ∈ T (λ) } is a basis of f_{st}^{λ}A_{K}.

Finally, by Theorem 3.7 we have that f_{st}^{λ}A_{K} ≅ C(λ)′, where the isomorphism is the linear extension of the map f_{sv}^{λ} ↦ f_{sv}^{λ} + A_{K}^{λ}, for v ∈ T (λ). Hence, C(λ) ≅ C(λ)′ ≅ f_{st}^{λ}A_{K}, as required. □

Recall that rad C(λ) is the radical of the bilinear form on C(λ) and that D(λ) = C(λ)/ rad C(λ).

Using Corollary 3.10 and Theorem 3.7, the basis {f_{st}^{λ}} gives an explicit decomposition of A_{K} into a direct sum of cell modules. Abstractly this also follows from Corollary 2.9 and the general theory of cellular algebras because a cellular algebra is semisimple if and only if C(λ) = D(λ), for all λ ∈ Λ; see [5, Theorem 3.4].

3.11. Corollary. Suppose that A_{K} is a cellular algebra with a family of JM–elements which separate T (Λ). Then C(λ) = D(λ), for all λ ∈ Λ, and

A_{K} ≅ ⨁_{λ∈Λ} C(λ)^{⊕|T (λ)|}.

Fix s ∈ T (λ) and, for notational convenience, set f_{t}^{λ} = f_{st}^{λ} so that C(λ) has basis { f_{t}^{λ} | t ∈ T (λ) } by Corollary 3.10. Note that f_{t}^{λ} = a_{t}^{λ} + ∑_{v▷t} b_{v}a_{v}^{λ}, for some b_{v} ∈ K, by Lemma 3.3(a).

For λ ∈ Λ let G(λ) = det (⟨a_{s}^{λ}, a_{t}^{λ}⟩_{λ})_{s,t∈T (λ)} be the Gram determinant of the bilinear form ⟨ , ⟩_{λ} on the cell module C(λ). Note that G(λ) is well–defined only up to multiplication by ±1 as we have not specified an ordering on the rows and columns of the Gram matrix.

3.12. Theorem. Suppose that A_{K} is a cellular algebra with a family of JM–elements which separate T (Λ). Let λ ∈ Λ and suppose that s, t ∈ T (λ). Then

⟨f_{s}^{λ}, f_{t}^{λ}⟩_{λ} = ⟨a_{s}^{λ}, f_{t}^{λ}⟩_{λ} = \begin{cases}γ_{t}, & ifs = t, \\ 0, & otherwise. \end{cases}

Consequently, G(λ) = ∏_{t∈T (λ)} γ_{t}.

Proof. By Theorem 3.7, {f_{st}^{λ}} is a cellular basis of A_{K} and, by Corollary 3.10, we may take { f_{t}^{λ} | t ∈ T (λ) } to be a basis of C(λ). By Theorem 3.7 again, f_{us}^{λ}f_{tv}^{λ} = δ_{st}γ_{t}f_{uv}^{λ}, so that ⟨f_{s}^{λ}, f_{t}^{λ}⟩_{λ} = δ_{st}γ_{t} by Corollary 3.10 and the definition of the inner product on C(λ). Using Proposition 3.4(b) and the associativity of the inner product on C(λ), we see that

⟨a_{s}^{λ}, f_{t}^{λ}⟩_{λ} = ⟨a_{s}^{λ}, f_{t}^{λ}F_{t}⟩_{λ} = ⟨a_{s}^{λ}F_{t}^{∗}, f_{t}^{λ}⟩_{λ} = ⟨a_{s}^{λ}F_{t}, f_{t}^{λ}⟩_{λ} = ⟨f_{s}^{λ}, f_{t}^{λ}⟩_{λ}.

So we have proved the first claim in the statement of the Theorem.

Finally, the transition matrix between the two bases {a_{t}^{λ}} and {f_{t}^{λ}} of C(λ) is unitriangular (when suitably ordered), so we have that

G(λ) = det (⟨a_{s}^{λ}, a_{t}^{λ}⟩_{λ}) = det (⟨f_{s}^{λ}, f_{t}^{λ}⟩_{λ}) = ∏_{t∈T (λ)} γ_{t},

as required. □

3.13. Remark. Extending the bilinear forms ⟨ , ⟩_{λ} to the whole of A_{K} (using Corollary 3.11), we see that the seminormal basis {f_{st}^{λ}} is an orthogonal basis of A_{K} with respect to this form.

In principle, we can use Theorem 3.12 to compute the Gram determinants of the cell modules of any cellular algebra A which has a separable family of JM–elements. In practice, of course, we need to find formulae for the structure constants { γ_{t} | t ∈ T (λ) } of the basis {f_{st}^{λ}}. In all known examples, explicit formulae for γ_{t} can be determined inductively once the actions of the generators of A on the seminormal basis have been determined. In turn, the action of A on its seminormal basis is determined by its action on the original cellular basis {a_{st}^{λ}}. In effect, Theorem 3.12 gives an effective recipe for computing the Gram determinants of the cell modules of A.

By definition the scalars γ_{t} are elements of the field K, for t ∈ T (λ). Surprisingly, their product must belong to R.

3.14. Corollary. Suppose that λ ∈ Λ. Then ∏_{t∈T (λ)} γ_{t} ∈ R.

Proof. By definition, the inner products ⟨a_{s}^{λ}, a_{t}^{λ}⟩_{λ} all belong to R, so G(λ) ∈ R. The result now follows from Theorem 3.12. □

As G(λ) ≠ 0 by Theorem 3.12 and Corollary 3.9, it follows that each cell module is irreducible.

3.15. Corollary. Suppose that λ ∈ Λ. Then the cell module C(λ) = D(λ) is irreducible.

We close this section by describing the primitive idempotents in A_{K}.

3.16. Theorem. Suppose that A_{K} is a cellular algebra with a family of JM–elements which separate T (Λ). Then

a) If t ∈ T (λ) and λ ∈ Λ then F_{t} = \frac{1}{γ_{t}} f_{tt}^{λ} and F_{t} is a primitive idempotent in A_{K}.

b) If λ ∈ Λ then F_{λ} = ∑_{t∈T (λ)} F_{t} is a primitive central idempotent in A_{K}.

c) { F_{t} | t ∈ T (Λ) } and { F_{λ} | λ ∈ Λ } are complete sets of pairwise orthogonal idempotents in A_{K}; in particular,

1_{A_{K}} = ∑_{λ∈Λ} F_{λ} = ∑_{t∈T (Λ)} F_{t}.

Proof. By Corollary 3.9, γ_{t} ≠ 0 for all t ∈ T (λ), so the statement of the Theorem makes sense. Furthermore, \frac{1}{γ_{t}} f_{tt}^{λ} is an idempotent by Theorem 3.7. By Corollary 3.15 the cell module C(λ) is irreducible and by Corollary 3.10, C(λ) ≅ f_{tt}^{λ}A_{K} = F_{t}A_{K}. Hence, F_{t} is a primitive idempotent.

To complete the proof of (a) we still need to show that F_{t} = \frac{1}{γ_{t}} f_{tt}^{λ}. By Theorem 3.7 we can write F_{t} = ∑_{ν∈Λ} ∑_{x,y∈T (ν)} r_{xy}f_{xy}^{ν}, for some r_{xy} ∈ K. Suppose that u, v ∈ T (μ), for some μ ∈ Λ. Then, by Proposition 3.4 and Theorem 3.7,

δ_{vt}f_{uv}^{μ} = f_{uv}^{μ}F_{t} = ∑_{ν∈Λ} ∑_{x,y∈T (μ)} r_{xy}f_{uv}^{μ}f_{xy}^{ν} = ∑_{y∈T (μ)} r_{vy}γ_{v}f_{uy}^{μ}.

By Corollary 3.9 γ_{v} ≠ 0, so comparing both sides of this equation shows that

r_{vy} = \begin{cases}\frac{1}{γ_{t}} , & if v = t = y, \\ 0, & otherwise. \end{cases}

As v is arbitrary we have F_{t} = \frac{1}{γ_{t}} f_{tt}^{λ}, as claimed.

This completes the proof of (a). Parts (b) and (c) now follow from (a) and the multiplication formula in Theorem 3.7. □

3.17. Corollary. Suppose that A_{K} is a cellular algebra with a family of JM–elements which separate T (Λ). Then

L_{i} = ∑_{t∈T (Λ)} c_{t}(i)F_{t}

and ∏_{c∈𝒞 (i)} (L_{i} − c) is the minimum polynomial for L_{i} acting on A_{K}.

Proof. By part (c) of Theorem 3.16,

L_{i} = L_{i} ∑_{t∈T (Λ)} F_{t} = ∑_{t∈T (Λ)} L_{i}F_{t} = ∑_{t∈T (Λ)} c_{t}(i)F_{t},

where the last equality follows from Proposition 3.4(c).

For the second claim, observe that ∏_{c∈𝒞 (i)} (L_{i} − c) · f_{st}^{λ} = 0 by Proposition 3.4(c), for all λ ∈ Λ and all s, t ∈ T (λ). If we omit the factor (L_{i} − d), for some d ∈ 𝒞 (i), then we can find an s ∈ T (μ), for some μ, such that c_{s}(i) = d so that ∏_{c≠d}(L_{i} − c)F_{s} ≠ 0. Hence, ∏_{c∈𝒞 (i)} (L_{i} − c) is the minimum polynomial for the action of L_{i} on A_{K}. □

The examples at the end of section 2 show that the number of JM–elements is not uniquely determined. Nonetheless, we are able to characterize the subalgebra of A_{K} which they generate.

3.18. Corollary. Suppose that A_{K} is a cellular algebra with a family of JM–elements which separate T (Λ). Then {L_{1}, …, L_{M}} generate a maximal abelian subalgebra of A_{K}.

Proof. As the JM–elements commute, by definition, the subalgebra ℒ_{K} of A_{K} which they generate is certainly abelian. By Theorem 3.16 and Corollary 3.17, ℒ_{K} is the subalgebra of A spanned by the primitive idempotents { F_{t} | t ∈ T (Λ) }. As the primitive idempotents of A_{K} span a maximal abelian subalgebra of A_{K}, we are done. □

4. The non–separated case

Up until now we have considered those cellular algebras A_{K} which have a family of JM–elements which separate T (Λ). By Corollary 2.9 the separation condition forces A_{K} to be semisimple. In this section we still assume that A = A_{R} has a family of JM–elements which separate T (Λ) over R but rather than studying the semisimple algebra A_{K} we extend the previous constructions to non–separated algebras over a field.

In this section let R be a discrete valuation ring with maximal ideal π. We assume that A_{R} has a family of JM–elements which separate T (Λ) over R. In addition, we assume that the elements { c − c′ | c ≠ c′ ∈ 𝒞 (i) for 1 ≤ i ≤ M } are invertible in R.

Let K be the field of fractions of R. Then A_{K} is semisimple by Corollary 2.9 and all of the results of the previous section apply to A_{K}. Let k = R/π be the residue field of K. Then A_{k} = A ⊗_{R} k is a cellular algebra with cellular basis given by the image of the cellular basis of A in A_{k}. We abuse notation and write {a_{st}^{λ}} for the cellular basis of all three algebras A = A_{R}, A_{K} and A_{k}, it will always be clear from the context which algebra these elements belong to.

Note that, in general, the JM–elements do not separate T (Λ) over k, so the arguments of the previous section do not necessarily apply to the algebra A_{k}.

If r ∈ R let \overline{r} = r + π be its image in k = R/π. More generally, if a = ∑ r_{st}f_{st}^{λ} ∈ A_{R} then we set \overline{a} = ∑ \overline{r_{st}}f_{st}^{λ} ∈ A_{k}.

If 1 ≤ i ≤ M and t ∈ T (λ) define the residue of i at t to be r_{t}(i) = \overline{c_{t}(i)}. Similarly, set r_{λ}(i) = \overline{c_{λ}(i)}, for 1 ≤ i ≤ M, and let ℛ_{λ} = \{ r_{λ}(i) | 1 ≤ i ≤ M \}.

By (2.4) the action of the JM–elements on A_{k} is given by

a_{st}^{λ}L_{i} ≡ r_{t}(i)a_{st}^{λ} + ∑_{v▷t} r_{tv}a_{sv}^{λ} (mod A_{k}^{λ}),

where r_{tv} ∈ k (and otherwise the notation is as in (2.4)). There is an analogous formula for the action of L_{i} on a_{st}^{λ} from the left.

We use residues modulo π to define equivalence relations on T (Λ) and on Λ.

4.1. Definition (Residue classes and linkage classes).

a) Suppose that s, t ∈ T (Λ). Then s and t are in the same residue class, and we write s ≈ t, if r_{s}(i) = r_{t}(i), for 1 ≤ i ≤ M. b) Suppose that λ, μ ∈ Λ. Then λ and μ are residually linked, and we write λ ∼ μ, if

there exist elements λ_{0} = λ, λ_{1}, …, λ_{r} = μ and elements s_{j}, t_{j} ∈ T (λ_{j}) such that s_{j−1} ≈ t_{j}, for i = 1, …, r.

It is easy to see that ≈ is an equivalence relation on T (Λ) and that ∼ is an equivalence relation on Λ. If s ∈ T (Λ) let 𝕋_{s} ∈ T (Λ)/ ≈ be its residue class. If 𝕋 is a residue class let 𝕋(λ) = 𝕋 ∩ T (λ), for λ ∈ Λ. By (2.6), the residue classes T (Λ)/ ≈ parameterize the irreducible ℒ_{k}–modules.

Let 𝕋 be a residue class T (Λ) and define

F_{𝕋} = ∑_{t∈𝕋} F_{t}.

By definition, F_{𝕋} is an element of A_{K}. We claim that, in fact, F_{𝕋} ∈ A_{R}.

The following argument is an adaptation of Murphy’s proof of [17, Theorem 2.1].

4.2. Lemma. Suppose that 𝕋 is a residue equivalence class in T (Λ). Then F_{𝕋} is an idempotent in A_{R}.

Proof. We first note that F_{𝕋} is an idempotent in A_{K} because it is a linear combination of orthogonal idempotents by Theorem 3.16(a). The hard part is proving that F_{𝕋} ∈ A_{R}.

Fix an element t ∈ 𝕋(μ), where μ ∈ Λ, and define

F′_{t} = ∏_{i=1}^{M} ∏_{c∈𝒞} \frac{L_{i} − c}{c_{t}(i) − c}.

By definition, F′_{t} ∈ A_{R}. Observe that the numerator of F′_{t} depends only on 𝕋. The denominator d_{t} = ∏_{i=1}^{M} ∏_{c∈𝒞} (c_{t}(i) − c) of F′_{t} depends on t. Let s ∈ T (λ). Then, by Proposition 3.4(d) and Theorem 3.16(a),

F′_{t}F_{s} = \begin{cases}\frac{d_{s}}{d_{t}} F_{s}, & if s ∈ 𝕋, \\ 0, & otherwise. \end{cases}

Consequently, F′_{t} = ∑_{λ∈Λ} ∑_{s∈𝕋(λ)} \frac{d_{s}}{d_{t}} F_{s}, by Theorem 3.16(c).

Now, if s ∈ 𝕋(λ) then d_{s} ≡ d_{t} (mod π) since s ≈ t. Therefore, 1 − \frac{d_{s}}{d_{t}} is a non–zero element of π since d_{s} ≠ d_{t} (as the JM–elements separate T (Λ) over R). Let e_{s} ∈ R be the denominator of F_{s} and choose N such that e_{s} ∈ π^{N}, for all s ∈ 𝕋. Then (1 − \frac{d_{s}}{d_{t}} )^{N} \frac{1}{e_{s}} ∈ R,

so that (1 − \frac{d_{s}}{d_{t}} )^{N}F_{s} ∈ A_{R}, for all s ∈ 𝕋. We now compute

(F_{𝕋} − F′_{t})^{N} = ( ∑_{λ∈Λ} ∑_{s∈𝕋(λ)} (1 − \frac{d_{s}}{d_{t}} )F_{s})^{N}

= ∑_{λ∈Λ} ∑_{s∈𝕋(λ)} (1 − \frac{d_{s}}{d_{t}} )^{N}F_{s},

where the last line follows because the F_{s} are pairwise orthogonal idempotents in A_{K}. Therefore, (F_{𝕋} − F′_{t})^{N} ∈ A_{R}.

To complete the proof we evaluate (F_{𝕋} − F′_{t})^{N} directly. First, by Theorem 3.16(a),

F′_{t}F_{𝕋} = ∑_{λ∈Λ} ∑_{s∈𝕋(λ)} \frac{d_{s}}{d_{t}} F_{s}F_{𝕋} = ∑_{λ∈Λ} ∑_{s∈𝕋(λ)} \frac{d_{s}}{d_{t}} F_{s} = F′_{t}.

Similarly, F_{𝕋}F′_{t} = F′_{t}. Hence, using the binomial theorem, we have

(F_{𝕋} − F′_{t})^{N} = ∑_{i=0}^{N}(−1)^{i}\binom{N}{i}(F′_{t})^{i}F_{𝕋}^{N−i}

= F_{𝕋} + ∑_{i=1}^{N}(−1)^{i}\binom{N}{i}(F′_{t})^{i}

= F_{𝕋} + (1 − F′_{t})^{N} − 1.

Hence, F_{𝕋} = (F_{𝕋} − F′_{t})^{N} − (1 − F′_{t})^{N} + 1 ∈ A_{R}, as required. □

By the Lemma, F_{𝕋} ∈ A_{R}. Therefore, we can reduce F_{𝕋} modulo π to obtain an element of A_{k}. Let G_{𝕋} = \overline{F_{𝕋}} ∈ A_{k} be the reduction of F_{𝕋} modulo π. Then G_{𝕋} is an idempotent in A_{k}.

Recall that if s ∈ T (Λ) then 𝕋_{s} is its residue class.

4.3. Definition. Let 𝕋 be a residue class of T (Λ).

a) Suppose that s, t ∈ 𝕋(λ). Define g_{st}^{λ} = G_{𝕋_{s}}a_{st}^{λ}G_{𝕋_{t}} ∈ A_{k}.

b) Suppose that Γ ∈ Λ/ ∼ is a residue linkage class in Λ. Let A_{k}^{Γ} be the subspace of A_{k} spanned by { g_{st}^{λ} | s, t ∈ T (λ) and λ ∈ Γ }.

Note that G_{𝕋}^{∗} = G_{𝕋} and that (g_{st}^{λ})^{∗} = g_{ts}^{λ}, for all s, t ∈ T (λ) and λ ∈ Λ. By Theorem 3.16, if 𝕊 and 𝕋 are residue classes in T (Λ) then G_{𝕊}G_{𝕋} = δ_{𝕊𝕋}G_{𝕋}.

4.4. Proposition. Suppose that s, t ∈ T (λ), for some λ ∈ Λ, that u ∈ T (Λ) and fix i with 1 ≤ i ≤ M. Let 𝕋 ∈ 𝕋(Λ)/ ≈. Then, in A_{k},

a) L_{i}g_{st}^{λ} = r_{s}(i)g_{st}^{λ},

b) g_{st}^{λ}L_{i} = r_{t}(i)g_{st}^{λ},

c) G_{𝕋}g_{st}^{λ} = δ_{𝕋_{s}𝕋} g_{st}^{λ},
d) g_{st}^{λ}G_{𝕋} = δ_{𝕋𝕋_{t}} g_{st}^{λ}.

We can now generalize the seminormal basis of the previous section to the algebra A_{k}.

4.5. Theorem. Suppose that A_{R} has a family of JM–elements which separate T (Λ) over R. a) { g_{st}^{λ} | s, t ∈ T (λ) and λ ∈ Λ } is a cellular basis of A_{k}.

b) Let Γ be a residue linkage class of Λ. Then A_{k}^{Γ} is a cellular algebra with cellular basis { g_{st}^{λ} | s, t ∈ T (λ) and λ ∈ Γ }.

c) The residue linkage classes decompose A_{k} into a direct sum of cellular subalgebras;

that is,

A_{k} = ⨁_{Γ∈Λ/∼} A_{k}^{Γ}.

Proof. Let Γ be a residue linkage class in Λ and suppose that λ ∈ Γ. Then, exactly as in the proof of Lemma 3.3(a), we see that if s, t ∈ T (λ) then g_{st}^{λ} = a_{st}^{λ} plus a linear combination of more dominant terms. Therefore, the elements {g_{st}^{λ}} are linearly independent because {a_{st}^{λ}} is a basis of A_{k}. Hence, {g_{st}^{λ}} is a basis of A_{k}. We prove the remaining statements in the Theorem simultaneously.

Suppose that λ, μ ∈ Λ and that s, t ∈ 𝕋(λ) and u, v ∈ 𝕋(μ). Then

g_{st}^{λ}g_{uv}^{μ} = G_{𝕋_{s}}a_{st}^{λ}G_{𝕋_{t}}G_{𝕋_{u}}a_{uv}^{μ}G_{𝕋_{v}} = \begin{cases}G_{𝕋_{s}}a_{st}^{λ}G_{𝕋_{t}}a_{uv}^{μ}G_{𝕋_{v}}, & if t ≈ u \\ 0, & otherwise. \end{cases}

Observe that t ≈ u only if λ ∼ μ. Suppose then that λ ∼ μ and let Γ be the residue linkage class in Λ which contains λ and μ. Then, because {a_{wx}^{ν}} is a cellular basis of A_{k}, we can write

a_{st}^{λ}G_{𝕋_{t}}a_{uv}^{μ} = ∑_{\begin{matrix}ν∈Λ \\ ν≥λ,ν≥μ \end{matrix}} ∑_{\begin{matrix}w,x∈T (ν) \\ w⊵s,x⊵v \end{matrix}} r_{wx}g_{wx}^{ν},

for some r_{wx} ∈ k such that if ν = λ then r_{wx} ≠ 0 only if w = s, and if ν = μ then r_{wx} ≠ 0 only if x = v. Therefore, using Proposition 4.4, we have

g_{st}^{λ}g_{uv}^{μ} = ∑_{\begin{matrix}ν∈Λ \\ ν≥λ,ν≥μ \end{matrix}} ∑_{\begin{matrix}w,x∈T (ν) \\ w⊵s,x⊵v \end{matrix}} r_{wx}G_{𝕋_{s}}g_{wx}^{ν}G_{𝕋_{v}}

= ∑_{\begin{matrix}ν∈Γ \\ ν≥λ,ν≥μ \end{matrix}} ∑_{\begin{matrix}w,x∈T (ν) \\ w⊵s,x⊵v \end{matrix}} r_{wx}g_{wx}^{ν}.

Consequently, we see that if λ ∼ μ ∈ Γ then g_{st}^{λ}g_{uv}^{μ} ∈ A_{k}^{Γ}; otherwise, g_{st}^{λ}g_{uv}^{μ} = 0. All of the statements in the Theorem now follow. □

Arguing as in the proof of Theorem 3.16(a) it follows that G_{𝕋} = ∑ r_{st}g_{st}^{λ}, where r_{st} is non–zero only if s, t ∈ 𝕋(λ) for some λ ∈ Λ.

We are not claiming in Theorem 4.5 that the subalgebras A_{k}^{Γ} of A_{k} are indecomposable. We call the indecomposable two–sided ideals of A_{k} the blocks of A_{k}. It is a general fact that each irreducible module of an algebra is a composition factor of a unique block, so the residue linkage classes induce a partition of the set of irreducible A_{k}–modules. By the general theory of cellular algebras, all of the composition factors of a cell module are contained in the same block; see [5, 3.9.8] or [14, Cor. 2.22]. Hence, we have the following.

4.6. Corollary. Suppose that A_{R} has a family of JM–elements which separate T (Λ) over R and that λ, μ ∈ Λ. Then C(λ) and C(μ) are in the same block of A_{k} only if λ ∼ μ.

Let Γ ∈ Λ/ ∼ be a residue linkage class. Then ∑_{λ∈Γ} F_{λ} ∈ A_{R} by Lemma 4.2 and Theorem 3.16(b). Set G_{Γ} = \overline{∑_{λ∈Γ} F_{λ}} ∈ A_{k}. The following result is now immediate from Theorem 4.5 and Theorem 3.16.

4.7. Corollary. Suppose that A_{R} has a family of JM–elements which separate T (Λ) over R.

a) Let Γ be a residue linkage class. Then G_{Γ} is a central idempotent in A_{k} and the identity element of the subalgebra A_{k}^{Γ}. Moreover,

A_{k}^{Γ} = G_{Γ}A_{k}G_{Γ} ≅ End_{A_{k}}(G_{Γ}A_{k}).

b) { G_{Γ} | Γ ∈ Λ/ ∼ } and { G_{𝕋} | 𝕋 ∈ T (Λ)/ ≈ } are complete sets of pairwise orthogonal idempotents of A_{k}. In particular,

1_{A_{k}} = ∑_{Γ∈Λ/∼} G_{Γ} = ∑_{𝕋∈T (Λ)/≈} G_{𝕋}.

Observe that the right ideals G_{𝕋}A_{k} are projective A_{k}–modules, for all 𝕋 ∈ T (Λ)/ ≈. Of course, these modules need not (and, in general, will not) be indecomposable.

Let ℛ(i) = \{ \overline{c} | c ∈ 𝒞 (i) \}, for 1 ≤ i ≤ M. If 𝕋 is a residue class in T (Λ) then we set r_{𝕋}(i) = r_{t}(i), for t ∈ 𝕋 and 1 ≤ i ≤ M.

4.8. Corollary. Suppose that A_{R} has a family of JM–elements which separate T (Λ) over R. Then

L_{i} = ∑_{𝕋∈T (Λ)/≈} r_{𝕋}(i)G_{𝕋}

and ∏_{r∈ℛ(i)} (L_{i} − r) is the minimum polynomial for L_{i} acting on A_{k}.

Proof. That L_{i} = ∑_{𝕋∈T (Λ)/≈} r_{𝕋}(i)G_{𝕋} follows from Corollary 4.7(b) and Proposition 4.4. For the second claim, for any s, t ∈ T (λ) we have that

∏_{r∈ℛ(i)} (L_{i} − r) · g_{st}^{λ} = 0

by Proposition 4.4, so that ∏_{r∈ℛ(i)} (L_{i} − r) = 0 in A_{k}. If we omit a factor (L_{i} − r_{0}) from this product then ∏_{r≠r_{0}}(L_{i} − r)g_{st}^{λ} ≠ 0 whenever s, t ∈ T (λ) and r_{0} = r_{s}(i). Hence, the product over ℛ(i) is the minimum polynomial of L_{i}. □

As our final general result we note that the new cellular basis of A_{k} gives us a new ‘not quite orthogonal’ basis for the cell modules of A_{k}. Given λ ∈ Λ fix s ∈ T (λ) and define g_{t}^{λ} = g_{st}^{λ} + A_{k}^{λ} for t ∈ T (λ).

4.9. Proposition. Suppose that A_{R} has a family of JM–elements which separate T (Λ) over R. Then { g_{t}^{λ} | t ∈ T (λ) } is a basis of C(λ). Moreover, if t, u ∈ T (λ) then

⟨g_{t}^{λ}, g_{u}^{λ}⟩_{λ} = \begin{cases}⟨a_{t}^{λ}, g_{u}^{λ}⟩_{λ}, & ift ≈ u, \\ 0, & ift ≉ u. \end{cases}

Proof. That { g_{t}^{λ} | t ∈ T (λ) } is a basis of C(λ) follows from Theorem 4.5 and the argument of Lemma 3.3(a). For the second claim, if t, u ∈ T (λ) then

⟨g_{t}^{λ}, g_{u}^{λ}⟩_{λ} = ⟨a_{t}^{λ}G_{𝕋_{t}}, g_{u}^{λ}⟩_{λ} = ⟨a_{t}^{λ}, g_{u}^{λ}G_{𝕋_{t}}⟩_{λ}

by the associativity of the inner product since G_{𝕋_{t}}^{∗} = G_{𝕋_{t}}. The result now follows from Proposition 4.4(d). □

In the semisimple case Theorem 3.12 reduces the Gram determinant of a cell module to diagonal form. This result reduces it to block diagonal form. Murphy has considered this block decomposition of the Gram determinant for the Hecke algebras of type A [19].

We now apply the results of this section to give a basis for the blocks of several of the algebras considered in section 2.

4.10. Theorem. Let k be a field and suppose that A_{R} is one of the following algebras: a) the group algebra R𝔖_{n} of the symmetric group;

b) the Hecke algebra ℋ_{R,q}(𝔖_{n}) of type A;

c) the Ariki–Koike algebra ℋ_{R,q,u} with q ≠ 1;

d) the degenerate Ariki–Koike algebra H_{R,u}; Then A has a family of JM–elements which separate T (Λ) over R and Theorem 4.5 gives a basis for the block decomposition of A_{k} into a direct sum of indecomposable subalgebras.

The cellular bases and the families of JM–elements for each of these algebras are given in the examples of Section 2. As k𝔖_{n} ≅ ℋ_{k,1}(𝔖_{n}), we use the Murphy basis for the symmetric group. Note that the Hecke algebras of type A should not be considered as the special case r = 1 of the Ariki–Koike algebras because the JM–elements that we use for these two algebras are different. Significantly, for the Ariki–Koike case we must assume that q ≠ 1 as the JM–elements that we use do not separate T (Λ) over R when q = 1.

Before we can begin proving this result we need to describe how to choose a modular system (R, K, k) for each of the algebras above. In all cases we start with a field k and a non–zero element q ∈ k and we let R be the localization of the Laurent polynomial ring k[t, t^{−1}] at the maximal ideal generated by (q − t). Then R is discrete valuation ring with maximal ideal π generated by the image of (q − t) in R. By construction, k ≅ R/π and t is sent to q by the natural map R ⟶ k = R/π. Let K be the field of fractions of R.

First consider the case of the Hecke algebra ℋ_{k,q}(𝔖_{n}). As we have said, this includes the symmetric group as the special case q = 1. We take A_{R} = ℋ_{R,t}(𝔖_{n}), A_{K} = ℋ_{K,t}(𝔖_{n}), and A_{k} = ℋ_{R,t}(𝔖_{n}) ⊗_{R} k. Then ℋ_{K,t}(𝔖_{n}) is semisimple and ℋ_{k,q}(𝔖_{n}) ≅ ℋ_{R,t}(𝔖_{n}) ⊗_{R} k.

Next, consider the Ariki–Koike algebra ℋ_{k,q,u} with parameters q ≠ 0, 1 and u = (u_{1}, …, u_{m}) ∈ k^{m}. Let v_{s} = u_{s} +(q −t)^{ns}, for s = 1, …, m, and set v = (v_{1}, …, v_{m}). We consider the triple of algebras A_{R} = ℋ_{R,t,v}, A_{K} = ℋ_{K,t,v} and A_{k} = ℋ_{k,q,u}. Once again, A_{K} is semisimple and A_{k} ≅ A_{R} ⊗_{R} k. The case of the degenerate Ariki–Koike algebras is similar and we leave the details to the reader.

The indexing set Λ for each of the algebras considered in Theorem 4.10 is the set of m– multipartitions of n, where we identify the set of 1–multipartitions with the set of partitions. If λ is an m–multipartition let [λ] be the diagram of λ; that is,

[λ] = { (s, i, j) | 1 ≤ s ≤ r and 1 ≤ j ≤ λ_{i}^{(s)} } .

Given a node x = (s, i, j) ∈ [λ] we define its content to be

c(x) = \begin{cases}[j − i]_{t}, & if A_{R} = ℋ_{R,t}(𝔖_{n}), \\ v_{s}t^{j−i}, & if A_{R} = ℋ_{R,t,v}, \\ v_{s} + (j − i), & if A_{R} = H_{R,v}. \end{cases}

We set 𝒞_{λ} = \{ c(x) | x ∈ [λ] \} and ℛ_{λ} = \{ \overline{c(x)} | x ∈ [λ] \}.

Unravelling the definitions, it is easy to see, for each of the algebras that we are considering, that if λ ∈ Λ and t ∈ T (λ) then 𝒞_{λ} = { c_{t}(i) | 1 ≤ i ≤ M }.

To prove Theorem 4.10 we need to show that the residue linkage classes correspond to the blocks of each of the algebras above. Hence, Theorem 4.10 is a Corollary of the following Proposition.

4.11. Proposition. Let A be one of the algebras considered in Theorem 4.10. Suppose that λ, μ ∈ Λ. The following are equivalent:

a) C(λ) and C(μ) belong to the same block of A_{k};

b) λ ∼ μ;

c) ℛ_{λ} = ℛ_{μ}.

Proof. First suppose that C(λ) and C(μ) are in the same block. Then λ ∼ μ by Corollary 4.6, so that (a) implies (b). Next, if (b) holds then, without loss of generality, there exist s ∈ T (λ) and t ∈ T (μ) with s ≈ t; however, then ℛ_{λ} = ℛ_{μ}. So, (b) implies (c). The implication ‘(c) implies (a)’ is the most difficult, however, the blocks of all of the algebras that we are considering have been classified and the result can be stated uniformly by saying that the cell modules C(λ) and C(μ) belong to the same block if and only if ℛ_{λ} = ℛ_{μ}; see [14, Cor. 3.58] for ℋ_{k,q}(𝔖_{n}), [6] for the Ariki–Koike algebras, and [12, Cor. 9.6.2] for the degenerate Ariki–Koike algebras. Therefore, (a) and (c) are equivalent. This completes the proof. □

If B is an algebra let Z(B) be its centre. It is well known for each algebra A in Theorem 4.10 the symmetric polynomials in the JM–element belongs to the centre of A. In the present context this has the following uniform explanation.

4.12. Proposition. Suppose that A has a family of JM–elements which separate T (Λ) over R and that for λ ∈ Λ there exist scalars c_{λ}(i), for 1 ≤ i ≤ M, such that

{ c_{λ}(i) | 1 ≤ i ≤ M } = { c_{t}(i) | 1 ≤ i ≤ M } ,

for any t ∈ T (λ). Then any symmetric polynomial in L_{1}, …, L_{M} belongs to the centre of A_{k}.

Proof. Suppose that X_{1}, …, X_{M} are indeterminates over R and let p(X_{1}, …, X_{M}) ∈ R[X_{1}, …, X_{M}] be a symmetric polynomial. Recall that L_{i} = ∑_{t} c_{t}(i)F_{t} in A_{K}, by Corollary 3.17. Therefore,

p(L_{1}, …, L_{M}) = ∑_{t∈T (Λ)} p(c_{t}(1), …, c_{t}(M))F_{t} = ∑_{λ∈Λ} p(c_{λ}(1), …, c_{λ}(M))F_{λ}.

The first equality follows because the F_{t} are pairwise orthogonal idempotents by Theorem 3.16. By Theorem 3.16(c) the centre of A_{K} is spanned by the elements { F_{λ} | λ ∈ Λ }, so this shows that p(L_{1}, …, L_{M}) belongs to the centre of A_{K}. However, p(L_{1}, …, L_{M}) belongs to A_{R} so, in fact, p(L_{1}, …, L_{M}) belongs to the centre of A_{R}. Now, \overline{Z(A_{R})} is contained in the centre of A_{k} and any symmetric polynomial over k can be lifted to a symmetric polynomial over R. Thus, it follows that the symmetric polynomials in the JM–elements of A_{k} are central in A_{k}. □

All of the algebras in Theorem 4.10 satisfy the conditions of the Proposition because, using the notation above, if t ∈ T (λ) then 𝒞_{λ} = { c_{t}(i) | 1 ≤ i ≤ M } for any of these algebras. Notice, however, that the (cyclotomic) Schur algebras considered in section 2 and the Brauer and BMW algebras do not satisfy the assumptions of Proposition 4.12.

Our final result gives the block decomposition of the Schur algebras. Let Λ_{m,n} be the set of m–multipartitions of n and let S_{R,t,v}(Λ_{m,n}) be the corresponding cyclotomic q–Schur algebra [3], where t and v are as above.

4.13. Corollary. Let k be a field and suppose that A is one of the following k–algebras: a) the q–Schur algebra S_{R,q}(n);

b) the cyclotomic q–Schur S_{R,t,v}(Λ_{m,n}) algebra with q ≠ 1. Then A has a family of JM–elements which separate T (Λ) over R and Theorem 4.5 gives a basis for the block decomposition of A_{k} into a direct sum of indecomposable subalgebras.

Proof. Once again it is enough to show that two cell modules C(λ) and C(μ) belong to the same block if and only if λ ∼ μ. By Schur–Weyl duality, the blocks of S_{k,q}(n) are in bijection with the blocks of ℋ_{k,q}(n) [14, 5.37–5.38] and the blocks of S_{k,q,u}(Λ_{m,n}) are in bijection with the blocks of ℋ_{k,q,u} [15, Theorem 5.5]. Hence the result follows from Proposition 4.11. □

Acknowledgements

I thank Marcos Soriano for many discussions about seminormal forms of Hecke algebras and for his detailed comments and suggestions on this paper. This paper also owes a debt to Gene Murphy as he pioneered the use of the Jucys–Murphy elements in the representation theory of symmetric groups and Hecke algebras.

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APPENDIX. CONSTRUCTING IDEMPOTENTS FROM TRIANGULAR ACTIONS

Marcos Soriano^{1}

Fachbereich Mathematik, Im Welfengarten 1, Universität Hannover, Deutschland

            E-mail address: soriano@math.uni-hannover.de

Abstract. We give a general construction of a complete set of orthogonal idempotents starting from a set of elements acting in an (upper) triangular fashion. The construction is inspired in the Jucys-Murphy elements (in their various appearances in several cellular algebras).

1. Setup and notation.

The construction of idempotents presented here is based only on matrix arithmetic. However, whenever possible, we will mention the more suggestive notation from combinatorial representation theory.

Let Λ be an R-algebra, where R is an arbitrary integral domain. The starting point is a representation ρ of Λ via matrices over R, i.e., an R-free (left) Λ-module M. Let d be the R-rank of M and set \underline{d} := \{1, …, d\}.

1.1. Remark. We do not make now any additional assumptions on R, Λ or M. We have in mind such examples as Λ being a cellular R-algebra and M a single cell (“Specht”) module M, which would give rise to “Young’s Orthogonal Form” for M, as well as the case M = Λ itself, e.g. for questions of semisimplicity.

Assume that with respect to a certain basis (of “tableaux”)

T := {t_{1}, …, t_{d}} ⊂ M

there is a finite set of elements L := {L_{1}, …, L_{n}} ⊂ Λ (the “Jucys-Murphy” elements) acting in an upper triangular way, i.e.,

ρ(L_{i}) = (\begin{matrix}r_{i}^{1} & ∗ & ⋯ & ∗ \\ 0 & r_{i}^{2} & ... & ⋮ \\ ⋮ & ... & ... & ∗ \\ 0 & ⋯ & 0 & r_{i}^{d} \end{matrix}) , ∀i ∈ \underline{n}

for certain diagonal entries \{r_{i}^{j}\}, i ∈ \underline{n}, j ∈ \underline{d} (the “residues” or “contents”). Call

(r_{1}^{j}, r_{2}^{j}, …, r_{n}^{j})

the residue sequence corresponding to the basis element t_{j}. ¿From now on, we identify L_{i} with its representing matrix, thus suppressing ρ. Note that we do not make any assumption on ⟨L⟩ being central in Λ or that L consists of pairwise commuting elements.

We finally need some notation related to matrices. We denote by \{E_{ij}\}_{i,j∈\underline{d}} the canonical matrix units basis of Mat_{d}(R), whose elements multiply according to E_{ij}E_{kl} = δ_{jk}E_{il}. The subring of Mat_{d}(R) consisting of upper triangular matrices contains a nilpotent ideal with R-basis {E_{ij}}_{1≤i<j≤d} which we denote by N. We define the support of a matrix A = (a_{ij}) ∈ Mat_{d}(R) in the obvious way,

1The author thanks Andrew Mathas for his generosity and hospitality.

supp(A) := \{(i, j) ∈ \underline{d} × \underline{d} | a_{ij} ≠ 0\} .

To any i ∈ \underline{d} we associate the following subset of \underline{d}^{2}:

𝔲_{i} := \{(k, l) ∈ \underline{d}^{2} | k ≤ i ≤ l\},

and extend this definition to any non–empty subset J ⊆ \underline{d} via 𝔲_{J} := ⋃_{i∈J} 𝔲_{i}. If J is non–empty then a matrix A has shape J if supp(A) ⊆ 𝔲_{J} and the sequence (a_{ii})_{i∈\underline{d}} of diagonal entries is the characteristic function of the subset J, i.e.,

a_{ii} = \{ \begin{matrix}1 , & if i ∈ J \\ 0 , & if i ∉ J . \end{matrix}

In particular, A ∈ ∑_{i∈J} E_{ii} + N and A is upper triangular. For example, the matrices of shape {i} have the form

( \begin{matrix}0 & ··· & 0 & ∗ & ∗ & ··· & ∗ \\ & ... & ⋮ & ⋮ & & & ⋮ \\ & & 0 & ∗ & & & ∗ \\ & & & 1 & ∗ & ··· & ∗ \\ & & & & 0 & ··· & 0 \\ & & & & & ... & ⋮ \\ & & & & & & 0 \end{matrix} ) .

2. An observation.

Let us pause to consider a single upper triangular matrix

Z = (\begin{matrix}ζ_{1} & ∗ & ⋯ & ∗ \\ 0 & ζ_{2} & ... & ⋮ \\ ⋮ & ... & ... & ∗ \\ 0 & ⋯ & 0 & ζ_{d} \end{matrix}) ∈ Mat_{d}(R) .

Note that by the Cayley-Hamilton theorem, the matrix Z satisfies the polynomial ∏_{i=1}^{d}(X− ζ_{i}). Assume that Z has shape J for some non–empty J ⊆ \underline{d} of cardinality k = |J|. Then Z satisfies the polynomial (X − 1)^{k} · X^{d−k}. What if k = 1? Then the Cayley-Hamilton equation for Z reads

0 = Z^{d−1} · (Z − 1) ⇔ Z^{d} = Z^{d−1} .

This implies (by induction) Z^{d+j} = Z^{d} for all j ≥ 1. In particular, the element F := Z^{d} is an idempotent.

Of course, this is just a special case of “lifting” idempotents, and can be extended (cf. [1], Section I.12) to the following ring theoretical version (Lemma 2.4). We introduce some notation first. Let N ≥ 2 be a natural number (corresponding to the nilpotency degree in Lemma 2.4; for N = 1 there is nothing to do). Consider the following polynomial in two (commuting) indeterminates

(X + Y)^{2N−1} = ∑_{i=0}^{2N−1} \binom{2N − 1}{i}X^{i}Y^{2N−1−i}

= ∑_{i=0}^{N−1} \binom{2N − 1}{i}X^{2N−1−i}Y^{i} + ∑_{i=0}^{N−1} \binom{2N − 1}{i}X^{i}Y^{2N−1−i}
=: ε_{N}(X, Y) + ε_{N}(Y, X)

(using the symmetry of the binomial coefficients). Note that ε_{N}(X, Y) has integer coefficients. Since N > 1,

(2.1)  ε_{N}(X, Y) ≡ X^{2N−1} mod (XY)

and ε_{N}(X, Y) ≡ 0 mod (X^{N}). This implies that

(2.2)  ε_{N}(X, Y) · ε_{N}(Y, X) ≡ 0 mod (XY)^{N} .

Specialise to a single indeterminate by setting ε_{N}(X) := ε_{N}(X, 1 − X) and observe that

(2.3)  1 = 1^{2N−1} = (X + (1 − X))^{2N−1} = ε_{N}(X) + ε_{N}(1 − X) .

Now we are ready to formulate the

2.4. Lemma. Let H be a ring and N a nilpotent two-sided ideal of nilpotency degree N. If x^{2} ≡ x mod N, then e := ε_{N}(x) is an idempotent with e ≡ x mod N.

Proof. Note that x^{2} ≡ x mod N ⇔ x − x^{2} = x(1 − x) ∈ N, implying

e = ε_{N}(x) ≡ x^{2N−1} ≡ x mod N

by equation (2.1). On the other hand, combining equations (2.2) and (2.3),

e − e^{2} = e(1 − e) = ε_{N}(x)ε_{N}(1 − x) ≡ 0 mod (x(1 − x))^{N} .

But (x(1 − x))^{N} ∈ N^{N} = 0, thus the equality e − e^{2} = 0 holds and e is indeed an idempotent. □

3. The separating condition and directedness.

We consider first a simple version of the idempotent construction that is relevant to semisimplicity questions. For i ∈ \underline{d} we denote by ˆ \underline{i}the set \underline{d} \ \{i\}. Let us assume now that for all i ∈ \underline{d} the following separating condition is satisfied:

(S) ∀j ∈ ˆ \underline{i} ∃k ∈ \underline{n} such that r_{k}^{i} − r_{k}^{j} ∈ R^{×} .

In particular, both residues are different. Of course, k = k(j) = k(j, i) may not be unique, but we assume a fixed choice made for all possible pairs of indices. Then we define (1 denotes the identity matrix)

Z_{i} := ∏_{j∈ˆ\underline{i}} \frac{L_{k} − r_{k}^{j} 1}{r_{k}^{i} − r_{k}^{j}} .

The product can be taken in any order, the essential fact being only to achieve that the matrix Z_{i} is of the form Z_{i} = E_{ii} + N_{i} for some upper triangular nilpotent matrix N_{i}. Just note that for the j-th factor F in the definition of Z_{i} we have

F_{ii} = \frac{r_{k}^{i} − r_{k}^{j}}{r_{k}^{i} − r_{k}^{j}} = 1 and F_{jj} = \frac{r_{k}^{j} − r_{k}^{j}}{r_{k}^{i} − r_{k}^{j}} = 0 .

Now, using the observation of §2, we obtain a set of idempotents E_{i} := Z_{i}^{d}. Our first assertion is

3.1. Lemma. The idempotent E_{i} has shape {i}.

Proof. Any matrix of the form (E_{ii} + N)^{d} with N ∈ N has shape {i}. To see this, use the non–commutative binomial expansion for U = (E_{ii} + N)^{d}, i.e., express U as a sum of terms X_{1} ⋯ X_{d}, where X_{j} ∈ {E_{ii}, N}. In the case when X_{1} or X_{d} equals E_{ii}, this summand has the appropriate form,

either ( \begin{matrix}0 & ··· & 0 & 0 & 0 & ··· & 0 \\ & ... & ⋮ & ⋮ & ⋮ & & ⋮ \\ & & 0 & 0 & 0 & ··· & 0 \\ & & & 0 & ∗ & ··· & ∗ \\ & & & & 0 & ··· & 0 \\ & & & & & ... & ⋮ \\ & & & & & & 0 \end{matrix} ) or ( \begin{matrix}0 & ··· & 0 & ∗ & 0 & ··· & 0 \\ & ... & ⋮ & ⋮ & ⋮ & & ⋮ \\ & & 0 & ∗ & 0 & ··· & 0 \\ & & & 0 & 0 & ··· & 0 \\ & & & & 0 & ··· & 0 \\ & & & & & ... & ⋮ \\ & & & & & & 0 \end{matrix} ) .

If all X_{j} = N, we have the (only) summand of the form N^{d} = 0 (by nilpotency), with no contribution. Similarly, if all X_{j} = E_{ii}, we obtain one summand E_{ii}. Thus we are left with the summands having X_{1} = X_{d} = N and X_{j} = E_{ii} for some 1 < j < d. But the support of any matrix in N E_{ii} N is contained in the set {(k, s) ∈ \underline{d}^{2} | k < i < s\}, as one sees by matrix unit gymnastics (running indices are underlined):

( ∑_{\underline{k}<\underline{j}} a_{kj}E_{kj}) · E_{ii} · ( ∑_{\underline{r}<\underline{s}} b_{rs}E_{rs}) = ( ∑_{\underline{k}<\underline{j}} a_{kj}E_{kj})( ∑_{i<\underline{s}} b_{is}E_{is}) = ∑_{1≤\underline{k}<i<\underline{s}≤d} a_{ki}b_{is}E_{ks} .

This finishes the proof of the lemma, as all summands add up to give U − E_{ii} nilpotent with support contained in 𝔲_{i}. □

Lemma 3.1 has an important consequence: the one-sided “directed” orthogonality of the obtained idempotents.

3.2. Definition. Let H be an arbitrary ring. Call a finite set {e_{1}, …, e_{d}} of idempotents in H directed, if e_{j}e_{i} = 0 whenever j > i.

3.3. Lemma. The set of idempotents \{E_{i}\}_{i∈\underline{d}} is directed.

Proof. Directedness is an immediate consequence of the fact that E_{i} has shape {i}. □

4. Gram-Schmidt orthogonalisation of directed systems of idempotents.

We can now proceed inductively and construct a complete set of orthogonal idempotents out of {E_{i}}. The inductive step goes as follows:

4.1. Lemma. Let H be an arbitrary ring. Assume we are given two finite sets of idempotents in H (one of them possibly empty)

𝔼 = {e_{1}, …, e_{k}} and 𝔽 = {f_{k+1}, f_{k+2}, …, f_{d}}

for some k ≥ 0 with the following properties:

a) 𝔼 consists of pairwise orthogonal idempotents ,

b) 𝔽 is directed ,

c) 𝔼 is orthogonal to 𝔽, i.e., ef = 0 = fe for e ∈ 𝔼, f ∈ 𝔽. Set F := ∑_{i=1}^{k} e_{i} + f_{k+1}. Then the sets of idempotents

𝔼̃ = {e_{1}, …, e_{k}, f_{k+1}} and 𝔽̂ = {(1 − F)f_{k+2}, …, (1 − F)f_{d}}

satisfy conditions (1)–(3).

Proof. First observe that F is an idempotent, by orthogonality. If j ≥ k + 2 we have (by the orthogonality of 𝔼 and 𝔽 and the directedness of 𝔽) that

f_{j} · F = f_{j} · (e_{1} + … + e_{k} + f_{k+1}) = ∑_{i=1}^{k} f_{j}e_{i} + f_{j}f_{k+1} = 0 + 0 = 0 .

This implies that f̂_{j} := (1 − F)f_{j} is an idempotent because

f̂_{j}^{2} = (f_{j} − F f_{j})(f_{j} − F f_{j}) = f_{j} − F f_{j} − \underbrace{f_{j}F}_{=0} f_{j} + F \underbrace{f_{j}F}_{=0} f_{j} = (1 − F)f_{j} = f̂_{j}.

Similarly, the set {f̂_{s}}_{k+2≤s≤d} is directed because for j > i > k + 1

f̂_{j} · f̂_{i} = (f_{j} − F f_{j})(f_{i} − F f_{i}) = \underbrace{f_{j}f_{i}}_{=0} − \underbrace{f_{j}F}_{=0} f_{i} − F \underbrace{f_{j}f_{i}}_{=0} +F \underbrace{f_{j}F}_{=0} f_{i} = 0 .

Since 𝔼 is orthogonal to f_{k+1}, 𝔼̃ consists obviously of pairwise orthogonal idempotents. Thus, we are left with checking orthogonality between f_{k+1} and 𝔽̂. Let j ≥ k + 2, then

f_{k+1} · f̂_{j} = f_{k+1}(1 − F)f_{j} = \underbrace{f_{k+1}(1 − f_{k+1})}_{=0} f_{j} − ∑_{i=1}^{k} \underbrace{f_{k+1}e_{i}f_{j}}_{=0} = 0 ,

as well as f̂_{j} · f_{k+1} = (1 − F)f_{j}f_{k+1} = 0 by directedness. □

Thus, keeping the notations from §1 and §3, we obtain the following

4.2. Proposition. A set L = {L_{1}, …, L_{n}} of “Jucys-Murphy operators” satisfying the separating condition (S) for all i ∈ \underline{d} gives rise to a complete set of orthogonal idempotents {e_{1}, …, e_{d}}.

Proof. Starting from 𝔼 = ∅ and 𝔽 = \{E_{i} = Z_{i}^{d}\}_{i∈\underline{d}}, we obtain — using Lemma 4.1 d times — a set {e_{1}, …, e_{d}} of orthogonal idempotents. Note that the idempotents e_{i} have again shape {i} (check this in the inductive step from Lemma 4.1 by considering the form of the matrix 1−F). Completeness of the set {e_{1}, …, e_{d}} now follows easily, since we obviously have by Lemma 3.1:

e := e_{1} + … + e_{d} = 1+N ,

for some (upper triangular) nilpotent matrix N. Thus, e − 1 is an idempotent and nilpotent matrix, implying that N = 0. □

Note that the proof gives, at the same time, a perfectly valid algorithm for constructing the complete set of orthogonal idempotents in question.

5. Linkage classes.

¿From now on, we assume that R is a local commutative ring with maximal ideal 𝔪. This includes the case of R being a field (when 𝔪 = 0).

Fix k ∈ \underline{n} and j ∈ \underline{d}. We may assume w.l.o.g. that not all residues r_{k}^{i}, i ∈ \underline{d}, are zero (replace L_{k} by 1+L_{k} if necessary ^{2}). We say that i ∈ \underline{d} is linked to j via L_{k}, if r_{k}^{i} − r_{k}^{j} ∈ 𝔪. Set

𝕃_{k}(j) := \{i ∈ \underline{d} | i is linked to j via L_{k}\} .

2Note that this does not change the property of the considered set of Jucys-Murphy operators of being central in Λ or, rather, consist of pairwise commuting elements.

Observe that j ∈ 𝕃_{k}(j) since 0 ∈ 𝔪.

5.1. Definition. The linkage class of j ∈ \underline{d} with respect to L = \{L_{1}, …, L_{n}\} is the set

𝕃(j) := ⋂_{k∈\underline{n}} 𝕃_{k}(j) .

5.2. Remark. Linkage classes with respect to L partition the set \underline{d} (of “tableaux”) into, say, l disjoint sets J_{1}, …, J_{l}. In view of the fact that R \ 𝔪 = R^{×}, the assumption of the separating condition (S) from §3 for all i ∈ \underline{d} just translates into the condition of all linkage classes being singletons.

Consider a fixed linkage class J. For all j ∈ \underline{d} \ J we assume that a fixed choice of k ∈ \underline{n} and i ∈ \underline{d} has been made such that

r_{k}^{i} − r_{k}^{j} ∈ R^{×} = R \ 𝔪 .

Then we define

Z_{J} := ∏_{j∉J} \frac{L_{k} − r_{k}^{j} 1}{r_{k}^{i} − r_{k}^{j}}

(the product can be taken in any order). Note that — by Lemma 3.1 — Z_{J}^{d} has shape J.

6. A general orthogonalisation algorithm for idempotents.

6.1. Proposition. A set L = \{L_{1}, …, L_{n}\} partitioning \underline{d} into l linkage classes gives rise to a complete set {e_{1}, …, e_{l}} of orthogonal idempotents.

Again, the proof of the proposition contains an algorithm for constructing {e_{1}, …, e_{l}}.

Proof. Let J_{1}, …, J_{l} denote the linkage classes and set U_{i} := Z_{J_{i}}^{d}, a matrix of shape J_{i}. We start the orthogonalisation procedure by setting 𝔼_{0} := ∅ and 𝔽_{l} := {ε_{d}(U_{i})}_{1≤i≤l}. Note that 𝔽_{l} consists of idempotents by Lemma 2.4. Assuming that two sets of idempotents 𝔼_{k} = {e_{1}, …, e_{k}} (pairwise orthogonal) and 𝔽_{l−k} = {f_{k+1}, …, f_{l}} with 𝔼_{k} orthogonal to 𝔽_{l−k} have been already constructed, we set 𝔼_{k+1} := 𝔼_{k} ∪{f_{k+1}} and have to modify 𝔽_{l−k} appropriately. The goal is that 𝔽_{l−k−1} consists of idempotents orthogonal to 𝔼_{k+1}.

Set F := ∑_{e∈𝔼_{k+1}} e and consider first f̃_{j} := ε_{d}((1 − F)f_{j}) for all j ≥ k + 2. Since e(1 − F) = 0 for e ∈ 𝔼_{k+1} and ε_{d}(X) ∈ X^{d} · ℤ[X, Y], 𝔼_{k+1} is left orthogonal to the idempotent f̃_{j}, j ≥ k + 2. Similarly, multiplication from the right by (1 − F) and application of the polynomial ε_{d} forces right orthogonality to hold, while keeping left orthogonality. I.e., the set 𝔽_{l−k−1} = {f̂_{k+2}, …, f̂_{d}} with f̂_{j} := ε_{d}(f̃_{j}(1 − F)) has the desired properties.

Thus, after l steps, we end up with an orthogonal set of idempotents {e_{1}, …, e_{l}}. Observe that the inductive step described above does not change the shape of the idempotents, implying that e_{i} has shape J_{i} as the original idempotent U_{i}. This fact, in addition to J_{1}, …, J_{l} partitioning \underline{d}, leads to the equation

e_{1} + … + e_{l} = 1+N

with N a nilpotent and idempotent matrix, thus implying N = 0 and the completeness of 𝔼_{l}. □

6.2. Remark. Retracing all steps in the proof of Proposition 6.1, we see that the constructed idempotents e_{i} belong to R[L_{1}, …, L_{n}], the R-subalgebra of Λ generated by L. Thus, if the elements from L do commute pairwise, this will still hold for the set of idempotents 𝔼 := 𝔼_{l}.

In particular, assuming that L is a set of central Jucys-Murphy elements for the module M = Λ, we obtain a set 𝔼 of central orthogonal idempotents. Thus, for example, the block decomposition of Λ in the case of R being a field must be a refinement of the decomposition into linkage classes induced by L.

We leave the adaptation of the presented methods to particular classes or examples for Λ, R, M and L to the reader’s needs.

References

[1] Walter Feit. The Representation Theory of Finite Groups. North-Holland, 1982