EXPLICIT IRREDUCIBLE REPRESENTATIONS OF THE IWAHORI-HECKE ALGEBRA OF TYPE F_{4}

ARUN RAM AND D. E. TAYLOR

Abstract. A general method for computing irreducible representations of Weyl groups and Iwahori-Hecke algebras was introduced by the first author in [8]. In that paper the representations of the algebras of types A_{n}, B_{n}, D_{n} and G_{2} were computed and it is the purpose of this paper to extend these computations to F_{4}. The main goal here is to compute irreducible representations of the Iwahori-Hecke algebra of type F_{4} by only using information in the character table of the Weyl group.

1. Introduction

In his thesis [6] P.N. Hoefsmit wrote down explicit irreducible representations of the Iwahori-Hecke algebras HA_{n−1}, HB_{n}, and HD_{n}, of types A_{n−1}, B_{n}, and D_{n}, respectively. Hoefsmit’s thesis was never published and H. Wenzl [9] independently discovered these representations in the type A_{n−1} case. The irreducible representations of Hoefsmit are analogues of the “seminormal” representations of the Weyl groups of types A_{n−1}, B_{n} and D_{n} which were written down by A. Young [10]. The Iwahori-Hecke algebras depend on parameters p and q and one can recover the representations of Young by setting p and q equal to 1 in Hoefsmit’s representations.

In this paper we shall extend Hoefsmit’s result and determine explicit realizations of all the irreducible representations of the Iwahori-Hecke algebra HF_{4}. The matrix entries of these representations are well defined when p = q = 1 and, when one sets p = q = 1, our representations specialize to give explicit realizations of all the irreducible representations of the Weyl group of type F_{4}. The final results are tabulated in the last section of this paper. Our numbering scheme for the irreducible characters of HF_{4} follows Geck [5].

In analogy with Hoefsmit, our representations of HF_{4} are in “seminormal” form with respect to the chain of subalgebras

HF_{4} ⊇ HB_{3} ⊇ HA_{2} ⊇ HA_{1},

which means that we choose the irreducible representations φ^{k} of HF_{4} so that, for all h ∈ HB_{3}, the matrices φ^{k}(h) are block diagonal matrices where the blocks φ^{μ}(h) are determined by the irreducible representations φ^{μ} of HB_{3}. Similarly, for all h ∈ HA_{2}, the matrices φ^{k}(h) are block diagonal where the blocks are determined by the irreducible representations φ^{λ}(h) of HA_{2}. In this way we construct the irreducible representations of HF_{4} inductively, by using the branching rules for restricting representations from HF_{4} to HB_{3} and from HB_{3} to HA_{2}. These branching rules can be calculated easily from the character tables of the corresponding Weyl groups.

1991 Mathematics Subject Classification. 20F55, 20F28.

The research of the first author was supported in part by an Australian Research Council Research Fellowship, National Science Foundation grant DMS-9622985, and the Mathematical Sciences Research Institute.

The matrix entries of our representations are rational functions in the variables p and q. These rational functions are quotients of polynomials in ℤ[p, q, p^{−1}, q^{−1}] and the denominators contain only the polynomials

[2]_{p}, [2]_{q}, [2]_{pq}, [2]_{pq^{−1}}, [2]_{p^{2}q}, [2]_{p^{2}q^{−1}}, [3]_{p}, and [3]_{q}, (1.1)

where [2]_{x} = x + x^{−1} and [3]_{x} = x^{2} + 1 + x^{−2}. This means that our representations are well defined over any field F such that p, q ∈ F and none of the polynomials in (1.1) are equal to 0.

The research in this paper was begun while the first author was visiting Sydney University on an Australian Research Council Research fellowship. Arun Ram thanks Sydney University and especially G. Lehrer for their wonderful hospitality during the year that he spent in Sydney. The first author is also grateful for support from a postdoctoral fellowship at the Mathematical Sciences Research Institute in Berkeley where the writing of this paper was completed.

2. Preliminaries

Let p and q be indeterminates. The Iwahori-Hecke algebra HF_{4} is the associative algebra with 1 over the field ℂ(p, q) generated by T_{1}, T_{2}, T_{3}, T_{4} with relations

T_{1}T_{2}T_{1} = T_{2}T_{1}T_{2}
T_{3}T_{4}T_{3} = T_{4}T_{3}T_{4}
T_{2}T_{3}T_{2}T_{3} = T_{3}T_{2}T_{3}T_{2}
T_{i}T_{j} = T_{j}T_{i},  if j ≠ i ± 1,
T_{i}^{2} = (p − p^{−1})T_{i} + 1, for i = 1, 2,
T_{i}^{2} = (q − q^{−1})T_{i} + 1, for i = 3, 4.

This is the Iwahori-Hecke algebra corresponding to the Weyl group W F_{4}. The Weyl group W F_{4} is generated by s_{1}, s_{2}, s_{3}, s_{4} which satisfy the same relations as the T_{i} except with p = q = 1. Let HA_{1}, HA_{2}, and HB_{3} be the subalgebras of HF_{4} such that

HA_{1} is generated by T_{1},
HA_{2} is generated by T_{1} and T_{2}, and
HB_{3} is generated by T_{1}, T_{2} and T_{3}.

These are the Iwahori-Hecke algebras corresponding to the Weyl groups W A_{1} =< s_{1} >, W A_{2} =< s_{1}, s_{2} > and W B_{3} =< s_{1}, s_{2}, s_{3} >, respectively.

REPRESENTATIONS OF THE IWAHORI-HECKE ALGEBRA HF_{4} 3

Our goal in this paper is to compute explicit representations of HF_{4} using only the information in the character tables of the Weyl groups W A_{1}, W A_{2}, W B_{3} and W F_{4}. We shall use the following notations.

(a) d_{λ} will denote the dimension of the irreducible representation indexed by λ;

(b) χ^{λ} will denote the character of the irreducible representation of the Weyl group W indexed by λ;

(c) Id_{λ} will denote the d_{λ} × d_{λ} identity matrix;

(d) T_{w}, w ∈ W, will denote the usual basis of the Iwahori-Hecke algebra H given by T_{w} = T_{i_{1}} ⋯ T_{i_{k}} if w = s_{i_{1}} ⋯ s_{i_{k}} is a reduced word for w.

(e) If A and B are matrices then A ⊕ B and A ⊗ B will denote the standard operations of direct sum and tensor product of matrices.

We shall need the following well known facts:

FACT 1. The irreducible representations of the Iwahori-Hecke algebra are indexed in the same way as the corresponding Weyl group. Thus,

(a) The irreducible representations of HF_{4} are indexed by k ∈ {1, 2, …, 25} (in the same manner as in [5] and in the same order as in the table on p. 412 of [2]).

(b) The irreducible representations of HB_{3} are indexed by pairs of partitions (α, β) such that |α| + |β| = 3.

(c) The irreducible representations of HA_{2} are indexed by partitions λ of 3.

(d) The irreducible representations of HA_{1} are indexed by partitions γ of 2.

FACT 2. [2] §10.11. The dimension of an irreducible Iwahori-Hecke algebra representation is the same as that of the corresponding representation of the Weyl group and the branching rules for Iwahori-Hecke algebras are the same as for the corresponding Weyl groups. Thus the branching rules for the inclusions HF_{4} ⊇ HB_{3} ⊇ HA_{2} can be calculated directly from the character tables of the corresponding Weyl groups. We have tabulated these branching rules in Tables 4.2 and 4.3.

FACT 3. [2] §10.9 and [4] (9.21). Let H be an Iwahori-Hecke algebra and let W be the corresponding Weyl group. If λ is an index for an irreducible representation of the Iwahori-Hecke algebra H then the minimal central idempotent corresponding to λ can be written in the form

z_{λ} = ∑_{w∈W} z_{w}^{λ}T_{w},

where z_{w}^{λ} ∈ ℂ(p, q) are elements which are well defined when p = q = 1. Furthermore, at p = q = 1,

z_{λ}|_{p=q=1} = \frac{χ^{λ}(1)}{|W |} ∑_{w∈W} χ^{λ}(w^{−1})w, (2.1)

where χ^{λ} is the character of the irreducible representation of W indexed by λ.

FACT 4. Let H be an Iwahori-Hecke algebra and let W be the corresponding Weyl group. Let R be the root system corresponding to W and let

r_{s} = a reflection in a short root,
r_{l} = a reflection in a long root,
N_{s} = the number of positive short roots in R, and
N_{l} = the number of positive long roots in R.

If there is only one root length then we declare all roots to be short. For each λ indexing an irreducible representation of H let χ^{λ} be the character of the corresponding irreducible representation of the Weyl group and define

c(λ) = χ^{λ}(w_{0})p^{c(λ,s)}q^{c(λ,ℓ)} (2.2)

where

c(λ, s) = \frac{N_{s}χ^{λ}(r_{s})}{χ^{λ}(1)} and c(λ, l) = \frac{N_{l}χ^{λ}(r_{l})}{χ^{λ}(1)}.

Let φ^{λ} be a realization of the irreducible representation indexed by λ and let Id_{λ} be the d_{λ} ×d_{λ} identity matrix, where d_{λ} is the dimension of φ^{λ}. Then we have the following result [7], [5], [8]:

(a) If w_{0} is central in W then φ^{λ}(T_{w_{0}}) = c(λ) Id_{λ},

(b) If w_{0} is not central in W then φ^{λ}(T_{w_{0}}^{2}) = c(λ)^{2} Id_{λ} .

3. Seminormal representations

We shall compute the irreducible representations of HF_{4} inductively: the representations of HA_{1} are one dimensional and one can immediately write them down, then we compute irreducible representations of HA_{2}, then HB_{3}, and finally HF_{4}. At each step we use the information from the previous cases since we construct the representations such that upon restriction to any of these subalgebras they are in block diagonal form with diagonal blocks determined by the previous calculations. The irreducible representations of HA_{2} are easy to derive and the irreducible representations of HB_{3} can be derived in a similar fashion to the way that we complete the calculations for HF_{4} below. Thus, in our description below we shall assume that the irreducible representations of HA_{1}, HA_{2}, and HB_{3} are already known and we shall describe how to obtain the irreducible representations of HF_{4}. The irreducible “seminormal” representations of HA_{1}, HA_{2}, and HB_{3} are tabulated in Section 4 below.

Let k be an index for an irreducible representation of HF_{4}. The branching rule

φ^{k} ↓_{HB_{3}} ≅ φ^{μ^{(1)}} ⊕ φ^{μ^{(2)}} ⊕ ⋯ ⊕ φ^{μ^{(ℓ)}}

describing the restriction of representations of HF_{4} to HB_{3} can be computed from the character table of the corresponding Weyl groups. We shall say that the irreducible representation φ^{k} of HF_{4} is in seminormal form if

φ^{k}(h) = φ^{μ^{(1)}}(h) ⊕ φ^{μ^{(2)}}(h) ⊕ ⋯ ⊕ φ^{μ^{(ℓ)}}(h), for all h ∈ HB_{3}. (3.1)

We require the two sides of (3.1) to be equal as matrices.

We shall compute irreducible representations of HF_{4} which are in seminormal form. Assuming that the irreducible representations of HB_{3} are known, the seminormal condition implies that to determine the irreducible representations of HF_{4} it is only necessary to determine the matrices φ^{k}(T_{4}) for each k.

Suppose that φ^{k} and ψ^{k} are two solutions to this problem, i.e. φ^{k} and ψ^{k} are both realizations of the irreducible representation of HF_{4} indexed by k and we have

φ^{k}(h) = ψ^{k}(h) = φ^{μ^{(1)}}(h) ⊕ φ^{μ^{(2)}}(h) ⊕ ⋯ ⊕ φ^{μ^{(ℓ)}}(h),

for all h ∈ HB_{3}. Then there is a matrix P ∈ GL(d_{k}), where d_{k} is the dimension of φ^{k}, such that P φ^{k}(h)P^{−1} = ψ^{k}(h), for all h ∈ HF_{4}. By Schur’s lemma this matrix is unique up to constant multiples. On the other hand we have

P (φ^{μ^{(1)}}(h) ⊕ φ^{μ^{(2)}}(h) ⊕ ⋯ ⊕ φ^{μ^{(ℓ)}}(h))P^{−1} = P φ^{k}(h)P^{−1} = ψ^{k}(h)
= φ^{μ^{(1)}}(h) ⊕ φ^{μ^{(2)}}(h) ⊕ ⋯ ⊕ φ^{μ^{(ℓ)}}(h),

for all h ∈ HB_{3}. By inspection of the table of branching rules from HF_{4} to HB_{3} one sees that the summands φ^{μ^{(i)}} are all distinct irreducible representations of HB_{3}. Hence, Schur’s lemma implies that

P = p_{1} Id_{μ^{(1)}} ⊕p_{2} Id_{μ^{(2)}} ⊕ ⋯ ⊕ p_{ℓ} Id_{μ^{(ℓ)}}, (3.2)

where the p_{i} are nonzero constants. Replacing P by p_{1}^{−1}P we may suppose that p_{1} = 1. Conversely, any choice of p_{i} ≠ 0, p_{1} = 1, in the equation (3.2) defines a matrix P such that P φ^{k}P^{−1} is a seminormal representation. Thus we have the following result.

Proposition 3.3. If φ^{k} is in seminormal form then the matrix φ^{k}(T_{4}) is determined up to the choice of ℓ − 1 free parameters where ℓ is the number of irreducible summands in φ^{k} on restriction to HB_{3}.

Let w_{0,1}, w_{0,2}, w_{0,3}, and w_{0,4} be the longest elements in the Weyl groups W A_{1}, W A_{2}, W B_{3}, and W F_{4}, respectively. Define elements

D_{1} = T_{w_{0,1}} = T_{1},
D_{2} = T_{w_{0,2}}^{2} = (T_{1}T_{2}T_{1})^{2},
(3.4)  D_{3} = T_{w_{0,3}} = (T_{3}T_{2}T_{1})^{3},
D_{4} = T_{w_{0,4}} = (T_{4}T_{w_{0,3}})^{3}T_{w_{0,2}}^{−2}.

in HF_{4}.

Lemma 3.5. If φ^{k} is in seminormal form then the matrices φ^{k}(D_{j}) are uniquely determined, for all 1 ≤ k ≤ 25, 1 ≤ j ≤ 4.

Proof. This follows from Fact 4.

The matrices φ^{k}(D_{j}) are tabulated in 5.4.

Let σ be a permutation matrix such that

σφ^{k}(h)σ^{−1} = φ^{(1^{3})}(h) ⊕ ⋯ ⊕ φ^{(1^{3})}(h)
⊕ φ^{(21)}(h) ⊕ ⋯ ⊕ φ^{(21)}(h)

⊕ φ^{(3)}(h) ⊕ ⋯ ⊕ φ^{(3)}(h) (3.6)

= ⨁_{λ⊢3} φ^{λ}(h)^{⊕m_{λ}}

for all h ∈ HA_{2}. The constant m_{λ} is the number of times the matrix φ^{λ}(h) appears. Since σφ^{k}(T_{4})σ^{−1} commutes with all of the matrices in (3.6), it follows from Schur’s lemma that

σφ^{k}(T_{4})σ^{−1} = (T_{(3)}^{k} ⊗ Id_{(3)}) ⊕ (T_{(21)}^{k} ⊗ Id_{(21)}) ⊕ (T_{(1^{3})}^{k} ⊗ Id_{(1^{3})})

= ⨁_{λ⊢3} T_{λ}^{k} ⊗ Id_{λ},

where, for each λ, T_{λ}^{k} is an m_{λ} × m_{λ} matrix and Id_{λ} is the d_{λ} × d_{λ} identity matrix. Note that

T_{λ}^{k} ⊗ Id_{λ} = σφ^{k}(z_{λ}T_{4})σ^{−1}, (3.7)

where z_{λ} is the minimal central idempotent in HA_{2} corresponding to λ. We can use the same method to write

σφ^{k}(D_{3})σ^{−1} = ⨁_{λ⊢3} D_{λ}^{k} ⊗ Id_{λ},

where D_{3} = T_{w_{0,3}}, as given in (3.4).

To determine the matrices φ^{k}(T_{4}) it is sufficient to determine the matrices T_{λ}^{k}. The matrices D_{λ}^{k} are completely determined by Lemma 3.5 and can easily be determined from the tables in 5.4. The relations (T_{4}D_{3})^{3} = D_{4}D_{2}^{2} and the relation T_{4}^{2} = (q − q^{−1})T_{4} + 1 imply that

(T_{λ}^{k}D_{λ}^{k})^{3} = c(k)c(λ)^{2} Id_{m_{λ}} and (T_{λ}^{k})^{2} = (q − q^{−1})T_{λ}^{k} + Id_{m_{λ}}, (3.8)

where c(k) and c(λ) are the constants given in equation (2.2).

3.1. Determining the diagonal entries of φ^{k}(T_{4}). We shall determine the diagonal entries of the matrices the matrices T_{λ}^{k} by determining the traces of the matrices

T_{λ}^{k}(D_{λ}^{k})^{−2}, T_{λ}^{k}(D_{λ}^{k})^{−1}, T_{λ}^{k}, T_{λ}^{k}D_{λ}^{k}, and T_{λ}^{k}(D_{λ}^{k})^{2}.

Proposition 3.9. Fix an index k for an irreducible representation of HF_{4} and let λ be an index for an irreducible representation of HA_{2}. Let T_{λ}^{k}, D_{λ}^{k} and z_{λ} be as above and let χ^{k} and χ^{λ} be the irreducible characters of the Weyl groups W F_{4} and W A_{2} which correspond to k and λ, respectively. Let c(k) and c(λ) be the constants defined in (2.2). Then

(a)  Tr(T_{λ}^{k}) = \frac{1}{12} ∑_{w∈W A_{2}} χ^{λ}(w^{−1}) ((q − q^{−1})χ^{k}(w) + (q + q^{−1})χ^{k}(ws_{4})) ,

(b)  Tr(T_{λ}^{k}D_{λ}^{k}) = \frac{χ^{k}(w_{0,4})c(k)^{\frac{1}{3}} c(λ)^{\frac{2}{3}}}{6} ∑_{w∈W A_{2}} χ^{λ}(w^{−1})χ^{k}(ws_{4}w_{0,3}),

(c)  Tr((T_{λ}^{k}D_{λ}^{k})^{2}) = \frac{c(k)^{\frac{2}{3}} c(λ)^{\frac{4}{3}}}{6} ∑_{w∈W A_{2}} χ^{λ}(w^{−1})χ^{k}(w(s_{4}w_{0,3})^{2}).

Proof. (a) ¿From the second equation in (3.8) we have that each eigenvalue of T_{λ}^{k} is either q or −q^{−1} and consequently Tr(T_{λ}^{k}) = t_{1}q − t_{2}q^{−1} for some positive integers t_{1} and t_{2}. These constants are determined as follows. Using (3.7) we get that

t_{1} − t_{2} = Tr(T_{λ}^{k})|_{p=q=1} = \frac{1}{χ^{λ}(1)} Tr(T_{λ}^{k} ⊗ Id_{λ})|_{p=q=1}

= \frac{1}{χ^{λ}(1)} Tr(σφ^{k}(z_{λ}T_{4})σ^{−1})|_{p=q=1} = \frac{1}{χ^{λ}(1)} Tr(φ^{k}(z_{λ}T_{4}))|_{p=q=1}.

Then we use (2.1) to obtain

t_{1} − t_{2} = \frac{1}{χ^{λ}(1)}χ^{k} (\frac{χ^{λ}(1)}{6} ∑_{w∈W A_{2}} χ^{λ}(w^{−1})ws_{4}) = \frac{1}{6} ∑_{w∈W A_{2}} χ^{λ}(w^{−1})χ^{k}(ws_{4}).

If Id_{λ}^{k} is the identity matrix of the same dimension as T_{λ}^{k} then

t_{1} + t_{2} = Tr(Id_{λ}^{k})|_{p=q=1} = \frac{1}{χ^{λ}(1)} Tr(Id_{λ}^{k} ⊗ Id_{λ})|_{p=q=1}

= \frac{1}{χ^{λ}(1)} Tr(φ^{k}(z_{λ}))|_{p=q=1} = \frac{1}{6} ∑_{w∈W A_{2}} χ^{λ}(w^{−1})χ^{k}(w).

These two equations determine t_{1} and t_{2} and thus Tr(T_{λ}^{k}) is determined.

(b) It follows from Fact 4 and the first equation in (3.8) that the eigenvalues of T_{λ}^{k}D_{λ}^{k} are all of the form ω^{i}c(λ)^{\frac{2}{3}} c(k)^{\frac{1}{3}} where ω is a primitive cube root of unity. Hence

Tr(T_{λ}^{k}D_{λ}^{k}) = ηc(λ)^{\frac{2}{3}} c(k)^{\frac{1}{3}}

for some constant η ∈ ℂ. By setting p and q equal to 1 we have

ηχ^{k}(w_{0,4}) = Tr(T_{λ}^{k}D_{λ}^{k})|_{p=q=1} = \frac{1}{χ^{λ}(1)} Tr(T_{λ}^{k}D_{λ}^{k} ⊗ Id_{λ})|_{p=q=1}

= \frac{1}{χ^{λ}(1)} Tr(φ^{k}(z_{λ}T_{4}T_{w_{0,3}}))|_{p=q=1}

as in (3.1). Using (2.1) we get

ηχ^{k}(w_{0,4}) = \frac{1}{6} ∑_{w∈W A_{2}} χ^{λ}(w^{−1})χ^{k}(ws_{4}w_{0,3}).

The proof of (c) is similar to that of (b) once one notes that Fact 4 and the first equation in (3.8) imply that the eigenvalues of the matrix (T_{λ}^{k}D_{λ}^{k})^{2} are all of the form ω^{2i}c(λ)^{\frac{4}{3}} c(k)^{\frac{2}{3}} .

Lemma 3.10. Given matrices T and D such that T^{2} = (q − q^{−1})T + Id and (T D)^{3} = c Id where c is a constant, we have

(a) Tr(T D^{−1}) = (q − q^{−1}) Tr(D^{−1}) + c^{−1} Tr((T D)^{2}).

(b) Tr(T D^{2}) = c Tr(D^{−1}) − (q − q^{−1}) Tr((T D)^{2}).

(c) Tr(T D^{−2}) = (q − q^{−1}) Tr(D^{−2}) + c^{−1}(q − q^{−1}) Tr(T D) + c^{−1} Tr(D).

Proof. (a) Writing the given equations in the form T = (q − q^{−1}) Id +T^{−1} and (T D)^{−1} = c^{−1}(T D)^{2}, we have

Tr(T D^{−1}) = (q − q^{−1}) Tr(D^{−1}) + Tr(T^{−1}D^{−1})
= (q − q^{−1}) Tr(D^{−1}) + c^{−1} Tr(T DT D).

(b) Similarly, from the fact that T^{−2} = Id −(q − q^{−1})T^{−1},

Tr(T D^{2}) = Tr(DT D) = c Tr(T^{−1}D^{−1}T^{−1}) = c Tr(T^{−2}D^{−1})
= c Tr(D^{−1}) − c(q − q^{−1}) Tr(T^{−1}D^{−1})
= c Tr(D^{−1}) − (q − q^{−1}) Tr(T DT D).
(c)  Tr(T D^{−2}) = (q − q^{−1}) Tr(D^{−2}) + Tr(T^{−1}D^{−2})
= (q − q^{−1}) Tr(D^{−2}) + Tr(D^{−1}T^{−1}D^{−1})
= (q − q^{−1}) Tr(D^{−2}) + c^{−1} Tr(T DT)
= (q − q^{−1}) Tr(D^{−2}) + c^{−1} Tr(T^{2}D)
= (q − q^{−1}) Tr(D^{−2}) + c^{−1}(q − q^{−1}) Tr(T D) + c^{−1} Tr(D).

Assume that T_{λ}^{k} has dimension at most 5 and write D_{λ}^{k} = diag(d_{1}, d_{2}, …, d_{r}). The diagonal entries of D_{λ}^{k} are determined by Proposition 3.5 and one can check directly that these diagonal entries are always all distinct. Let S be a subset of {1, 2, …, r}\{i} such that S and its complement have at most 2 elements. Then the diagonal entries of T_{λ}^{k} are given by

(3.11)  (T_{λ}^{k})_{ii} = Tr(T E_{ii}) where, for each 1 ≤ i ≤ r,
E_{ii} = (∏_{\begin{matrix}j∈S \\ j≠i \end{matrix}} \frac{D_{λ}^{k} − d_{j}}{d_{i} − d_{j}} )(∏_{\begin{matrix}j∉S \\ j≠i \end{matrix}} \frac{(D_{λ}^{k})^{−1} − d_{j}^{−1}}{d_{i}^{−1} − d_{j}^{−1}} ).

These values can be evaluated explicitly by expanding E_{ii} in terms of (D_{λ}^{k})^{j} and using Lemma 3.10 and Proposition 3.9 to evaluate the traces Tr(T_{λ}^{k}(D_{λ}^{k})^{j}).

Formula (3.11) suffices for computing the diagonal entries of the matrices T_{λ}^{k}, and thus of the matrices φ^{k}(T_{4}), for all k except k = 25. The matrix T_{(21)}^{25} has dimension 6 and formula (3.11) is not applicable. The diagonal entries of the matrix φ^{25}(T_{4}) are computed as follows. Since the matrices T_{(1^{3})}^{25} and T_{(3)}^{25} are each of dimension two we use formula (3.11) to determine their diagonal entries. By Lemma 3.10 and Proposition 3.9 we can determine the traces of the matrices

T_{(21)}^{25}(D_{(21)}^{25})^{−2}, T_{(21)}^{25}(D_{(21)}^{25})^{−1}, T_{(21)}^{25}, T_{(21)}^{25}D_{(21)}^{25}, T_{(21)}^{25}(D_{(21)}^{25})^{2},

and these traces give five linear relations that the diagonal entries of T_{(21)}^{25} must satisfy. Finally, we use the formula

0 = Tr(φ^{25}(T_{4}T_{3}T_{2}T_{1})) = ∑_{i} φ^{25}(T_{4})_{ii}φ^{25}(T_{3})_{ii}φ^{25}(T_{2})_{ii}φ^{25}(T_{1})_{ii}

to determine the diagonal entries of φ^{25}(T_{4}) completely. This last formula is a consequence of the following lemma.

Lemma 3.12. (a) Tr(φ^{25}(T_{4}T_{3}T_{2}T_{1})) = 0.

(b) The diagonal entries of the matrix φ^{k}(T_{4}T_{3}T_{2}T_{1}) satisfy

φ^{k}(T_{4}T_{3}T_{2}T_{1})_{ii} = φ^{k}(T_{4})_{ii}φ^{25}(T_{3})_{ii}φ^{k}(T_{2})_{ii}φ^{25}(T_{1})_{ii}.

Proof. (a) Since the Coxeter number for the Weyl group W F_{4} is 12 (see [1]) we have that (T_{4}T_{3}T_{2}T_{1})^{6} = T_{w_{0,4}}. Then it follows from Fact 4 that the eigenvalues of the matrix φ^{25}(T_{4}T_{3}T_{2}T_{1}) must be of the form ω^{j}c(25) where ω is a primitive 6th root of unity and c(25) is the constant given in (2.2). It follows that Tr(φ^{25}(T_{4}T_{3}T_{2}T_{1})) = ηc(25) for some constant η. Then, from the character table of the Weyl group W F_{4}, we have

η = ηc(25)|_{p=q=1} = Tr(φ^{25}(T_{4}T_{3}T_{2}T_{1}))|_{p=q=1} = χ^{25}(s_{4}s_{3}s_{2}s_{1}) = 0.

(b) Let h ∈ HB_{3}. Let z_{λ}, λ ⊢ 3, be the minimal central idempotents in HA_{2}. Since φ^{k} is in seminormal form the matrices φ^{k}(z_{λ}) are diagonal matrices with 1’s and 0’s on the diagonal, their sum is the identity matrix and they are mutually orthogonal. It follows that there is a single partition λ such that

φ^{k}(T_{4}h)_{ii} = φ^{k}(z_{λ}T_{4}h)_{ii} = (φ^{k}(z_{λ}T_{4})φ^{k}(z_{λ}h))_{ii}

It follows from the seminormal condition and the fact that the branching rules for restricting representations from HF_{4} to HB_{3} are multiplicity free that the matrix φ^{k}(z_{λ}h) is a diagonal matrix. Thus the diagonal entries of the matrix φ^{k}(T_{4}h) satisfy

φ^{k}(T_{4}h)_{ii} = φ^{k}(T_{4})_{ii}φ^{k}(h)_{ii},

for all h ∈ HB_{3}. Since the irreducible representations of HB_{3} and HA_{2} that we are using are also chosen to be in seminormal form, their representations also satisfy a similar identity. The result then follows by induction.

3.2. Computing the off-diagonal entries of φ^{k}(T_{4}).

Proposition 3.13. Let T = (t_{ij}) and D = diag(d_{1}, d_{2}, …, d_{r}) be r × r matrices such that the diagonal entries of D are distinct, none of the entries t_{ij} of T are 0, and

T^{2} = (q − q^{−1})T + Id and (T D)^{3} = c Id,

for some constant c. For distinct indices i, j, k define u_{ij} = u_{ji} = t_{ij}t_{ji} and v_{ijk} = t_{ij}t_{jk}/t_{ik}. Then

(a)  ∑_{j≠i} u_{ij} = −t_{ii}^{2} + (q − q^{−1})t_{ii} + 1,
(b)  ∑_{j≠i} u_{ij}d_{j} = −t_{ii}^{2}d_{i} + ct_{ii}d_{i}^{−2} − c(q − q^{−1})d_{i}^{−2},
(c)  ∑_{j≠i,k} v_{ijk} = −t_{ii} − t_{kk} + (q − q^{−1}), for i ≠ k,
(d)  ∑_{j≠i,k} v_{ijk}d_{j} = −t_{ii}d_{i} − t_{kk}d_{k} + cd_{i}^{−1}d_{k}^{−1}, for i ≠ k.

Proof. Equations (a) and (b) are obtained by comparing the (i, i) entries on each side of the matrix equations T^{2} = (q − q^{−1})T + q and T DT = cD^{−1}T^{−1}D^{−1}. Equations (c) and (d) are obtained by comparing the (i, k) entries.

The equations

v_{ijk} = \frac{v_{1ij}v_{1jk}v_{1ki}}{u_{ik}} , v_{1ji} = \frac{u_{ij}}{v_{1ij}} , t_{ij} = \frac{t_{1j}v_{1ij}}{t_{1i}} , (3.14)

imply that all of the values in Proposition 3.13 are determined once we know u_{ij} and v_{1ij} for i < j and t_{1i} for 1 < i ≤ r.

In view of the relations (3.8) we may apply Proposition 3.13 to the matrices T_{λ}^{k} and D_{λ}^{k}. Equations (a) and (b) of Proposition 3.13 give

4 equations in the single variable u_{12} when dim(T_{λ}^{k}) = 2,

6 equations in the 3 variables u_{ij} when dim(T_{λ}^{k}) = 3,

8 equations in the 6 variables u_{ij} when dim(T_{λ}^{k}) = 4,

12 equations in the 15 variables u_{ij} when dim(T_{λ}^{k}) = 6.

These equations are sufficient to determine the products u_{ij} = t_{ij}t_{ji} for all T_{λ}^{k} except T_{(21)}^{25} where we have dim(T_{(21)}^{25}) = 6.

If dim(T_{λ}^{k}) = 2, 3, or 4 we use the linear equations in (a) and (b) of Proposition 3.13 to solve for the u_{ij}. Then we use the equations in (3.14) to write the equations in (c) and (d) of Proposition 3.13 in terms of the variables t_{1i}, and v_{1ij}, i < j. After doing this we are able to use the subset of the resulting equations which are linear in the v_{1ij} to uniquely determine the values of the v_{1ij}, i < j. This determines the T_{λ}^{k} up to the choice of the t_{1i}. Finally, the equations resulting from the condition T_{3}T_{4}T_{3} = T_{4}T_{3}T_{4} force certain relations between the T_{λ}^{k} for fixed k and different λ. For each fixed k we picked out a few nice equations resulting from this condition to determine the T_{λ}^{k} completely for all λ. This completely determined the representations φ^{k} for all k except k = 25.

The case of φ^{25} is slightly more complex. We used the same methods as above to determine the matrices T_{λ}^{25} in terms of the variables t_{1i} for each λ except T_{(21)}^{25}. In the case of T_{(21)}^{25} we have dim(T_{(21)}^{25}) = 6 and the system of 12 equations obtained from (a) and (b) of Proposition 3.13 is a rank 11 system in the 15 unknowns u_{ij}. These linear equations can be used to write 11 of the u_{ij} variables in terms of the other 4. Next we chose the nicest equations resulting from (c) and (d) of Proposition 3.13 and the condition T_{4}T_{3}T_{4} = T_{3}T_{4}T_{3} and used Maple [3] to solve these equations. These equations are quite nontrivial and we found that we needed to choose these equations carefully in order to stay within the bounds of the capability of Maple. In this way we determined the matrices T_{λ}^{25}, for all λ, and thus determined φ^{25} completely.

12 ARUN RAM AND D. E. TAYLOR

4. Branching rules

4.1. The branching rules from HA_{2} to HA_{1}.

φ^{λ}  dim Restriction to HA_{1}
φ^{(3)}  1  φ^{(2)}
φ^{(2,1)}  2  φ^{(2)} ⊕ φ^{(1^{2})}
φ^{(1^{3})}  1  φ^{(1^{2})}

4.2. The branching rules from HB_{3} to HA_{2}.

φ^{μ}  dim Restriction to HA_{2}
φ^{(3),∅}  1  φ^{(3)}
φ^{(1^{3}),∅}  1  φ^{(1^{3})}
φ^{∅,(3)}  1  φ^{(3)}
φ^{∅,(1^{3})}  1  φ^{(1^{3})}
φ^{(21),∅}  2  φ^{(21)}
φ^{∅,(21)}  2  φ^{(21)}
φ^{(2),(1)}  3  φ^{(3)} ⊕ φ^{(21)}
φ^{(1^{2}),(1)}  3  φ^{(21)} ⊕ φ^{(1^{3})}
φ^{(1),(2)}  3  φ^{(3)} ⊕ φ^{(21)}
φ^{(1),(1^{2})}  3  φ^{(21)} ⊕ φ^{(1^{3})}

REPRESENTATIONS OF THE IWAHORI-HECKE ALGEBRA HF_{4} 13

4.3. The branching rules from HF_{4} to HB_{3}. The bands in this table separate the orbits of the group of field automorphisms ⟨ α_{p}, α_{q} ⟩, see 5.4.

φ^{k} dim  Restriction to HB_{3}
φ^{1}  1  φ^{(3),∅}
φ^{2}  1  φ^{(1^{3}),∅}
φ^{3}  1  φ^{∅,(3)}
φ^{4}  1  φ^{∅,(1^{3})}
φ^{5}  2  φ^{(21),∅}
φ^{6}  2  φ^{∅,(21)}
φ^{7}  2  φ^{(3),∅} ⊕ φ^{∅,(3)}
φ^{8}  2  φ^{(1^{3}),∅} ⊕ φ^{∅,(1^{3})}
φ^{9}  4  φ^{(21),∅} ⊕ φ^{∅,(21)}
φ^{10}  9  φ^{(3),∅} ⊕ φ^{(21),∅} ⊕ φ^{(2),(1)} ⊕ φ^{(1),(2)}
φ^{11}  9  φ^{(21),∅} ⊕ φ^{(1^{3}),∅} ⊕ φ^{(1^{2}),(1)} ⊕ φ^{(1),(1^{2})}
φ^{12}  9  φ^{(2),(1)} ⊕ φ^{(1),(2)} ⊕ φ^{∅,(3)} ⊕ φ^{∅,(21)}
φ^{13}  9  φ^{(1^{2}),(1)} ⊕ φ^{(1),(1^{2})} ⊕ φ^{∅,(21)} ⊕ φ^{∅,(1^{3})}
φ^{14}  6  φ^{(1^{2}),(1)} ⊕ φ^{(1),(2)}
φ^{15}  6  φ^{(2),(1)} ⊕ φ^{(1),(1^{2})}
φ^{16} 12  φ^{(2),(1)} ⊕ φ^{(1^{2}),(1)} ⊕ φ^{(1),(2)} ⊕ φ^{(1),(1^{2})}
φ^{17}  4  φ^{(3),∅} ⊕ φ^{(2),(1)}
φ^{18}  4  φ^{(1^{3}),∅} ⊕ φ^{(1^{2}),(1)}
φ^{19}  4  φ^{(1),(2)} ⊕ φ^{∅,(3)}
φ^{20}  4  φ^{(1),(1^{2})} ⊕ φ^{∅,(1^{3})}
φ^{21}  8  φ^{(21),∅} ⊕ φ^{(2),(1)} ⊕ φ^{(1^{2}),(1)}
φ^{22}  8  φ^{(1),(2)} ⊕ φ^{(1),(1^{2})} ⊕ φ^{∅,(21)}
φ^{23}  8  φ^{(3),∅} ⊕ φ^{(2),(1)} ⊕ φ^{(1),(2)} ⊕ φ^{∅,(3)}
φ^{24}  8  φ^{(1^{3}),∅} ⊕ φ^{(1^{2}),(1)} ⊕ φ^{(1),(1^{2})} ⊕ φ^{∅,(1^{3})}
φ^{25} 16 φ^{(21),∅} ⊕ φ^{(2),(1)} ⊕ φ^{(1^{2}),(1)} ⊕ φ^{(1),(2)} ⊕ φ^{(1),(1^{2})} ⊕ φ^{∅,(21)}

5. Seminormal Representations for HA_{1}, HA_{2}, HB_{3}, and HF_{4}

5.1. The Iwahori-Hecke algebra HA_{1}. The irreducible representations φ^{λ} of HA_{1} are indexed by the partitions λ of 2 and we have

φ^{(2)}(T_{1}) = (p) and φ^{(1^{2})}(T_{1}) = (−p^{−1}).

5.2. The Iwahori-Hecke algebra HA_{2}. The irreducible representations φ^{λ} of HA_{2} are indexed by the partitions of 3 and can be given explicitly by

φ^{(3)}(T_{1}) = (p),  φ^{(3)}(T_{2}) = (p),
φ^{(21)}(T_{1}) = diag(p, −p^{−1}),  φ^{(21)}(T_{2}) = M_{2}(p, α),
φ^{(1^{3})}(T_{1}) = (−p^{−1}),  φ^{(1^{3})}(T_{2}) = (−p^{−1}),

where

M_{2}(p, α) = −\frac{1}{[2]_{p}} (\begin{matrix}p^{−2} & α([2]_{p} − 1) \\ \frac{1}{α}([2]_{p} + 1) & −p^{2} \end{matrix})

and [2]_{p} = p + p^{−1}. The variable α is a free parameter, see Lemma 3.3.

5.3. The Iwahori-Hecke algebra HB_{3}. The irreducible representations φ^{μ} = φ^{α,β} of HB_{3}(p^{2}, q^{2}) are indexed by pairs of partitions μ = (α, β) such that |α| + |β| = 3. Let diag(A, B, …, C) denote the block diagonal matrix with the matrices A, B, …, C in order on the diagonal. Then, using the notation

[2]_{x} = x + x^{−1},  [3]_{x} = x^{2} + 1 + x^{−2},  and [0]_{x} = x − x^{−1},

irreducible seminormal representations of the Iwahori-Hecke algebra HB_{3} can be given explicitly as follows:

φ^{(3),∅}(T_{1}) = (p),  φ^{(3),∅}(T_{2}) = (p),  φ^{(3),∅}(T_{3}) = (q),
φ^{(1^{3}),∅}(T_{1}) = (−p^{−1}),  φ^{(1^{3}),∅}(T_{2}) = (−p^{−1}), φ^{(1^{3}),∅}(T_{3}) = (q),
φ^{∅,(3)}(T_{1}) = (p),  φ^{∅,(3)}(T_{2}) = (p),  φ^{∅,(3)}(T_{3}) = (−q^{−1}),
φ^{∅,(1^{3})}(T_{1}) = (−p^{−1}),  φ^{∅,(1^{3})}(T_{2}) = (−p^{−1}), φ^{∅,(1^{3})}(T_{3}) = (−q^{−1}),
φ^{(21),∅}(T_{1}) = diag(p, −p^{−1}), φ^{(21),∅}(T_{2}) = M_{2}(p, 1), φ^{(21),∅}(T_{3}) = diag(q, q),
φ^{∅,(21)}(T_{1}) = diag(p, −p^{−1}), φ^{∅,(21)}(T_{2}) = M_{2}(p, 1), φ^{∅,(21)}(T_{3}) = diag(−q^{−1}, −q^{−1}).

φ^{(2),(1)}(T_{1}) = diag(p, p, −p^{−1}),  φ^{(2),(1)}(T_{2}) = diag(p, M_{2}(p, 1)),
φ^{(1^{2}),(1)}(T_{1}) = diag(p, −p^{−1}, −p^{−1}),  φ^{(1^{2}),(1)}(T_{2}) = diag(M_{2}(p, 1), −p^{−1}),
φ^{(1),(2)}(T_{1}) = diag(p, p, −p^{−1}),  φ^{(1),(2)}(T_{2}) = diag(p, M_{2}(p, 1)),
φ^{(1),(1^{2})}(T_{1}) = diag(p, −p^{−1}, −p^{−1}),  φ^{(1),(1^{2})}(T_{2}) = diag(M_{2}(p, 1), −p^{−1}).

φ^{(2),(1)}(T_{3}) = diag(M_{(2),(1)}, q),
φ^{(1^{2}),(1)}(T_{3}) = diag(q, M_{(1^{2}),(1)}),
φ^{(1),(2)}(T_{3}) = diag(M_{(1),(2)}, −q^{−1}),
φ^{(1),(1^{2})}(T_{3}) = diag(−q^{−1}, M_{(1),(1^{2})}),

where

M_{(2),(1)} = \frac{1}{[3]_{p}} (\begin{matrix}q + p^{−2}[0]_{q} & −[2]_{p}[2]_{p/q} \\ −[2]_{p^{2}q} & −q^{−1} + p^{2}[0]_{q} \end{matrix}) ,

M_{(1^{2}),(1)} = \frac{1}{[3]_{p}} (\begin{matrix}−q^{−1} + p^{−2}[0]_{q} & −[2]_{p^{2}/q} \\ −[2]_{p}[2]_{p^{2}q} & q + p^{2}[0]_{q} \end{matrix}) ,

M_{(1),(2)} = \frac{1}{[3]_{p}} (\begin{matrix}−q^{−1} + p^{−2}[0]_{q} & [2]_{p}[2]_{pq} \\ [2]_{p^{2}/q} & q + p^{2}[0]_{q} \end{matrix}) ,

M_{(1),(1^{2})} = \frac{1}{[3]_{p}} (\begin{matrix}q + p^{−2}[0]_{q} & [2]_{p^{2}q} \\ [2]_{p}[2]_{p/q} & −q^{−1} + p^{2}[0]_{q} \end{matrix}) .

5.4. The Iwahori-Hecke algebra HF_{4}. Let α_{p} be the automorphism of ℚ(p, q) which fixes q and sends p to −p^{−1}. Similarly, let α_{q} be the automorphism which fixes p and sends q to −q^{−1}. These field automorphisms act on the entries of the matrices φ^{λ}(T_{i}) and thereby permute the representations φ^{λ}. The representation resulting from the application of a field automorphism to a representation in seminormal form may no longer be seminormal. In order to bring the representation back to seminormal form it may be necessary to conjugate by a permutation matrix π. The orbits of the irreducible representations under the action of α_{p} and α_{q} and the permutations π for conjugating back to seminormal form are given in the following table. If φ is a representation of HF_{4} and π is a permutation then we shall let π ∘ φ denote the representation determined by (π ∘ φ)(h) = πφ(h)π^{−1}, for all h ∈ HF_{4}.

φ^{k}  Orbit of ⟨ α_{p}, α_{q} ⟩
φ^{1}  φ^{2} = α_{p}φ^{1}, φ^{3} = α_{q}φ^{1}, and φ^{4} = α_{p}α_{q}φ^{1}
φ^{5}  φ^{6} = α_{q}φ^{5}
φ^{7}  φ^{8} = α_{p}φ^{7}
φ^{10}  φ^{11} = π_{11} ∘ (α_{p}φ^{10}), where π_{11} = (1, 3)(4, 6)(7, 9)
φ^{12} = π_{12} ∘ (α_{q}φ^{10}), where π_{12} = (1, 7)(2, 8)(3, 9)
φ^{13} = π_{13} ∘ (α_{p}α_{q}φ^{10}), where π_{13} = (1, 9)(2, 8)(3, 7)(4, 6)
φ^{14}  φ^{15} = π_{15} ∘ (α_{p}φ^{14}), where π_{15} = (1, 3)(4, 6)
φ^{17}  φ^{18} = π_{18} ∘ (α_{p}φ^{17}), where π_{18} = (2, 4)
φ^{19} = π_{19} ∘ (α_{q}φ^{17}), where π_{19} = (1, 4, 3, 2)
φ^{20} = π_{20} ∘ (α_{p}α_{q}φ^{17}), where π_{20} = (1, 4)(2, 3)
φ^{21}  φ^{22} = π_{22} ∘ (α_{q}φ^{21}), where π_{22} = (1, 7, 5, 3)(2, 8, 6, 4)
φ^{23}  φ^{24} = π_{24} ∘ (α_{p}φ^{23}), where π_{24} = (2, 4)(5, 7)

Let w_{0,1}, w_{0,2}, w_{0,3} and w_{0,4} be the longest elements in the Weyl groups W A_{1}, W A_{2}, W B_{3} and W F_{4}, respectively. Let

D_{1} = T_{w_{0,1}} = T_{1},
D_{2} = T_{w_{0,2}}^{2} = (T_{1}T_{2}T_{1})^{2},
D_{3} = T_{w_{0,3}} = (T_{3}T_{2}T_{1})^{3},
D_{4} = T_{w_{0,4}} = (T_{4}T_{w_{0,3}})^{3}T_{w_{0,2}}^{−2}

in HF_{4}. The following tables give the values of φ^{k}(D_{j}), for one representative from each equivalence class of representations. The rest of the matrices φ^{k}(D_{j}) are easily obtained by applying the automorphisms α_{p} and α_{q} and conjugating by a permutation π as indicated in 5.4 above.

φ^{1}(T_{1}) = (p),_{φ^{1}(T_{w_{0,2}}^{2}) = (p^{6}),}_{φ^{1}(T_{w_{0,3}}) = (p^{6}q^{3}),}_{φ^{1}(T_{w_{0,4}}) = (p^{12}q^{12}),} φ^{5}(T_{1}) = diag(p, −p^{−1}),

φ^{5}(T_{w_{0,2}}^{2}) = Id,
φ^{5}(T_{w_{0,3}}) = q^{3} Id,
φ^{5}(T_{w_{0,4}}) = q^{12} Id,

φ^{7}(T_{1}) = p Id,
φ^{7}(T_{w_{0,2}}^{2}) = p^{6} Id,
φ^{7}(T_{w_{0,3}}) = diag(p^{6}q^{3}, −p^{6}q^{−3}),
φ^{7}(T_{w_{0,4}}) = p^{12} Id,

φ^{9}(T_{1}) = diag(p, −p^{−1}, p, −p^{−1}),
φ^{9}(T_{w_{0,2}}^{2}) = Id,
φ^{9}(T_{w_{0,3}}) = diag(q^{3}, q^{3}, −q^{−3}, −q^{−3}),
φ^{9}(T_{w_{0,4}}) = Id,
φ^{10}(T_{1}) = diag(p, p, −p^{−1}, p, p, −p^{−1}, p, p, −p^{−1},),
φ^{10}(T_{w_{0,2}}^{2}) = diag(p^{6}, 1, 1, p^{6}, 1, 1, p^{6}, 1, 1),
φ^{10}(T_{w_{0,3}}) = diag(p^{6}q^{3}, q^{3}, q^{3}, −p^{2}q, −p^{2}q, −p^{2}q, p^{2}q^{−1}, p^{2}q^{−1}, p^{2}q^{−1}),
φ^{10}(T_{w_{0,4}}) = p^{4}q^{4} Id,

φ^{14}(T_{1}) = diag(p, −p^{−1}, −p^{−1}, p, p, −p^{−1}),
φ^{14}(T_{w_{0,2}}^{2}) = diag(1, 1, p^{−6}, p^{6}, 1, 1),
φ^{14}(T_{w_{0,3}}) = diag(−p^{−2}q, −p^{−2}q, −p^{−2}q, p^{2}q^{−1}, p^{2}q^{−1}, p^{2}q^{−1}),
φ^{14}(T_{w_{0,4}}) = Id,

φ^{16}(T_{1}) = diag(p, p, −p^{−1}, p, −p^{−1}, −p^{−1}, p, p, −p^{−1}, p, −p^{−1}, −p^{−1}),
φ^{16}(T_{w_{0,2}}^{2}) = diag(p^{6}, 1, 1, 1, 1, p^{−6}, p^{6}, 1, 1, 1, 1, p^{−6}),
φ^{16}(T_{w_{0,3}}) = diag(−p^{2}q, −p^{2}q, −p^{2}q, −p^{−2}q, −p^{−2}q, −p^{−2}q, p^{2}q^{−1}, p^{2}q^{−1}, p^{2}q^{−1},
p^{−2}q^{−1}, p^{−2}q^{−1}, p^{−2}q^{−1}),
φ^{16}(T_{w_{0,4}}) = Id,

φ^{17}(T_{1}) = diag(p, p, p, −p^{−1}),
φ^{17}(T_{w_{0,2}}^{2}) = diag(p^{6}, p^{6}, 1, 1),
φ^{17}(T_{w_{0,3}}) = diag(p^{6}q^{3}, −p^{2}q, −p^{2}q, −p^{2}q),
φ^{17}(T_{w_{0,4}}) = −p^{6}q^{6} Id,

φ^{21}(T_{1}) = diag(p, −p^{−1}, p, p, −p^{−1}, p, −p^{−1}, −p^{−1}),
φ^{21}(T_{w_{0,2}}^{2}) = diag(1, 1, p^{6}, 1, 1, 1, 1, p^{−6}),
φ^{21}(T_{w_{0,3}}) = diag(q^{3}, q^{3}, −p^{2}q, −p^{2}q, −p^{2}q, −p^{−2}q, −p^{−2}q, −p^{−2}q),
φ^{21}(T_{w_{0,4}}) = −q^{6} Id,

φ^{23}(T_{1}) = diag(p, p, p, −p^{−1}, p, p, −p^{−1}, p),
φ^{23}(T_{w_{0,2}}^{2}) = diag(p^{6}, p^{6}, 1, 1, p^{6}, 1, 1, p^{6}),
φ^{23}(T_{w_{0,3}}) = diag(p^{6}q^{3}, −p^{2}q, −p^{2}q, −p^{2}q, p^{2}q^{−1}, p^{2}q^{−1}, p^{2}q^{−1}, −p^{6}q^{−3}),
φ^{23}(T_{w_{0,4}}) = −p^{6} Id,

φ^{25}(T_{1}) = diag(p, −p^{−1}, p, p, −p^{−1}, p, −p^{−1}, −p^{−1}, p, p, −p^{−1}, p, −p^{−1}, −p^{−1}, p, −p^{−1}),
φ^{25}(T_{w_{0,2}}^{2}) = diag(1, 1, p^{6}, 1, 1, 1, 1, p^{−6}, p^{6}, 1, 1, 1, 1, p^{−6}, 1, 1),
φ^{25}(T_{w_{0,3}}) = diag(q^{3}, q^{3}, −p^{2}q, −p^{2}q, −p^{2}q, −p^{−2}q, −p^{−2}q, −p^{−2}q, p^{2}q^{−1}, p^{2}q^{−1}, p^{2}q^{−1},
p^{−2}q^{−1}, p^{−2}q^{−1}, p^{−2}q^{−1}, −q^{−3}, q^{−3}),
φ^{25}(T_{w_{0,4}}) = − Id .

Using the methods described in the previous sections we have produced matrices φ^{k}(T_{i}) giving the 25 irreducible representations φ^{k} of HF_{4}. The following tables give the values of φ^{k}(T_{i}), for one representative from each equivalence class of representations. The rest of the matrices φ^{k}(T_{i}) are obtained by applying the automorphisms α_{p} and α_{q} and conjugating by a permutation π as indicated in 5.4 above.

We shall use the notations

[2]_{x} = x + x^{−1},  [3]_{x} = x^{2} + 1 + x^{−2},  and [0]_{x} = x − x^{−1},

and the notation

φ^{k}(T_{i})^{[a_{1},a_{2},... ,a_{r}]}

will denote the r × r submatrix of φ^{k}(T_{i}) which is formed by the intersection of the a_{1}, …, a_{r}th rows and columns. The notation diag(A, B, …, C) will denote the block diagonal matrix with the matrices A, B, …, C in order along the diagonal. The matrix M_{2}(x, y) will be as given in 5.2, the matrices M_{α,β} are as given in 5.3 and the variables α, β, ξ, θ, and η are free parameters, see Lemma 3.3. Any entries of the matrices φ^{k}(T_{i}) which are not given explicitly below are taken to be 0.

The representations φ^{1} and φ^{5}.

φ^{1}(T_{1}) = (p),_{φ^{1}(T_{2}) = (p),}_{φ^{1}(T_{3}) = (q),}_{φ^{1}(T_{4}) = (q),} φ^{5}(T_{1}) = diag(p, −p^{−1}),

φ^{5}(T_{2}) = M_{2}(p, 1),
φ^{5}(T_{3}) = diag(q, q),
φ^{5}(T_{4}) = diag(q, q).

REPRESENTATIONS OF THE IWAHORI-HECKE ALGEBRA HF_{4} 19

The representations φ^{7} and φ^{9}.

φ^{7}(T_{1}) = diag(p, p),
φ^{7}(T_{2}) = diag(p, p),
φ^{7}(T_{3}) = diag(q, −q^{−1}),
φ^{7}(T_{4}) = M_{2}(q, α),

φ^{9}(T_{1}) = diag(p, −p^{−1}, p, −p^{−1}),
φ^{9}(T_{2}) = diag(M_{2}(p, 1), M_{2}(p, 1)),
φ^{9}(T_{3}) = diag(q, q, −q^{−1}, q^{−1}),
φ^{9}(T_{4})^{[1,3]} = φ^{9}(T_{4})^{[2,4]} = M_{2}(q, α).

The representation φ^{10}.

φ^{10}(T_{1}) = diag(p, p, −p^{−1}, p, p, −p^{−1}, p, p, −p^{−1}),
φ^{10}(T_{2}) = diag(p, M_{2}(p, 1), p, M_{2}(p, 1), p, M_{2}(p, 1)),
φ^{10}(T_{3}) = diag(q, q, q, M_{(2,1)}, q, M_{(1,2)}, q),
φ^{10}(T_{4})^{[1,4,7]} = M_{10},
φ^{10}(T_{4})^{[2,5,8]} = φ^{10}(T_{4})^{[3,6,9]} = N_{10},

where

M_{10} = \frac{1}{[2]_{q}[2]_{p^{2}q}} (\begin{matrix}p^{−2}q^{−1}[2]_{q}[0]_{q} & −[2]_{q}[2]_{p^{2}q^{2}}ξη^{−1} & −[2]_{q}[2]_{p^{2}q^{2}}ξ \\ −[2]_{p^{2}q^{−1}}ηξ^{−1} & [2]_{p^{2}q} + p^{2}q[2]_{q}[0]_{q} & −[2]_{p^{2}q^{−1}}η \\ −[2]_{p^{2}q}ξ^{−1} & −[2]_{p^{2}q}η^{−1} & q^{2}[2]_{p^{2}q} \end{matrix})

and

N_{10} = \frac{1}{[2]_{q}[2]_{pq^{−1}}} (\begin{matrix}pq^{−1}[2]_{q}[0]_{q} & −[2]_{q}[2]_{pq^{−2}}θη^{−1} & −[2]_{q}[2]_{pq^{−2}}θ \\ −[2]_{pq}ηθ^{−1} & [2]_{pq^{−1}} + p^{−1}q[2]_{q}[0]_{q} & −[2]_{pq}η \\ −[2]_{pq^{−1}}θ^{−1} & −[2]_{pq^{−1}}θ^{−1} & q^{2}[2]_{pq^{−1}} \end{matrix}) .

20 ARUN RAM AND D. E. TAYLOR

The representation φ^{14}.

φ^{14}(T_{1}) = diag(p, −p^{−1}, −p^{−1}, p, p, −p^{−1}),
φ^{14}(T_{2}) = diag(M_{2}(p, 1), −p^{−1}, p, M_{2}(p, 1)),
φ^{14}(T_{3}) = diag(q, M_{(1^{2},1)}, M_{(1,2)}, −q^{−1}),
φ^{14}(T_{4})^{[3]} = −q^{−1},
φ^{14}(T_{4})^{[4]} = q,
φ^{14}(T_{4})^{[1,5]} = M_{14},
φ^{14}(T_{4})^{[2,6]} = M_{14},

where

M_{14} = \frac{1}{[2]_{p^{2}q^{−1}}} (\begin{matrix}1 + p^{2}q^{−1}[0]_{q} & −[3]_{p}α \\ (1 − [2]_{p^{2}q^{−2}})α^{−1} & −1 + p^{−2}q[0]_{q} \end{matrix}) .

REPRESENTATIONS OF THE IWAHORI-HECKE ALGEBRA HF_{4} 21

The representation φ^{16}.

φ^{16}(T_{1}) = diag(p, p, −p^{−1}, p, −p^{−1}, −p^{−1}, p, p, −p^{−1}, p, −p^{−1}, −p^{−1}),
φ^{16}(T_{2}) = diag(p, M_{2}(p, 1), M_{2}(p, 1), −p^{−1}, p, M_{2}(p, 1), M_{2}(p, 1) − p^{−1}),
φ^{16}(T_{3}) = diag(M_{(2,1)}, q, q, M_{(1^{2},1)}, M_{(1,2)}, −q^{−1}, −q^{−1}, M_{(1,1^{2})}),
φ^{16}(T_{4})^{[1,7]} = M_{16}(ξ),
φ^{16}(T_{4})^{[6,12]} = M_{16}(η),
φ^{16}(T_{4})^{[2,4,8,10]} = φ^{16}(T_{4})^{[3,5,9,12]} = N_{16},

where

M_{16}(α) = \frac{1}{[2]_{q}} (\begin{matrix}1 + q^{−1}[0]_{q} & −3α \\ −[3]_{q^{2}}/α[3]_{q} & −1 + q[0]_{q} \end{matrix}) ,

N_{16} = \frac{1}{[2]_{p}[2]_{q}} (\begin{matrix}f_{16}(p, q) & \frac{3[2]_{pq}ξθ}{[2]_{p^{2}/q}η} & \frac{3[2]_{pq}ξ}{[2]_{p^{2}/q}} & \frac{3[2]_{pq}ξθ}{[2]_{p^{2}q}} \\ \frac{[3]_{p^{2}}[2]_{p/q}η}{[2]_{p^{2}q}ξθ} & −f_{16}(−p^{−1}, q) & \frac{[3]_{p^{2}}[2]_{p/q}η}{[2]_{p^{2}/q}θ} & \frac{3[2]_{p/q}η}{[2]_{p^{2}q}} \\ \frac{[3]_{q^{2}}[2]_{p/q}}{[3]_{q}[2]_{p^{2}q}ξ} & \frac{[3]_{q^{2}}[2]_{p/q}θ}{[3]_{q}[2]_{p^{2}/q}η} & −f_{16}(p, −q^{−1}) & −\frac{3[2]_{p/q}θ}{[2]_{p^{2}q}} \\ \frac{[3]_{p^{2}}[3]_{q^{2}}[2]_{pq}}{3[3]_{q}[2]_{p^{2}q}ξθ} & \frac{[3]_{q^{2}}[2]_{pq}}{[3]_{q}[2]_{p^{2}/q}η} & −\frac{[3]_{p^{2}}[2]_{pq}}{[2]_{p^{2}/q}θ} & f_{16}(−p^{−1}, −q^{−1}) \end{matrix}) ,

and

f_{16}(x, y) = \frac{−2x/y + xy + 1/xy − 1/xy^{3} − y/x − 1/x^{3}y^{3} + y/x^{3}}{[2]_{x^{2}y}} .

The representation φ^{17}.

φ^{17}(T_{1}) = diag(p, p, p, −p^{−1}),
φ^{17}(T_{2}) = diag(p, p, M_{2}(p, 1)),
φ^{17}(T_{3}) = diag(q, M_{(2,1)}, q),
φ^{17}(T_{4}) = diag(M_{17}, q, q),

where

M_{17} = \frac{1}{[2]_{p^{2}q}} (\begin{matrix}1 + p^{−2}q^{−1}[0]_{q} & −[3]_{p}α \\ (1 − [2]_{p^{2}q^{2}})α^{−1} & −1 + p^{2}q[0]_{q} \end{matrix}) .

22 ARUN RAM AND D. E. TAYLOR

The representation φ^{21}.

φ^{21}(T_{1}) = diag(p, −p^{−1}, p, p, −p^{−1}, p, −p^{−1}, −p^{−1}),
φ^{21}(T_{2}) = diag(M_{2}(p, 1), p, M_{2}(p, 1), M_{2}(p, 1), −p^{−1}),
φ^{21}(T_{3}) = diag(q, q, M_{(2,1)}, q, q, M_{(1^{2},1)}),
φ^{21}(T_{4})^{[3]} = φ^{21}(T_{4})^{[8]} = q,
φ^{21}(T_{4})^{[1,4,6]} = φ^{21}(T_{4})^{[2,5,7]} = M_{21},

where

M_{21} = \frac{1}{[2]_{p}[2]_{pq}[2]_{p/q}} (\begin{matrix}(q[2]_{p^{2}} + q^{−2}[0]_{q})[2]_{p} & −[2]_{q^{3}}[2]_{p}ξη^{−1} & −[2]_{q^{3}}[2]_{p}ξ \\ −[3]_{p}[2]_{pq}ηξ^{−1} & (p^{−1}q[2]_{p}[0]_{q} + 1)[2]_{pq} & −[3]_{p}[2]_{pq}η \\ −[3]_{p}[2]_{p/q}ξ^{−1} & −[3]_{p}[2]_{p/q}η^{−1} & (pq[2]_{p}[0]_{q} + 1)[2]_{p/q} \end{matrix}) .

The representation φ^{23}.

φ^{23}(T_{1}) = diag(p, p, p, −p^{−1}, p, p, −p^{−1}, p),
φ^{23}(T_{2}) = diag(p, p, M_{2}(p, 1), p, M_{2}(p, 1), p),
φ^{23}(T_{3}) = diag(q, M_{(2,1)}, q, M_{(1,2)}, −q^{−1}, −q^{−1}),
φ^{23}(T_{4})^{[3,6]} = φ^{23}(T_{4})^{[4,7]} = M_{2}(q, η/([2]_{q} − 1)θ),
φ^{23}(T_{4})^{[1,2,5,8]} = M_{23},

where

M_{23} = \frac{1}{[2]_{q}} (\begin{matrix}f_{23}(p, q) & \frac{[2]_{pq^{2}}ξ}{[2]_{p^{2}q}[2]_{pq}η} & \frac{[2]_{p/q}[2]_{pq^{2}}ξ}{[2]_{p^{2}q}[2]_{pq}^{2}θ} & \frac{[2]_{pq^{2}}ξ}{[2]_{p^{2}q}[2]_{pq}} \\ \frac{[3]_{p^{2}}[2]_{p}η}{[2]_{p^{2}q}[2]_{p/q}ξ} & g_{23}(p, q) & \frac{([2]_{q^{2}} − 1)[2]_{p}η}{[2]_{pq}[2]_{p^{2}q}θ} & \frac{−[3]_{p^{2}}[2]_{p}η}{[2]_{p/q}[2]_{p^{2}q}} \\ \frac{[3]_{q}[3]_{p^{2}}[2]_{p}θ}{[2]_{p/q}[2]_{p^{2}/q}ξ} & \frac{([2]_{q^{2}} − 1)[3]_{q}[2]_{p}θ}{[2]_{p/q}[2]_{p^{2}/q}η} & −g_{23}(p, −1/q) & \frac{[3]_{p^{2}}[2]_{p}θ}{[2]_{p/q}[2]_{p^{2}/q}} \\ \frac{[3]_{q}[2]_{p/q^{2}}}{[2]_{p/q}[2]_{p^{2}/q}ξ} & \frac{−[3]_{q}[2]_{p/q^{2}}}{[2]_{p/q}[2]_{p^{2}/q}η} & \frac{[2]_{p/q^{2}}}{[2]_{pq}[2]_{p^{2}/q}θ} & −f_{23}(p, −1/q) \end{matrix}) ,

where

f_{23}(x, y) = \frac{y^{4} − x^{4}y^{2} − x^{2}y^{2} − 1}{x^{3}y^{4}[2]_{xy}[2]_{x^{2}y}} and
g_{23}(x, y) = \frac{x^{4}y^{6} − x^{4}y^{2} + x^{2}y^{6} − x^{2}y^{4} + x^{2}y^{2} + y^{6} + y^{2} − 1}{xy^{4}[2]_{x/y}[2]_{x^{2}y}}.

REPRESENTATIONS OF THE IWAHORI-HECKE ALGEBRA HF_{4} 23

The representation φ^{25}.

φ^{25}(T_{1}) = diag(p, −p^{−1}, p, p, −p^{−1}, p, −p^{−1}, −p^{−1},
p, p, −p^{−1}, p, −p^{−1}, −p^{−1}, p, −p^{−1}),
φ^{25}(T_{2}) = diag(M_{2}(p, 1), p, M_{2}(p, 1), M_{2}(p, 1), −p^{−1},
p, M_{2}(p, 1), M_{2}(p, 1), −p^{−1}, M_{2}(p, 1)),
φ^{25}(T_{3}) = diag(q, q, M_{(2,1)}, q, q, M_{(1^{2},1)}, M_{(1,2)}, −q^{−1}, −q^{−1}, M_{(1,1^{2})}, −q^{−1}, −q^{−1}),
φ^{25}(T_{4})^{[3,9]} = M_{2}(q, α/([2]_{q} − 1)η),
φ^{25}(T_{4})^{[8,14]} = M_{2}(q, β/([2]_{q} − 1)θ),
φ^{25}(T_{4})^{[1,4,6,10,12,15]} = φ^{25}(T_{4})^{[2,5,7,11,13,16]} = M_{25},

where

M_{25} = \frac{1}{[2]_{q}} (\begin{matrix}f_{25}(p, q) & \frac{−[2]_{p^{2}/q}ξ}{[2]_{pq}[2]_{p/q}α} & \frac{−[2]_{p^{2}q}ξ}{[2]_{pq}[2]_{p/q}β} \\ \frac{−2[3]_{p}[2]_{p^{2}/q^{2}}α}{[2]_{p}[2]_{p/q}[2]_{p^{2}q}[2]_{p^{2}/q}ξ} & g_{25}(p, q) & \frac{[3]_{p}[2]_{p^{2}/q^{2}}α}{[2]_{p}[2]_{p/q}[2]_{p^{2}/q}β} \\ \frac{−2[3]_{p}[2]_{p^{2}q^{2}}β}{[2]_{p}[2]_{pq}[2]_{p^{2}q}[2]_{p^{2}/q}ξ} & \frac{[3]_{p}[2]_{p^{2}q^{2}}β}{[2]_{p}[2]_{pq}[2]_{p^{2}q}α} & g_{25}(−1/p, q) \\ \frac{−2[3]_{p}[2]_{p^{2}/q^{2}}[3]_{q}η}{[2]_{p}[2]_{pq}[2]_{p^{2}q}[2]_{p^{2}/q}ξ} & \frac{−([3]_{p} − [3]_{q} + 2)[3]_{q}η}{[2]_{p}[2]_{pq}[2]_{p^{2}q}α} & \frac{[3]_{p}[2]_{p^{2}/q^{2}}[3]_{q}η}{[2]_{p}[2]_{pq}[2]_{p^{2}/q}β} \\ \frac{−2[3]_{p}[2]_{p^{2}q^{2}}[3]_{q}θ}{[2]_{p}[2]_{p/q}[2]_{p^{2}q}[2]_{p^{2}/q}ξ} & \frac{[3]_{p}[2]_{p^{2}q^{2}}[3]_{q}θ}{[2]_{p}[2]_{p/q}[2]_{p^{2}q}α} & \frac{−([3]_{p} − [3]_{q} + 2)[3]_{q}θ}{[2]_{p}[2]_{p/q}[2]_{p^{2}/q}β} \\ \frac{[2]_{p^{2}q^{2}}[2]_{p^{2}/q^{2}}}{[2]_{pq}[2]_{p/q}[2]_{p^{2}q}[2]_{p^{2}/q}ξ} & \frac{[2]_{p^{2}q^{2}}}{[2]_{pq}[2]_{p/q}[2]_{p^{2}q}α} & \frac{[2]_{p^{2}/q^{2}}}{[2]_{pq}[2]_{p/q}[2]_{p^{2}/q}β} \end{matrix}

\begin{matrix}\frac{−[2]_{p^{2}q}ξ}{[2]_{pq}[2]_{p/q}η} & \frac{−[2]_{p^{2}/q}ξ}{[2]_{pq}[2]_{p/q}θ} & \frac{[3]_{q}[2]_{p^{2}q}[2]_{p^{2}/q}ξ}{[2]_{pq}[2]_{p/q}} \\ \frac{−([3]_{p} − [3]_{q} + 2)α}{[2]_{p}[2]_{p/q}[2]_{p^{2}/q}η} & \frac{[3]_{p}[2]_{p^{2}/q^{2}}α}{[2]_{p}[2]_{p/q}[2]_{p^{2}q}θ} & \frac{2[3]_{p}[3]_{q}α}{[2]_{p}[2]_{p/q}} \\ \frac{[3]_{p}[2]_{p^{2}q^{2}}β}{[2]_{p}[2]_{pq}[2]_{p^{2}/q}η} & \frac{−([3]_{p} − [3]_{q} + 2)β}{[2]_{p}[2]_{pq}[2]_{p^{2}q}θ} & \frac{2[3]_{p}[3]_{q}β}{[2]_{p}[2]_{pq}} \\ −g_{25}(p, −1/q) & \frac{−[3]_{p}[2]_{p^{2}/q^{2}}η}{[2]_{p}[2]_{pq}[2]_{p^{2}q}θ} & \frac{−2[3]_{p}[3]_{q}η}{[2]_{p}[2]_{pq}} \\ \frac{−[3]_{p}[2]_{p^{2}q^{2}}θ}{[2]_{p}[2]_{p/q}[2]_{p^{2}/q}η} & −g_{25}(−1/p, −1/q) & \frac{−2[3]_{p}[3]_{q}θ}{[2]_{p}[2]_{p/q}} \\ \frac{−[2]_{p^{2}q^{2}}}{[2]_{pq}[2]_{p/q}[2]_{p^{2}/q}[3]_{q}η} & \frac{−[2]_{p^{2}/q^{2}}}{[2]_{pq}[2]_{p/q}[2]_{p^{2}q}[3]_{q}θ} & −f_{25}(p, −1/q) \end{matrix} ) ,

and where

f_{25}(x, y) = −\frac{x^{4}y^{2} + x^{2} − x^{2}y^{4} + y^{2}}{x^{2}y^{4}[2]_{xy}[2]_{x/y}} , and
g_{25}(x, y) = −\frac{x^{6}y^{4} − x^{4}y^{6} − x^{4}y^{2} + x^{4} + x^{4}y^{4} + x^{2}y^{2} + x^{2} − x^{2}y^{6} + y^{2} − y^{6}}{x^{4}y^{4}[2]_{x}[2]_{x/y}[2]_{x^{2}y}} .

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[3] Bruce W. Char, Keith O. Geddes, Gaston H. Gonnet, Benton L. Leong, Michael B. Monagan, and Stephen M. Watt. Maple V Language Reference Manual. Springer-Verlag, New York, Berlin, Heidelberg, 1991.

[4] C.W. Curtis and I. Reiner. Methods of representation theory, Vol. I. With applications to finite groups and orders. Wiley, New York, 1981.

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Arun Ram, Department of Mathematics, Princeton University, Princeton, NJ 08544-1000

 E-mail address: rama@math.princeton.edu

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

 E-mail address: don@maths.su.oz.au