‘GOOD’ ELEMENTS OF FINITE COXETER GROUPS AND REPRESENTATIONS OF IWAHORI–HECKE ALGEBRAS ^{∗}

Meinolf Geck and Jean Michel

1 Introduction and main results

Let W = W (M) be a Coxeter group, that is, a group with a presentation

⟨s ∈ I | s^{2} = 1, \underbrace{sts …}_{m_{st} factors} = \underbrace{tst …}_{m_{st} factors} for s, t ∈ I, s ≠ t⟩

where I is a finite set and M = (m_{st})_{s,t∈I} is a symmetric matrix of positive integers with m_{ss} = 1 and m_{st} > 1 if s ≠ t. We shall assume throughout that M is chosen so that W is a finite group. The Artin-Tits braid group B = B(M) associated to W is the group defined by the presentation

⟨s ∈ I | \underbrace{sts …}_{m_{st} factors} = \underbrace{tst …}_{m_{st} factors} for s, t ∈ I, s ≠ t⟩.

Usually, we will use the same symbols for generators and elements of W and of B, excepted that we will use bold letters for elements of B. This is justified by the fact that there is a canonical section to the natural quotient map B → W given as follows. Let w ∈ W and choose a minimal expression w = s_{1} … s_{m}. Then it is well-known that the corresponding product w = s_{1} … s_{m} ∈ B is independent of the chosen minimal expression. Let B^{+} be the submonoid of B consisting of all words in the generators s ∈ I of B (words with no inverses); in this way, we can identify W with the subset B_{red}^{+} of B^{+} which consists of elements of the same length as their image in W.

Now let C be a conjugacy class in W, and let C_{min} be the set of elements of minimal length in C. It has been shown in [15] that the elements in C_{min} have remarkable properties with respect to conjugation in W and in the associated generic Iwahori-Hecke algebra H. (Actually these properties hold in B since the proof does not use the quadratic relations in H.) Here, we will show that there exist elements in C_{min} which, when considered as elements of B, also have remarkable properties with respect to taking powers of them. Our main application will be a general formula for the absolute values of the eigenvalues of the irreducible representations of H; this will allow us, in particular, to determine the previously unknown character table of the Iwahori-Hecke algebra of type E_{8} (the only Iwahori-Hecke algebra whose table was missing up to now, see the remarks following Proposition 1.3 below).

∗1991 Mathematics Subject Classification 20C20, 20F36

Let us now explain in more detail our main results. They are motivated by the results of Deligne [10] and Brieskorn–Saito [4] about the solution of the word problem and a normal form for elements in B^{+}. Following Brieskorn-Saito, such a normal form is given as follows. Let w ∈ B^{+}, and let J = J(w) be the set of all s ∈ I such that s divides w in B^{+} from the left. Then the least common multiple of these generators also divides w. Now this least common multiple is w_{J}, the (image in the braid monoid of the) longest element of the parabolic subgroup W_{J} of W generated by the elements J. Hence we can divide w by w_{J}, and proceed in the same way with the quotient. Thus, w can be written in the form w = w_{J_{1}} … w_{J_{r}} for suitable subsets J_{i} ⊆ I such that J_{i} = J(w_{J_{i}} … w_{J_{r}}) for i = 1, …, r.

Theorem 1.1 Let C be a conjugacy class in W, and let d be the order of the elements in C. Then there exists an element w ∈ C_{min}, a strictly decreasing sequence of subsets J_{1} ⊃ J_{2} ⊃ … ⊃ J_{r} of I, and positive integers d_{1}, d_{2}, …, d_{r} such that the relation

w^{d} = w_{J_{1}}^{d_{1}}w_{J_{2}}^{d_{2}} … w_{J_{r}}^{d_{r}} holds in B^{+}.

Note that the d_{i} are necessarily even (for the relation w^{d} = 1 to hold). Such an element w ∈ C_{min} will be called ‘good’. Moreover, if d is even then a good element w ∈ C_{min} can be chosen so that w^{d/2} is given by the analogous expression as above where each d_{i} is replaced by d_{i}/2. Such an element will then be called ‘very good’.

The point of the Theorem is the property that the subsets J_{i} form a decreasing sequence from which it follows that the elements w_{J_{i}}^{d_{i}} commute with each other (see the description of the centre of B^{+} in [10], [4]). These results can be seen as a generalization of the well-known fact that, if C is the class such that C_{min} is the set of Coxeter elements and h is the Coxeter number, then w^{h} = w_{I}^{2} for w ∈ C_{min}, and if h is even then there exists an element w ∈ C_{min} such that w^{h/2} = w_{I}. They also generalize results of [5] on the class of regular elements of order d when the length of elements of C_{min} is 2N/d (where N is the number of reflections in W). These elements are characterized by the fact that the relation in Theorem 1.1 holds with r = 1, and all elements of C_{min} are good. (However, they are not necessarily all very good when d is even.)

It is readily checked that, in order to prove Theorem 1.1, it is sufficient to consider only irreducible Coxeter groups. The proof will then be given case–by– case in §§ 2 and 3 where we use a reduction argument, as follows. Let C be a conjugacy class in W. We say that C is cuspidal if C ∩ W_{J} = ∅ for every proper subset J ⊂ I. It is then clear that Theorem 1.1 need only be proved for cuspidal classes of W. For each irreducible Coxeter group W we start with a list of conjugacy classes for which it is easy to verify that it contains all cuspidal classes of W. We then check the existence of good respectively very good elements in the classes in this list, which is all we need in order to prove Theorem 1.1.

In § 2 we consider Coxeter groups of classical type. For type A_{n} the only class in our list is the class of the Coxeter elements and Theorem 1.1 is covered by the above mentioned general result about Coxeter elements. For types B_{n} and D_{n} our list consists of those classes for which the ‘standard’ representatives of [15] don’t lie in proper parabolic subgroups, and check that these standard representatives are good respectively very good. For the exceptional and non-crystallographic types, we use once more the GAP [23] programs mentioned in [15] (see also [14]) to produce a complete list of all classes C such that C_{min} ∩W_{J} = ∅ for proper subsets J ⊂ I. An implementation in GAP of the Deligne normal form for elements in B^{+} written by the second named author is then applied to find all good respectively very good elements. These results are given in § 3 in the form of explicit tables.

We would also like to mention that, as was already the case with [15], it was the possibility of performing such computations in GAP on a large number of elements in relatively big groups like W (E_{8}) which led us to the precise statement of the Theorem 1.1.

The above strategy leaves open the question if our lists of classes consist precisely of the cuspidal ones. This is indeed so. First note that a class C is cuspidal if and only if C_{min} ∩ W_{J} = ∅ for every proper subset J ⊂ I. To see this, recall the definition of the relation ⟶ from [15]. Given w, w′ ∈ W and s ∈ I we write w ⟶^{s} w′ if w′ = sws and l(w′) ≤ l(w), and ⟶ is the transitive closure of this relation. For every w ∈ C there exists some w′ ∈ C_{min} such that w ⟶ w′, see [15], Theorem 1.1(a). On the other hand, the definition of the relation w ⟶ w′ easily implies that every generator in I appearing in a reduced expression for w′ also appears in a reduced expression for w. Hence, if C_{min} ∩ W_{J} = ∅ for all proper subsets J ⊂ I then C must be cuspidal.

In the course of the above case-by-case examination we then also verified the following result (suggested to us by G. Pfeiffer and later proved by a general argument by B. Howlett, private communication) which finally provides a simple criterion to check whether or not a class is cuspidal.

Lemma 1.2 A conjugacy class C in W is cuspidal if and only if there exists some w ∈ C_{min} such that all generators in C_{min} appear in a reduced expression for w.

Another strong property of cuspidal classes has been found by G. Pfeiffer (unpublished). He has shown that, for a cuspidal class C, any two elements in C_{min} are conjugate by cyclic shift. Here, we say that two elements w, w′ ∈ W are conjugate by cyclic shift if there exists a sequence w = w_{0}, w_{1}, …, w_{m} = w′ of elements of W and elements x_{i}, y_{i} in W such that w_{i} = x_{i}y_{i}, w_{i+1} = y_{i}x_{i}, and l(w_{i}) = l(x_{i}) + l(y_{i}) = l(w_{i+1}) for all i. We only remark that an interesting point about conjugacy by cyclic shift is that there is an obvious analogous version in B^{+} (where the length condition above is trivially satisfied). Now, if w, w′ ∈ W are conjugate by cyclic shift then the elements w, w′ are also conjugate by cyclic shift in B^{+} hence, in particular, are conjugate in B. (See [4, 10] for a general solution of the conjugacy problem in B^{+}.) These questions are thus related to the open problem of determining exactly in which way the sets {w | w ∈ C} (for C a conjugacy class in W) split up under conjugacy in B^{+}.

In §§ 4 and 5 we will apply the above results on good elements to the representation theory of the generic Iwahori-Hecke algebra H = H(M) associated with M. Let A = ZZ[v_{s}^{±1} | s ∈ I] be the ring of Laurent polynomials in indeterminates v_{s} such that v_{s} = v_{t} whenever s, t ∈ I are conjugate in W. Then H can be described as the quotient of the group algebra of the braid group B over A obtained by factoring out the relations

(s − v_{s}^{2})(s + 1) = 0 for s ∈ I.

For w ∈ B^{+}, we denote by Tw the image of w in H; we also denote it by T_{w} if w is the image in B_{red}^{+} of w ∈ W. Then the elements {T_{w} | w ∈ W } form an A-basis of H. If K denotes the field of fractions of A, then KH = K ⊗_{A} H is a semi-simple split K-algebra which is isomorphic to the group algebra of W over K. This implies, in particular, that every irreducible representation of KH is absolutely irreducible, and that there is a bijection between the irreducible characters of KH and those of W (for a full proof, see [12], [17], and the references there).

Proposition 1.3 Let ρ : KH → K^{m×m} be an irreducible representation of KH and w ∈ C_{min} for some conjugacy class C of W. Let d be the order of the elements in C, and ζ_{d} be a primitive d-th root of unity over the rationals. Then there exist integers e_{i} and rational numbers f_{s,i} such that the eigenvalues of ρ(T_{w}) (in a sufficiently large extension field of K) are of the form

ζ_{d}^{e_{i}} ∏_{s∈I} v_{s}^{f_{s,i}} for i = 1, …, m.

Moreover, the exponents f_{s,i} can be determined explicitly from the ordinary character table of W, various parabolic subgroups of W, and the corresponding induce/restrict matrices.

This again generalizes results of [5] for regular elements. A formula for the exponents f_{s,i} will be given in § 4. The problem of determining the exponents e_{i} seems to be more subtle, and is not solved in general.

This information on eigenvalues of irreducible representations of KH will serve as the missing tool to compute the values of the irreducible characters of the generic Iwahori-Hecke algebra of type E_{8}. In order to solve the problem concerning the roots of unity we also have to use relations among the character values, which are derived from the modular representation theory of Iwahori-Hecke algebras, as developed in [11, 13]. This then completes the computation of the character tables for all Iwahori-Hecke algebras of all types. For the classical types A_{n}, B_{n}, D_{n}, see [22, 16, 19, 20, 21]; for the exceptional types G_{2}, F_{4}, E_{6}, E_{7}, see [12, 13]; for the dihedral and non-crystallographic types I_{2}(m), H_{3} and H_{4}, see [9], §67C, and [17, 1], where explicit matrix representations are given from which the character tables can be computed directly. All the tables are accessible on a computer through the CHEVIE system [14].

2 Good elements in the cuspidal classes for classical types

In this section, we prove Theorem 1.1 for a finite Coxeter group W of classical type, see (2.1) and Corollary 2.7. This will be based on [15], § 2, where ‘standard’ representatives of minimal length in the various conjugacy classes of W were described. As a by-product, we also verify Lemma 1.2.

Let the matrix M, and hence B, W, I be determined by one of the following Dynkin diagrams.

[figure: A_{n−1}; s_{1}; s_{2}; s_{n−1}; n ≥ 1; B_{n}; t; s_{1}; s_{2}; s_{n−1}; n ≥ 1; D_{n}; s_{1}; u; s_{2}; s_{3}; s_{n−1}; n ≥ 2]

At first we settle the case of type A_{n−1}.

2.1 Proof of Theorem 1.1 (and Lemma 1.2) for type A_{n−1}.

In this case, W is isomorphic to the symmetric group S_{n}. The conjugacy classes of W are parameterized by partitions of n, corresponding to the cycle types of permutations. Let λ = (1 ≤ λ_{1} ≤ … ≤ λ_{r}) be such a partition, and C_{λ} be the corresponding class. The elements in C_{min} are precisely the Coxeter elements in the parabolic subgroup of W corresponding to λ (see [15], (2.6)). Hence, if r > 1 then C_{λ} has non-trivial intersection with the proper parabolic subgroup S_{λ_{1}} × … × S_{λ_{r}} and is non-cuspidal. We conclude that there is only one conjugacy class left which can possibly be cuspidal, and this is the class of the Coxeter elements in W. As already mentioned in the Introduction, Theorem 1.1 certainly is true for Coxeter elements, see [4], Lemma 5.8. (Note that this holds in finite Coxeter groups of arbitrary type.) Finally, the elements of minimal length in our class are precisely the various Coxeter elements in W (see [15]), hence Lemma 1.2 is true.

2.2 Standard representatives for types B_{n} and D_{n}.

Now let W, I be of type B_{n} or D_{n}. Recall that the conjugacy classes in these groups are classified in terms of pairs of partitions corresponding to the signed cycle type of elements (cf. [6]). In [15] it was shown that each class contains a standard representative of minimal length in so-called signed block form. Our aim will be to prove that these standard representatives in cuspidal classes are good respectively very good.

For this purpose, we have to introduce some notation, cf. [15], § 2, but note that here we use a slightly different labelling of the nodes of the Dynkin diagrams. This modification is suggested by G. Pfeiffer in [20], §4, and leads to a somewhat more uniform description of the standard representatives. For each i ≥ 1 let

t_{i} := \{ \begin{matrix}s_{i}s_{i−1} … s_{1}ts_{1} … s_{i−1}s_{i} & in type B_{n}, \\ s_{i}s_{i−1} … s_{2}us_{1}s_{2} … s_{i−1}s_{i} & in type D_{n}. \end{matrix}

For i = 0 we let t_{0} := t (in type B_{n}) respectively t_{0} := 1 (in type D_{n}). The elements t_{i} commute with each other. Note that in type D_{n} we have ut_{i} = t_{i}s_{1} for all i ≥ 1. Given m ≥ 0 and d ≥ 1 we define a positive block, respectively a negative block, of length d and starting at m by

\begin{aligned}b^{+}(m, d) & := & s_{m+1} … s_{m+d−1}, \\ b^{−}(m, d) & := & t_{m}s_{m+1} … s_{m+d−1}. \end{aligned}

Note that, if we are in type D_{n} then the same element s_{1} … s_{d−1} can occur as a positive or as a negative block of length d and starting at 0.

With one possible exception, two blocks b^{±}(m, d) and b^{±}(m′, d′) commute with each other if the underlying index sets {m, m + 1, …, m + d − 1} and {m′, m′ + 1, …, m′ + d′ − 1} are disjoint (cf. [15], (2.2)). The exception occurs when we are in type D_{n} and when one block is a ‘proper’ negative block and the other block is our ‘special’ block b^{−}(0, d) = s_{1} … s_{d−1} with d > 1. This will be discussed in more detail below (see Lemma 2.5 and the proof of Lemma 2.6).

In type B_{n} the classes of W are parametrized by pairs (α, β) of partitions such that |α| + |β| = n, corresponding to the signed cycle types of elements in W (see [6]). We assume that α corresponds to the negative cycles of the elements in that class, and β to the positive cycles.

In type D_{n} the classes are again parameterized by double partitions (α, β) of n as before but with an even number of parts of α, and where in the case that α is empty and all parts of β are even there are two classes of this type (see [6]).

Now let (α, β) be a double partition of n labelling a class in W. From [15], (2.7) and (2.8), see also [20], § 4.1, we extract the following uniform description of a standard representative w_{(α,β)} of minimal length in such a class. If α = (1 ≤ α_{1} ≤ … ≤ α_{r}) and β = (1 ≤ β_{1} ≤ … ≤ β_{s}) we let m_{i} := α_{1} + … + α_{i−1} and n_{j} := |α| + β_{1} + … + β_{j−1} for all i, j ≥ 1. Then

w_{(α,β)} := b^{−}(m_{1}, α_{1}) … b^{−}(m_{r}, α_{r})b^{+}(n_{1}, β_{1}) … b^{+}(n_{s}, β_{s}).

If we are in type D_{n} and if α is empty and all parts of β are even then a representative for the second class labelled by (−, β) is obtained by replacing the generator s_{1} by u in the above reduced expression for w_{(−,β)}.

Note also the following particularity for type D_{n}. Let the first part of α be d ≥ 1. If d > 1 then w_{(α,β)} starts with the special ‘negative’ block b^{−}(0, d) = s_{1} … s_{d−1} while if d = 1 then w_{(α,β)} starts with a proper negative block of the form t_{1}s_{2} . . ..

2.3 Cuspidal standard representatives for types B_{n} and D_{n}.

Let (α, β) be a double partition of n labelling a class of W as in (2.2). We say that a standard representative in this class is cuspidal if all generators in I appear in a reduced expression for it. Clearly, standard representatives in cuspidal classes must be cuspidal. Hence, in order to prove Theorem 1.1, we only need to consider classes with cuspidal standard representatives.

From the description in (2.2) it follows immediately that a standard representative with label (α, β) is cuspidal if and only if β (corresponding to the positive cycles) is empty. (In particular, we see that we don’t have to consider the splitting of classes in type D_{n} when dealing with cuspidal classes only.)

If β is empty we denote the corresponding class by C_{α} and the standard representative w_{(α,β)} simply by w_{α}.

The expressions defining all of the above elements are reduced in W. The corresponding elements in the braid monoid B^{+} will be denoted by bold letters t_{i}, b^{±}(m, d) and w_{α}. Note that also the above mentioned commutation rules for positive and negative with disjoint underlying index sets remain valid in B^{+}.

In order to check that these elements are good respectively very good we shall first need some preliminary results about powers of negative blocks.

Lemma 2.4 Assume that W is of type B_{n} or D_{n}. Let d ≥ 1, and m ≥ 0 (for type B_{n}), respectively m ≥ 1 (for type D_{n}). Consider the negative block b^{−}(m, d) of length d and starting at m.

(a) The order of b^{−}(m, d) in W is 2d.

(b) We have b^{−}(m, d)^{d} = t_{m}t_{m+1} … t_{m+d−1}.

Note that the equality in (b) also holds in B^{+} because the expressions on both sides are reduced.

Proof. At first assume that W is of type B_{n}. Let W′ be the subgroup of W generated by the elements t_{m}, s_{m+1}, …, s_{m+d−1}. Then it is easy to see that W′ is a Coxeter group of type B_{d}, and that b^{−}(m, d) is a Coxeter element in W′. Hence the order of b^{−}(m, d) is the Coxeter number of W′ which is equal to 2d. Also, b^{−}(m, d)^{d} equals the longest element in W′ which is the product of the successive conjugates of the first generator t_{m} by the following generators s_{m+1}, …, s_{m+d−1}, hence equals t_{m}t_{m+1} … t_{m+d−1} as required. Since this expression is reduced in W, the asserted equality in (b) also holds in B^{+}.

Now let W be of type D_{n}. Then W can be embedded as a normal subgroup of index 2 into a finite Coxeter group W′ of type B_{n}. In order to distinguish generators and elements of W and W′ we attach a prime to them if we are in W′. Thus W has canonical generators u, s_{1}, …, s_{n−1} and W′ has canonical generators denoted by t′, s′_{1}, …, s′_{n−1}. An embedding of W into W′ is given by u ↦ t′s′_{1}t′ and s_{i} ↦ s′_{i} for all i ≥ 1. Now t_{m} ↦ t′t′_{m}, see the definitions in (2.2), and note this would also work for m = 0. The image of b^{−}(m, d) in W′ can therefore be written as t′t′_{m}s′_{m+1} … s′_{m+d−1} = t′b^{−}(m, d)′. Since m ≥ 1 the two elements t′ and b^{−}(m, d)′ commute, and so their product has order 2d by (a). If we raise their product to the d-th power we obtain t′^{d}t′_{m} … t′_{m+d−1} by (b). This element can be rewritten again as a product of the generators of W, and we find t_{m}t_{m+1} … t_{m+d−1} as desired. Again, since this is reduced in W, we also have an equality of expressions in B^{+}. This completes the proof.

Lemma 2.5 Assume that W is of type D_{n}. Let e ≥ 2 and consider the special negative block b^{−}(0, e) = s_{1}s_{2} … s_{e−1}, see (2.2). To abbreviate notation, we write a(e) := b^{−}(0, e) and â(e) = us_{2} … s_{e−1}.

(a) The order of a(e) and of â(e) in W is e.

(b) We have t_{1} … t_{e−1} = a(e)â(e)a(e) …, where the product is taken over e consecutive factors a(e), â(e).

Note again that the equalities in (b) also hold in B^{+} because the expressions are reduced.

Proof. Again, we use an embedding of W into a Coxeter group W′ of type B_{n} as in the proof of Lemma 2.4. With the notation introduced there, we find

a(e) = s′_{1}s′_{2} … s′_{e−1},
â(e) = t′s′_{1}t′s′_{2} … s′_{e−1} = b^{−}(0, e)′t′.

The expression for â(e) is conjugate to that for a(e), which is a Coxeter element in the parabolic subgroup of type A_{e−1}. This proves (a).

Now we distinguish the cases where e is even or odd. For e odd we obtain:

a(e)â(e) … = s′_{1} … s′_{e−1}(b^{−}(0, e)′t′s′_{1}s′_{2} … s′_{e−1})^{(e−1)/2}
= s′_{1} … s′_{e−1}b^{−}(0, e)′^{e−1}
= t′b^{−}(0, e)′^{e}
= t′(t′_{0} … t′_{e−1}) by Lemma 2.4
= t′_{1} … t′_{e−1}.

Since the total number of factors in this product is even, we can introduce an even number of terms t′ and rewrite this expression in W. We obtain the desired expression, which is reduced. The proof for e even is completely analogous.

Lemma 2.6 Assume that W is of type B_{n} or D_{n}. Let α = (1 ≤ α_{1} ≤ … ≤ α_{r}) be a partition of n (with an even number of parts if W is of type D_{n}). Consider the standard representative w_{α} = b^{−}(m_{1}, α_{1}) … b^{−}(m_{r}, α_{r}) where m_{i} = α_{1} +…+α_{i−1} for all i.

(a) The order of w_{α} is 2d where d is the least common multiple of the parts of α.

(b) Let e_{i} := d/α_{i} for all i and t(m, d) := t_{m}t_{m+1} … t_{m+d−1} for m ≥ 0, d ≥ 1.

Then

w_{α}^{d} = t(m_{1}, α_{1})^{e_{1}} … t(m_{r}, α_{r})^{e_{r}}.

Proof. (a) If W is of type D_{n}, we consider an embedding into a Coxeter group of type B_{n} as in the proof of Lemma 2.4. In any case, the element w_{α} is a product of commuting negative cycles corresponding to the parts of α (see [6]). The order of any such negative cycle, corresponding to the part α_{i} say, is 2α_{i}. This implies that the order of w_{α} indeed is 2d.

(b) Assume at first that α_{1} = 1 if W is of type D_{n}, so that the ‘special’ block of the form s_{1} … s_{d−1} is not involved in w_{α}. We have already noted in (2.3) that then the various negative blocks involved in w_{α} commute with each other. Therefore, w_{α}^{d} equals the product of the terms b^{−}(m_{i}, α_{i})^{d} for i = 1, …, r. In order to evaluate such a term, we can use Lemma 2.4, and obtain directly the desired form.

Now suppose that we are in type D_{n} and α_{1} > 1. We have to take into account the fact that the first ‘negative’ block b^{−}(m_{1}, α_{1}) = a(α_{1}) does not commute with the proper negative blocks b^{−}(m_{i}, α_{i}) for i ≥ 2. Instead, we have the following relation, for d ≥ 0 and m ≥ e − 1 ≥ 1.

a(e)b^{−}(m, d) = s_{1}s_{2} … s_{e−1}(t_{m}s_{m+1} … s_{m+d−1})
= s_{1}t_{m}s_{m+1} … s_{m+d−1}(s_{2} … s_{e−1})
= t_{m}u(s_{m+1} … s_{m+d−1})(s_{2} … s_{e−1}) since s_{1}t_{m} = t_{m}u
= b^{−}(m, d)us_{2} … s_{e−1}
= b^{−}(m, d)â(e).

Similarly, we prove that â(e)b^{−}(m, d) = b^{−}(m, d)a(e). Note that these relations hold in B^{+} since the expressions on both sides are reduced.

Now, if we raise w_{α} to the d-th power, we can use these rules to collect all terms b^{−}(m_{i}, α_{i}) with i ≥ 2 to the right. Since the total number of parts of α is even, the number of such terms b^{−}(m_{i}, α_{i}) is odd. Hence we arrive at the following expression.

w_{α}^{d} = (a(α_{1})â(α_{1}) …)(b^{−}(m_{2}, α_{2})^{d} … b^{−}(m_{r}, α_{r})^{d})

with d consecutive factors a(α_{1}), â(α_{1}). Using Lemma 2.5 we compute that

(t_{1} … t_{α_{1}−1})^{e_{1}} = a(α_{1})â(α_{1}) … (d factors).

The terms b^{−}(m_{i}, α_{i})^{d} for i ≥ 2 can be rewritten as before. We obtain:

w_{α}^{d} = (t_{1} … t_{α_{1}−1})^{e_{1}}t(m_{2}, α_{2})^{e_{2}} … t(m_{r}, α_{r})^{e_{r}}.

Now note that t_{1} … t_{α_{1}−1} = t(0, α_{1}). Hence we have again reached the desired form.

The following Corollary summarizes our results and establishes Theorem 1.1 for types B_{n} and D_{n}.

Corollary 2.7 (Very good elements for types B_{n} and D_{n}.) Assume that W is of type B_{n} or D_{n}. Let α = (1 ≤ α_{1} ≤ … ≤ α_{r}) be a partition of n (with an even number of parts, if W is of type D_{n}) and d ≥ 1 be the least common multiple of the parts of α. Consider the standard representative w_{α} = b^{−}(m_{1}, α_{1}) … b^{−}(m_{r}, α_{r}) where m_{i} = α_{1} + … + α_{i−1} for all i. Then we have

w_{α}^{d} = w_{I_{r}}^{e_{r}}w_{I_{r−1}}^{e_{r−1}−e_{r}}w_{I_{r−2}}^{e_{r−2}−e_{r−1}} … w_{I_{1}}^{e_{1}−e_{2}} in B^{+},

where e_{i} = d/α_{i} for all i ≥ 1, I_{r} = I and I_{i−1} = I_{i} \{s_{m_{i}}, …, s_{m_{i}+α_{i}−1}} for all i > 1. In particular, w_{α} is very good. Moreover, all terms in the above decomposition of w_{α}^{d} commute with each other. Hence w_{α}^{2d} is given by the analogous expression as above where each exponent is multiplied by 2. Thus, w_{α} is also good.

Proof. By Lemma 2.6 we have that

w_{α}^{d} = t(m_{1}, α_{1})^{e_{1}} … t(m_{r}, α_{r})^{e_{r}},

where the e_{i} are defined as above. Since the parts of α are ordered by increasing length we have e_{1} ≥ e_{2} ≥ … ≥ e_{r}. Therefore, we can collect together terms of the form t(m_{i}, α_{i})^{e_{r}} for i = 1, …, r. We note that the product of the terms t(m_{i}, α_{i}) is reduced of length n^{2} (for type B_{n}) respectively n(n − 1) (for type D_{n}). Hence this product equals the longest element w_{I} of W. So we obtain

w_{α}^{d} = (t(m_{1}, α_{1})^{e_{1}−e_{r}} … t(m_{r−1}, α_{r−1})^{e_{r−1}−e_{r}})w_{I}^{e_{r}}.

Now we proceed in the same way with the first r − 1 factors in this product and obtain

w_{α}^{d} = (t(m_{1}, α_{1})^{e_{1}−e_{r−1}} … t(m_{r−1}, α_{r−2})^{e_{r−2}−e_{r−1}})w_{I_{r−1}}^{e_{r−1}−e_{r}}w_{I}^{e_{r}}

where I_{r−1} = I \ {s_{m_{r}}, …, s_{m_{r}+α_{r}−1}}. Continuing in this way completes the proof of the first assertion.

The elements w_{I_{i}} are products of various factors t(m_{i}, α_{i}). Hence the commutation of the terms w_{I_{i}} is a consequence of the commutation of the terms t(m_{i}, α_{i}).

We note that the number r of subsets I_{i} can become arbitrarily large. It is given by the number of distinct parts of α.

2.8 Proof of Lemma 1.2.

Let W be of type B_{n} or D_{n}. Let α be a partition labelling a class C_{α} with a cuspidal standard representative w_{α}. Assume, if possible, that C_{α} is not cuspidal. Then C_{α} would have non-trivial intersection with a proper parabolic subgroup of W. Such a subgroup is a direct product of possibly a factor of type B_{i} or D_{i}, and factors of type A_{j−1}, for various j ≥ 1. By [6], such a decomposition would result in at least one non-trivial positive cycle in the signed cycle type of the elements in C_{α}, which is impossible.

3 Good elements in cuspidal classes for exceptional types

In this section, we give explicit results on cuspidal classes and good elements for all exceptional irreducible finite Coxeter groups. These results were obtained using the computer algebra system GAP [23] as well as the CHEVIE programs for dealing with finite Coxeter groups and an implementation of Deligne’s normal form for elements in the braid group.

3.1 An algorithm to check Theorem 1.1 (and Lemma 1.2).

Let W be a fixed finite Coxeter group. At first we apply the method described in [15], § 3, Step 1 and 2, to find a (relatively small) subset M ⊆ W such that each element in M has minimal length in its conjugacy class and such that, for each element w ∈ W, there exists an element m ∈ M with w → m (where → is defined in [15]). Some more details on the programs are given in [14], (3.2). For example, when W is of type E_{8} with roughly 600000000 elements in total, the method described in [loc. cit.] yields a subset M containing around 50000 elements.

Next we compute representatives of the orbits of M under cyclic shift. (Note that the condition that w, w′ ∈ W are conjugate by cyclic shift is equivalent to the condition that w →′ and w′ → w.) Let us denote such a set of representatives by M′. Now we consider the intersection of M′ with a conjugacy class C. By construction, this intersection is contained in C_{min} and it is precisely a set of representatives of the orbits of C_{min} under cyclic shift.

We then keep only those classes for which every element m ∈ C_{min} ∩ M′ has the property that all generators of W appear in a reduced expression for m.

Now suppose that the elements in C have order d. We consider in turn each element m ∈ C_{min} ∩ M′, compute its orbit under cyclic shift, and calculate the normal form of w^{d} (for each w in that orbit) in the braid monoid B^{+}. In this way, we can first verify Lemma 1.2 and also obtain complete lists of good and very good representatives of minimal length in all classes under consideration. We can thus check that good elements always exist in C_{min}, and that there exist very good elements if d is even. Our tables then show that, if w is very good then all terms in the normal form for w^{d/2} commute with each other. This implies that the element is automatically good. Thus, Theorem 1.1 is verified.

The results we have obtained are summarized in the following tables. For each type, we give a table containing the cuspidal classes, the cardinality of C_{min}, good (resp. very good when d is even) representatives w, the normal form of w^{d} and the number of (very) good elements w ∈ C_{min} with the same normal form for w^{d}. We also mention the existence of good elements of even order which are not very good, whenever such elements exist. The cuspidal classes in the crystallographic types are labelled by their admissible diagrams, as defined in [6].

3.2 Type G_{2}.

Let I and B, W be determined by the following Dynkin diagram.

[figure: G_{2}; >; 1; 2]
cuspidal order |C_{min}|  good/very good power no. such
class  d  representative w w^{d} elements
G_{2}  6  2  12 w_{I}^{2}  2
A_{2}  3  2  1212 w_{I}^{2}  2
A_{1} + Ã_{1}  2  1  w_{I} = longest element w_{I}^{2}  1

Remark. In the above and the following tables, we abbreviate a product like s_{3}s_{4}s_{2} … by 342 . . .. We also abbreviate w_{{s_{2},s_{3},s_{4}}} (the w_{J} of the subset J = {s_{2}, s_{3}, s_{4}}) as w_{234}.

3.3 Type I_{2}(m), m ≥ 5.

Let I and B, W be determined by the following Dynkin diagram.

[figure: I_{2}(m); m; 1; 2]

If m is even the sets C_{min} are: {1}, {(s_{1}s_{2})^{m/2}} (central), {s_{1}}, {s_{2}}, and C_{i} = {(s_{1}s_{2})^{i}, (s_{2}s_{1})^{i}} for i = 1, …, m/2 − 1.

If m is odd the sets C_{min} are: {1}, {s_{1}, s_{2}} and C_{i} = {(s_{1}s_{2})^{i}, (s_{2}s_{1})^{i}} for i = 1, …, (m − 1)/2.

In both cases, the cuspidal classes are those with representatives of the form (s_{1}s_{2})^{i} for some i ≥ 1. The longest element in W is w_{I} = s_{1}s_{2}s_{1} … (with m factors). Hence we see that (s_{1}s_{2})^{i} is good in any case, and very good if its order is even.

12 Geck-Michel

3.4 Type H_{3}.

Let I and B, W be determined by the following Dynkin diagram.

[figure: H_{3}; 5; 1; 2; 3]
cusp. order |C_{min}|  good/very good power no. such
class  d  representative w w^{d} elements
6  10  4  123 w_{I}^{2}  4
8  6  6  12123 w_{I}^{2}  6
9  10  6  121232123 w_{I}^{6}  6
10  2  1  w_{I} = longest element w_{I}^{2}  1

Remark. The numbering in this and the table for H_{4} refer to the number of that class in the CHEVIE table.

3.5 Type F_{4}.

Let I and B, W be determined by the following Dynkin diagram.

[figure: F_{4}; >; 1; 2; 3; 4]
cuspidal order |C_{min}|  good/very good power no. such
class  d  representative w  w^{d}  elements
4A_{1}  2  1  w_{I} = longest element  w_{I}^{2}  1
D_{4}  6  8  2323432134 w_{I}^{2}.w_{34}^{4}  2
D_{4}(a_{1})  4  12  324321324321  w_{I}^{2}  12
C_{3} + A_{1}  6  8  1214321323 w_{I}^{2}.w_{12}^{4}  2
A_{2} + Ã_{2}  3  16  3214321323432132  w_{I}^{2}  16
F_{4}(a_{1})  6  16  32432132  w_{I}^{2}  16
F_{4}  12  8  4321  w_{I}^{2}  8
A_{3} + Ã_{1}  4  16  23234321324321 w_{I}^{2}.w_{23}^{2}  2
B_{4}  8  14  243213  w_{I}^{2}  14

3.6 Type H_{4}.

Let I and B, W be determined by the following Dynkin diagram.

[figure: H_{4}; 5; 1; 2; 3; 4]

Good elements and representations 13

cusp.order |C_{min}|  good/very good power no. such
class d  representative w w^{d} elements
11 30  8  1234 w_{I}^{2}  8
14 20 12  121234 w_{I}^{2}  12
15 15 18  12321234 w_{I}^{2}  18
17 12 22  1212321234 w_{I}^{2}  22
18 10 24  132121321234 w_{I}^{2}  24
19 10 34  12132121321234 w_{I}^{2}.w_{12}^{4} 4
21 10 12  1212132121321234 w_{I}^{2}.w_{12}^{8} 2
22 15 24  1212321234321234 w_{I}^{4}  24
23 20 38  121213212134321234 w_{I}^{6}  38
24 6  40  12121321213214321234 w_{I}^{2}  40
25 6  36  1212132121321234321234 w_{I}^{2}.w_{23}^{4} 4
26 5  24  123212132123432121321234 w_{I}^{2}  24
27 5  56  12124321213212343212132123 w_{I}^{2}.w_{12}^{2} 4
28 30 40  1212321213214321213214321234 w_{I}^{14}  40
29 4  60  121213212132124321213214321234 w_{I}^{2}  60
30 10 24  121321213212343212132123432121321234 w_{I}^{6}  24
31 10 36  12121321213212343212132123432121321234 w_{I}^{6}.w_{12}^{4} 2
32 3 40 1212132121321243212132124321213214321234 w_{I}^{2} 40
33 5  24 1212321213212343212132123432121321234321
·21321234 w_{I}^{4}  24
34 2  1  w_{I} =longest element w_{I}^{2}  1

3.7 Type E_{6}.

Let I and B, W be determined by the following Dynkin diagram.

[figure: E_{6}; s_{1}; s_{3}; s_{4}; s_{2}; s_{5}; s_{6}]
cuspidal order |C_{min}|  good/very good power no. such
class  d  representative w  w^{d}  elements
3A_{2}  3  80  131423145423143565423456  w_{I}^{2}  80
E_{6}(a_{2})  6  144  242314543165  w_{I}^{2}  16
A_{5} + A_{1}  6  48  23423454231456 w_{I}^{2}.w_{24}^{4}  2
E_{6}(a_{1})  9  80  13423456  w_{I}^{2}  80
E_{6}  12  32  423156  w_{I}^{2}  8

Remark. As the numbers show, in the class of regular elements E_{6}(a_{2}) (resp. E_{6}) there are 128 (resp. 24) good elements which are not very good. Also, in the class A_{5} + A_{1} there are 2 further elements such that w^{6} = w_{I}^{2}.w_{24}^{4}, 4 elements such that w^{6} = w_{I}^{2}.w_{34}^{4} and 4 elements such that w^{6} = w_{I}^{2}.w_{45}^{4} but none of them are very good.

14 Geck-Michel

3.8 Type E_{7}.

Let I and B, W be determined by the following Dynkin diagram.

[figure: E_{7}; s_{1}; s_{3}; s_{4}; s_{5}; s_{2}; s_{6}; s_{7}]
cuspidal order |C_{min}|  good/very good power no. such
class  d  representative w  w^{d}  elements
7A_{1}  2  1  w_{I} =longest element  w_{I}^{2}  1
D_{4} + 3A_{1}  6  32 2342345423456542345676542
·314567 w_{I}^{2}.w_{24567}^{4}  2
E_{7}(a_{4})  6  800  543654234765423143542  w_{I}^{2}  800
D_{6}(a_{2}) + A_{1} 6  708  12314231435423143546576 w_{I}^{2}w_{13}^{4}  18
24265427654231435426543 w_{I}^{2}.w_{24}^{4}  18
13142315423143654234567 w_{I}^{2}w_{34}^{4}  18
12342345423145654234567 w_{I}^{2}w_{45}^{4}  18
12314235423465423457654 w_{I}^{2}w_{56}^{4}  12
2A_{3} + A_{1} 4 360 2425423654234567654231435
·42654317 w_{I}^{2}.w_{257}^{2}  12
D_{6} + A_{1} 10 96 234234765423145 w_{I}^{2}.w_{24}^{8} 8
234231454234567 w_{I}^{2}w_{34}^{8}  8
124235423465437 w_{I}^{2}w_{45}^{8}  8
A_{5} + A_{2}  6  420 4354234576542314354265431 w_{I}^{2}.w_{2345}^{2}  32
E_{7}(a_{1})  14 160  427654231  w_{I}^{2}  160
A_{7}  8  316  23654234765423145 w_{I}^{2}.w_{257}^{2}.w_{2}^{4} 16
12342542365427654 w_{I}^{2}w_{257}^{2}w_{5}^{4}  12
E_{7}  18  64  7654231  w_{I}^{2}  64
E_{7}(a_{2})  12 280  65427654231 w_{I}^{2}.w_{257}^{2}  24
E_{7}(a_{3})  30 366  2342765423145 w_{I}^{6}.w_{24}^{4}  16
2343154234567 w_{I}^{6}w_{34}^{4}  16
1242354265437 w_{I}^{6}w_{45}^{4}  16

3.9 Type E_{8}.

Let I and B, W be determined by the following Dynkin diagram.

[figure: E_{8}; s_{1}; s_{3}; s_{4}; s_{5}; s_{6}; s_{2}; s_{7}; s_{8}]

The results in this case are given in the following tables. Remark. The class denoted D_{5}(a_{1})+A_{3} in those tables contains 216 other elements which are good but not very good (and have the same value for w^{d}).

Good elements and representations 15

cuspidal order |C_{min}|  good/very good power no. such
class  d  representative w  w^{d} elements
8A_{1}  2  1  w_{I} = longest element w_{I}^{2}  1
2D_{4}(a_{1})  4 15120 12314231435423143542654234
·5765423143548765423143542654765876 w_{I}^{2}  15120
D_{4} + 4A_{1} 6  56 12314231435423143542654231
·43542654318765423143542654317654234567 w_{I}^{2}w_{123456}^{4} 2
4A_{2}  3 4480 12314231435423143565423143
·542676542314354265431787654
·231435426543176542345678765 w_{I}^{2}  4480
E_{8}(a_{8})  6 4480 12314231435423145654231456
·76542345678765 w_{I}^{2}  4480
E_{7}(a_{4}) + A_{1} 6 11592
123142314354234565423145676542314354678765 w_{I}^{2}w_{13}^{4}  80
123142314354316542314567654234567876542345 w_{I}^{2}w_{24}^{4}  80
123142314354231435426542314567654234567876 w_{I}^{2}w_{34}^{4}  80
123423145423143546542314567654234567876543 w_{I}^{2}w_{45}^{4}  80
123142345423145654231435465765423456787654 w_{I}^{2}w_{56}^{4}  80
123142345431654317654318765423143542654765 w_{I}^{2}w_{67}^{4}  80
123142345431654234567654234567876542345678 w_{I}^{2}w_{78}^{4}  80
2D_{4}  6 4070
12314315431654317654231435426543176542345678 w_{I}^{2}w_{1356}^{4}  4
12314231435423456542314567654231435465768765 w_{I}^{2}w_{1367}^{4}  4
12314231436542345676542345678765423143546578 w_{I}^{2}w_{1378}^{4}  4
12314231435431654317654318765423143542654765 w_{I}^{2}w_{2467}^{4}  4
12314231435431654234567654234567876542345678 w_{I}^{2}w_{2478}^{4}  4
13142315431654317654318765423143542654376542 w_{I}^{2}w_{3467}^{4}  4
13142315423143654234567654234567876542345678 w_{I}^{2}w_{3478}^{4}  4
12342345423145654234567654234567876542345678 w_{I}^{2}w_{4578}^{4}  4
2A_{3} + 2A_{1} 4 1260 12314231435423143542654231
·4354265431765423143542654317876542345678 w_{I}^{2}w_{2345}^{2} 12
D_{8}(a_{3})  8 7748 12314231435423146542347654
·8765 w_{I}^{2}  7748
D_{6} + 2A_{1} 10 256
12342345423456542345676542345678 w_{I}^{2}w_{2456}^{8}  4
23423454234565423145676542345678 w_{I}^{2}w_{3456}^{8}  4
12423542346542345765423456876543 w_{I}^{2}w_{4567}^{8}  4
2A_{4}  5 7952 12314231454231456542314567
·6542314567876542345678 w_{I}^{2}  7952
E_{8}(a_{6}) 10 3370 123142314542345654765876 w_{I}^{2} 3370
E_{6}(a_{2}) + A_{2} 6 16374 12314231454231436542314354
·265431765423456787 w_{I}^{2}w_{2345}^{2} 128
E_{8}(a_{3})  12 2696  12314234542365476548 w_{I}^{2}  2696

16 Geck-Michel

cuspidal order |C_{min}|  good/very good  power no. such
class  d  representative w  w^{d}  elements
A_{5} + A_{2} + A_{1} 6 3752 12314231454231456542345676
·54234567876542345678 w_{I}^{2}w_{234578}^{2}w_{78}^{2} 16
D_{8}(a_{1}) 12 2040 1234234543654276548765 w_{I}^{2}w_{2478}^{4} 24
2342314542654376548765 w_{I}^{2}w_{3478}^{4}  12
1234234542345654765876 w_{I}^{2}w_{4578}^{4}  24
D_{8} 14 852 123423454234565768 w_{I}^{2}w_{2}^{12} 128
234231454234565768 w_{I}^{2}w_{3}^{12}  88
124235423456543768 w_{I}^{2}w_{4}^{12}  108
123425423654276548 w_{I}^{2}w_{5}^{12}  68
A_{7} + A_{1}  8 2080
1231423454231456542345676542345678 w_{I}^{2}w_{2345}^{2}w_{25}^{4} 16
2343543654231435426543176542345678 w_{I}^{2}w_{2345}^{2}w_{35}^{4} 12
E_{7} + A_{1} 18 192 1234234542345678 w_{I}^{2}w_{24}^{16} 16
2342314542345678 w_{I}^{2}w_{34}^{16}  16
1242354234654378 w_{I}^{2}w_{45}^{16}  16
A_{8}  9 2816
1234315423145654231435467687 w_{I}^{2}w_{13}^{4}  128
1234235423143542654234567687 w_{I}^{2}w_{24}^{4}  88
1231423145423145654234567687 w_{I}^{2}w_{34}^{4}  100
1242354236542314354267654387 w_{I}^{2}w_{45}^{4}  64
1234315423165423147654387654 w_{I}^{2}w_{56}^{4}  24
E_{8}(a_{4}) 18 732 12342542345678 w_{I}^{2}w_{24}^{4} 32
23431542345678 w_{I}^{2}w_{34}^{4}  32
12423542654378 w_{I}^{2}w_{45}^{4}  32
E_{8}(a_{2}) 20 624 123425465478 w_{I}^{2} 624
D_{5}(a_{1}) + A_{3} 12 15134 12314543654231435476542314
·35426543176542345678 w_{I}^{4}w_{123456}^{2}  24
E_{6} + A_{2}  12 840 12314231454231456542345678 w_{I}^{2}w_{2345}^{6}  64
E_{8}(a_{7})  12 2360  1231423454316542345678 w_{I}^{2}w_{2345}^{2}  64
E_{7}(a_{2}) + A_{1} 12 1758 123142314354316542345678 w_{I}^{2}w_{2345}^{2}w_{24}^{8}  8
131423154231436542345678 w_{I}^{2}w_{2345}^{2}w_{34}^{8}  8
123423454231456542345678 w_{I}^{2}w_{2345}^{2}w_{45}^{8}  8
E_{8}(a_{1})  24 320  1234254678  w_{I}^{2}  320
D_{8}(a_{2})  30 4996 13423542345654276542345678 w_{I}^{6}w_{2456}^{4}w_{4}^{20}  8
12342354234654276542345678 w_{I}^{6}w_{2456}^{4}w_{5}^{20}  8
24235423456543176542345678 w_{I}^{6}w_{3456}^{4}w_{4}^{20}  8
23423542346543176542345678 w_{I}^{6}w_{3456}^{4}w_{5}^{20}  8
12435423654234576542876543 w_{I}^{6}w_{4567}^{4}w_{5}^{20}  8
12423542365423476542876543 w_{I}^{6}w_{4567}^{4}w_{6}^{20}  8
E_{8}(a_{5})  15 1516  1231423454657658  w_{I}^{2}  1516
E_{8}  30 128  12347658  w_{I}^{2}  128

4 On the eigenvalues of representations on good

elements

Let KH be the Iwahori-Hecke algebra with parameters {v_{s}^{2} | s ∈ I} attached to the Coxeter group (W, I) as in § 1. Recall that, by specialization, we obtain a bijection between the irreducible characters of KH and those of W. This is done as follows. Let f : A → Q I, v_{s} ↦ 1 (s ∈ I). For any irreducible character χ of KH we have χ(T_{w}) ∈ A for all w ∈ W. Then the function w ↦ f(χ(T_{w})) defines an irreducible character of W, and this process establishes a bijection between the irreducible characters of KH and those of W (cf. [9], §68, or [13]). To simplify notation, we shall write χ(w) instead of f(χ(T_{w})) for w ∈ W.

In order to state the following basic (and well-known) result, we introduce some more notation. We shall denote by I′ ⊆ I a full set of representatives of elements in I under conjugation in W. We denote by N_{s} the cardinality of the conjugacy class of the element s ∈ I′ in W. Note that this also equals the total multiplicity of all generators in I, which are conjugate to s and which appear in a reduced expression for w_{I}, the longest element in W.

Lemma 4.1 Let ρ : KH → K^{m×m} be an irreducible representation of KH, with character χ. Let

c_{ρ} = ∏_{s∈I′} v_{s}^{N_{s}(χ(s)+χ(1))/χ(1)} ∈ K,

Then ρ(T_{w_{I}}^{2} ) = c_{ρ}^{2} · I_{m} (where I_{m} denotes the m × m-identity matrix).

We remark that, if all s ∈ I are conjugate in W, then an alternative description of the exponents in the above monomial in terms of highest and lowers powers of the parameters dividing the corresponding generic degrees is given in [18], (5.12).

Proof. The main ingredients of the argument to prove this are sketched in the corrections and additions to [2]. Let us briefly recall it. First we notice that T_{w_{I}}^{2} lies in the centre of KH, since it is the image of the central element w_{I}^{2} ∈ B^{+}. Hence ρ(T_{w_{I}}^{2} ) = c · I_{m} for some c ∈ K. Now we consider a reduced expression w_{I} = s_{1} … s_{N}, and compute that

c^{m} = det ρ(T_{w_{I}}^{2} ) = (∏_{i=1}^{N} det ρ(T_{s_{i}}))^{2}.

The quadratic relation for T_{s_{i}} shows that ρ(T_{s_{i}}) has eigenvalues v_{s_{i}}^{2} and −1, with multiplities a_{i} and b_{i} say.

The numbers a_{i} and b_{i} can be computed by specialization. We have χ(T_{s_{i}}) = a_{i}v_{s_{i}}^{2} −b_{i} and m = χ(T_{1}) = a_{i} +b_{i}. Now consider the specialization v_{s} ↦ 1 (s ∈ I). Then χ(s_{i}) = a_{i} − b_{i}, and hence we find that

a_{i} = \frac{1}{2}(m + χ(s_{i})) and b_{i} = \frac{1}{2}(m − χ(s_{i})).

Now, if s_{i} and s_{j} are conjugate in W then the respective multiplicities a_{i}, b_{i} and a_{j}, b_{j} are the same. Combining this with the above formula yields that

c^{m} = ∏_{s∈I′}(v_{s}^{2a_{s}}(−1)^{b_{s}})^{2N_{s}} = ∏_{s∈I′} v_{s}^{2N_{s}(m+χ(s))}.

This equation shows that c^{m} is a monomial in the indeterminates v_{s} (s ∈ I′). Since c ∈ K, we conclude that c itself is ± a monomial with exponents given by 2N_{s}(m + χ(s))/m, for s ∈ I′. Since ρ(T_{w_{I}}^{2} ) specializes to the identity matrix we conclude that the sign in ± must be +.

Finally, we have to show that c is a square. For this purpose, we observe that N_{s}χ(s)/m is the value at Class(s) of the central character of W corresponding to the character obtained by specialization from ρ, hence it is an integer. It follows that N_{s}(m + χ(s))/m = N_{s} + N_{s}χ(s)/m is an integer, for all s ∈ I′. Hence c is a square, and the proof is complete. (Note that the above argument proves in fact that c_{ρ} is a monomial in the v_{s}, even if our splitting field K had been chosen to be an algebraic closure of Q I(v_{s} | s ∈ S).)

The following result contains, in particular, a proof of the first (qualitative) statement in Proposition 1.3. An algorithm for explicitly determining the absolute values of the eigenvalues will be given below.

Corollary 4.2 Let ρ : KH → K^{m×m} be any representation of KH, and w ∈ W be an element of minimal length in some conjugacy class of W. Let d be the order of w. Then ρ(T_{w})^{d} is diagonalizable and the eigenvalues are monomials in v_{s}^{2} (s ∈ I′). Consequently, the eigenvalues of ρ(T_{w}) are d-th roots of unity times fractional powers of those monomials (in some sufficiently large extension field of K).

Proof. Let C be the conjugacy class of w. By [15], Theorem 1.1(b), the elements T_{x} (x ∈ C_{min}) are conjugate by units in KH. In particular, the matrices ρ(T_{x}) (x ∈ C_{min}) have the same eigenvalues (counting multiplicities). Since good elements exist in C_{min} by Theorem 1.1, we may therefore assume from the outset that our given element w ∈ W is good. Let

w^{d} = w_{J_{1}}^{d_{1}} … w_{J_{r}}^{d_{r}}

as in Theorem 1.1, where J_{1} ⊃ … ⊃ J_{r} is a decreasing chain of subsets, and the d_{i} are positive even integers. Since KH was defined as a quotient of the group algebra of the braid group B, we have a corresponding relation in KH:

T_{w}^{d} = T_{w_{J_{1}}}^{d_{1}} … T_{w_{Jr}}^{d_{r}} .

Now note that, since the subsets J_{i} form a decreasing sequence, the corresponding elements T_{w_{J_{i}}}^{2} commute with each other. It follows that, if we apply a representation ρ to the above relation, then the representing matrices on these elements can be simultaneously triangulized, and the eigenvalues of ρ(T_{w})^{d} are the products of the respective eigenvalues of the matrices ρ(T_{w_{J_{i}}}^{2} ) (raised to the (d_{i}/2)-th power). For each subset J_{i}, we consider the restriction of ρ to the corresponding parabolic sub-algebra KH_{J_{i}}. Then this restriction splits up into a direct sum of irreducible constituents, and Lemma 4.1 implies that T_{w_{J_{i}}}^{2} is represented by a diagonal matrix in each such irreducible constituent, with known eigenvalues. This completes the proof.

The above proof shows in fact how to obtain the exact shape of the absolute values of the eigenvalues. We shall make this more explicit in the following algorithmic description.

4.3 A formula for the absolute values of eigenvalues.

Let C be a conjugacy class in W, and w ∈ C_{min} a good element of order d say, such that we have

w^{d} = w_{J_{1}}^{d_{1}} … w_{J_{r}}^{d_{r}} in B^{+}

where J_{1} ⊃ … ⊃ J_{r} and d_{1}, …, d_{r} are positive even integers (cf. Theorem 1.1). For technical reasons, we shall assume that the last subset J_{r} in the above sequence is the empty set and d_{r} = 1; moreover, we can assume that J_{1} = I (possibly with d_{1} = 0).

Let ρ be an irreducible representation of KH. We want to compute the list of absolute values of the eigenvalues of ρ(T_{w}).

For this purpose we consider pairs γ = (J, ρ) where J ⊆ I is any subset of I and ρ is an irreducible representation of the parabolic subgroup W_{J} ⊆ W (always taken up to isomorphism). For such a pair γ, we denote by c_{γ} the monomial defined in Lemma 4.1 (with respect to an irreducible representation of KH_{J} which specializes to ρ). Now let γ = (J, ρ) and γ′ = (J′, ρ′) be two such pairs. We shall write γ ⟶^{m} γ′ if J′ ⊆ J and m is the multiplicity of ρ′ as a constituent in the restriction of ρ to W_{J′}. (Note that, by a specialization argument, this multiplicity is the same regardless of whether we consider representations of KH or of W.)

Now we consider chains

γ_{1} ⟶^{m_{1}} γ_{2} ⟶^{m_{2}} … ⟶^{m_{r−1}} γ_{r}

where γ_{i} = (J_{i}, ρ_{i}) for some irreducible representation ρ_{i} of W_{J_{i}} (1 ≤ i ≤ r), with ρ_{1} = ρ. Such a chain will be called (ρ, w)-adapted. Note that, by our above convention, we always have γ_{r} = (∅, 1). Therefore, if we add up all terms m_{1} … m_{r−1}, where the summation is over all (ρ, w)-adapted chains, we get exactly the degree of ρ. Now we have:

Proposition 4.4 The list of absolute values of the eigenvalues of ρ(T_{w}) is obtained by taking, for each (ρ, w)-adapted chain

γ_{1} ⟶^{m_{1}} γ_{2} ⟶^{m_{2}} … ⟶^{m_{r−1}} γ_{r},

the fractional monomial

c_{γ_{1}}^{d_{1}/d} … c_{γ_{r}}^{d_{r}/d}

with multiplicity m_{1} … m_{r−1}.

Note that, in order to perform these computations, we only need to know the induce/restrict matrices for the irreducible representations of the finite Coxeter group W and its parabolic subgroups, and these are readily available, for example via the system CHEVIE [14].

4.5 Examples in type E_{6}.

Let us illustrate the above algorithm by an example. We assume that (W, I) is of type E_{6}. All generators in I are conjugate, hence we can write v := v_{s} for all s ∈ I. Let w ∈ W be a good element of minimal length in some conjugacy class C of W. If C is a non-cuspidal class, the eigenvalues of T_{w} in any representation can be obtained inductively by restricting this representation to suitable proper parabolic sub-algebras of KH. So let us assume that C is one of the classes printed in the table in (3.7). If only one subset J_{i} occurs in the decomposition of w^{d} then the absolute values of the eigenvalues are already determined by Lemma 4.1. There is only one class left, A_{5} + A_{1}, for which we have d = 6 and

w^{6} = w_{I}^{2}w_{24}^{4}.

Table 1: Absolute values of eigenvalues for type E_{6}

ρ  Absolute values  ρ

Absolute values

(1, 0) v^{28} (1)  (24, 12) v^{10} (6), v^{12} (16), v^{14} (2)
(1, 36) v^{0} (1)  (30, 3) v^{16} (1), v^{18} (18), v^{20} (11)
(10, 9) v^{12} (1), v^{14} (8), v^{16} (1) (30, 15) v^{8} (11), v^{10} (18), v^{12} (1)
(6, 1) v^{22} (2), v^{24} (4)  (60, 8) v^{12} (8), v^{14} (44), v^{16} (8)
(6, 25) v^{4} (4), v^{6} (2)  (80, 7) v^{12} (12), v^{14} (56), v^{16} (12)
(20, 10) v^{12} (4), v^{14} (12), v^{16} (4) (90, 8) v^{12} (15), v^{14} (60), v^{16} (15)
(15, 5) v^{16} (1), v^{18} (8), v^{20} (6) (60, 5) v^{14} (4), v^{16} (42), v^{18} (14)
(15, 17) v^{8} (6), v^{10} (8), v^{12} (1) (60, 11) v^{10} (14), v^{12} (42), v^{14} (4)
(15, 4) v^{18} (10), v^{20} (5)  (64, 4) v^{15} (4), v^{17} (40), v^{19} (20)
(15, 16) v^{8} (5), v^{10} (10)  (64, 13) v^{9} (20), v^{11} (40), v^{13} (4)
(20, 2) v^{20} (10), v^{22} (10)  (81, 6) v^{40/3}(9), v^{46/3}(54), v^{52/3}(18)
(20, 20) v^{6} (10), v^{8} (10)  (81, 10) v^{32/3}(18), v^{38/3}(54), v^{44/3}(9)
(24, 6) v^{14} (2), v^{16} (16), v^{18} (6)

In Table 1 we give, for each irreducible representation ρ of KH, the list of absolute values of eigenvalues for this element w (with their multiplicities in brackets). The irreducible representation ρ is labeled by a pair (n, f) where n is the degree and f is minimal such that ρ appears as a constituent in the f-th symmetric power of the reflection representation. We see from this table that, indeed, fractional powers of the indeterminate v can occur.

Let’s explicitly work out the case where ρ is the unique irreducible representation of KH of degree 10 (the others being completely analogous). According to our above scheme we have three subsets: J_{1} = I ⊃ J_{2} = {s_{2}, s_{4}} ⊃ J_{3} = ∅, and the corresponding positive even integers d_{1} = 2, d_{2} = 4, d_{3} = 1. From the ordinary character table of W we compute that c_{ρ} = v^{36}. The parabolic subgroup W_{J_{2}} is of type A_{2}. It has two 1-dimensional irreducible representations, ind and sign, with monomials c_{ind} = v^{6} and c_{sgn} = 1, and a 2-dimensional irreducible representation, ref, with monomial c_{ref} = v^{3}.

In order to compute (ρ, w)-adapted chains, we have to consider the following 5 pairs: γ_{1} = (I, ρ), γ_{21} = (J_{2}, ind), γ_{22} = (J_{2}, sgn), γ_{23} = (J_{2}, ref), and γ_{3} = (J_{3}, 1). Using CHEVIE, we compute the following (ρ, w)-adapted chains:

γ_{1} ⟶^{1} γ_{21} ⟶^{1} γ_{3}, γ_{1} ⟶^{1} γ_{22} ⟶^{1} γ_{3}, γ_{1} ⟶^{4} γ_{23} ⟶^{2} γ_{3}.

So we find that the absolute values of the eigenvalues of ρ(T_{w}) are as follows:

v^{12}, v^{16} (once each), v^{14} (8 times).

Using the ordinary character table of W, we can compute that the eigenvalues of the representation of W obtained by specializing ρ are as follows: 1,1,-1,-1,-1,- 1,ζ_{6},ζ_{6}, ζ_{6}^{−1},ζ_{6}^{−1} (where ζ_{6} is a primitive 6-th root of unity). We conclude that the eigenvalues of ρ(T_{w}) itself are obtained as products of these 10 roots of unity with the above 10 monomials. But the problem is that we don’t know which root of unity has to be multiplied by which monomial. (From the table in [12], § 4, we see that the character value is −v^{16} + 2v^{14} − v^{12}, which implies that −v^{16}, −v^{12}, and the products of the eight remaining roots of unity with v^{14} are the correct eigenvalues.) So, if we want to compute the exact form of the eigenvalues (and not only their absolute values), we need to know more relations among the character values of HK. One source of such relations comes from the modular representation theory of Iwahori-Hecke algebras. This will be discussed in more detail in the next section.

5 On the computation of character tables

We keep the notation from the previous section, and let KH be the Iwahori-Hecke algebra associated with a finite Coxeter system (W, I). The character table of KH is defined to be the (square) matrix of character values (χ(T_{w_{C}})), where χ runs over the irreducible characters of KH and w_{C} runs over a set of representatives of minimal length in the various conjugacy classes C of W (see [15]). Since T_{x} and T_{y} are conjugate in KH for x, y ∈ C_{min} ([loc. cit.], Theorem 1.1(b)), we may assume, by Theorem 1.1, that each w_{C} is a good element. In the previous section, we have described an algorithm to compute the absolute values of the eigenvalues of ρ(T_{w_{C}}), for any irreducible representation ρ of KH. It is the purpose of this section to develop more techniques, derived especially from the modular representation theory of Iwahori-Hecke algebras, to actually find the trace of ρ(T_{w_{C}}). Our main application will be the computation of the character table of the Iwahori-Hecke algebra of type E_{8}. (We can also re-compute the tables for E_{6}, E_{7} which were already determined before in [12, 13]). With this application in mind, and for technical simplicity, we will assume throughout this section that all generators in I are conjugate in W. Then the parameters for KH are all equal, and we shall write v := v_{s} (s ∈ I).

5.1 First reductions.

Let us fix a class C in W, a good element w = w_{C} ∈ C_{min} of order d say, and an irreducible representation ρ : KH → K^{m×m} with character χ. From Proposition 4.4 we know the absolute values of the eigenvalues of ρ(T_{w}). Collecting terms with the same exponent in v, we can write

χ(T_{w}) = ∑_{i=1}^{k} a_{i}v^{n_{i}}

where the a_{i} are sums of d-th roots of unity over Q I and the n_{i} are (known) rational numbers. The problem that we have to solve is the determination of a_{1}, …, a_{k}.

There are some straightforward reductions of our problem.

(a) Restriction to parabolic sub-algebras. Assume that the class C is noncuspidal and that w ∈ W is contained in the parabolic subgroup W_{J} ⊂ W corresponding to the proper subset J ⊂ I. Assume also that the character table of the corresponding sub-algebra KH_{J} ⊂ KH is known (which we may, by an inductive hypothesis). Then, clearly, we have

χ(T_{w}) = ∑_{χ′} m_{χ,χ′}χ′(T_{w}),

where the sum is over all irreducible characters χ′ of KH_{J} and m_{χ,χ′} denotes the multiplicity of χ′ in the restriction of χ. As already remarked in (4.3), these multiplicities are the same for the corresponding specialized characters of the Coxeter group W, and are readily available.

(b) Integrality of character values. It is well-known that χ(T_{w}) ∈ ZZ[v^{2}] unless χ is one the two characters of degree 512 in type E_{7} or one of the four characters of degree 4096 in type E_{8}, in which cases we have χ(T_{w}) ∈ ZZ[v]. If χ is not one of these exceptional characters, we call χ a rational character. This implies that the terms a_{i}v^{n_{i}}, where n_{i}/2 (if χ is rational) respectively n_{i} (if χ is exceptional) is not an integer, must cancel out in the sum formula for χ(T_{w}). Hence all these terms need not be considered. From now on, we will assume that these terms are deleted from the sum formula for χ(T_{w}).

(c) Specialization to character values of W. As already remarked in § 4, specializing v ↦ 1 yields a bijection between the irreducible characters of KH and those of the Coxeter group W itself. Thus, we have a relation

∑_{i=1}^{k} a_{i} = χ(w).

The character tables of all finite Coxeter groups are known and contained in the CHEVIE system [14], for example. There is a particular special case where this already determines the character value χ(T_{w}). Assume that only the subset J_{1} = I occurs in the decomposition of w^{d}, that is, we have w^{d} = w_{I}^{d_{1}} for some d_{1} ≥ 0. Then k = 1, χ(T_{w}) = a_{1}v^{n_{1}}, and we conclude that

χ(T_{w}) = χ(w)c_{ρ}^{d_{1}/d} with c_{ρ} as in Lemma 4.1.

(d) Alvis-Curtis duality. Let χ′ be the Alvis-Curtis dual of χ. This is the character of KH obtained from χ by composition with the unique algebra automorphism of KH which maps T_{s} to −v^{2}T_{s}^{−1} for all s ∈ I. The values of χ′ can be computed from those of χ by the rule

χ′(T_{w}) = (−v^{2})^{l(w)}\overline{χ(T_{w})}, where \overline{f(v)} = f(1/v) for any f ∈ A.

Thus, if {χ, χ′} is a pair of dual characters, then it is sufficient to compute the character values only for one of them. If χ is self-dual (that is, χ = χ′), we obtain the relation

∑_{i=1}^{k} a_{i}v^{n_{i}} = ∑_{i=1}^{k}(−1)^{l(w)}a_{i}v^{2l(w)−n_{i}}.

(References for all of the above facts can be found in [13], Kapitel 2.)

Finally, there are some rules applying to special types of characters.

(e) Galois conjugation. Assume that χ is one of the non-rational characters in type E_{7} or E_{8}. In type E_{7}, the other non-rational character is obtained from χ by Curtis-Alvis duality, hence we can apply rule (d) above. In type E_{8}, we have to apply both Curtis-Alvis duality and the Galois conjugation A → A, v ↦ −v, in order to get the character values of the three other non-rational characters (once those for χ are known).

(f) Exterior powers of the reflection representation. Assume that χ is the character of the reflection representation of H_{K}, of degree |I|. Then, by [8], we explicitly know the matrices ρ(T_{s}), from which we can compute the matrix ρ(T_{w}). Moreover, it is shown in [8] that HK acts on the various exterior powers of ρ, yielding irreducible characters χ^{(i)} (0 ≤ i ≤ |I|) with values given by

χ^{(i)}(T_{w}) = v^{2(1−i)l(w)}Trace(ρ(T_{w})^{∧i}).

These include, in particular, the representation ind : T_{s} ↦ v^{2} (s ∈ I) for i = 0, and the representation sign : T_{s} ↦ −1 (s ∈ I) for i = |I|.

In general, the relations in (b)–(f) are not sufficient to determine χ(T_{w}) (for w in a cuspidal class of W). As an example, we should note that in type E_{8}, even after applying rule (b) above, we may still have several hundred unknowns a_{i} taken altogether for various χ (the maximum is 538 for the class D_{4} + 4A_{1}) and rules (c)–(f) give less than 300 relations. So we need a lot of additional relations.

5.2 Relations between several characters.

The relations between character values that we are now going to describe will involve several irreducible characters χ of HK. Therefore, we will refine the notation in (5.1) and write

χ(T_{w}) = ∑_{i=1}^{k_{χ}} a_{χ,i}v^{n_{χ,i}} ∈ ZZ[v],

where k_{χ}, a_{χ,i}, and n_{χ,i} have the same meaning as above.

(a) Orthogonality with generic degrees. The first such rule is obtained as follows. Let P = ∑_{w∈W} v^{2l(w)} ∈ ZZ[v] be the Poincaré polynomial of W, and D_{χ} ∈ Q I[v] be the generic degree associated with χ (see [2]). Then we have the relation

∑_{χ} D_{χ}χ(T_{w}) = \{ \begin{matrix}0 & for 1 ≠ w ∈ W \\ P & for w = 1. \end{matrix}

The generic degrees are known in all cases, and can be found in the tables in [7], for example.

(b) Φ_{e}-modular specialization. The second such set of rules comes from the modular representation theory of Iwahori-Hecke algebras. Let e ≥ 1 and K_{e} ⊇ Q I be the field extension generated by the 2e-th roots of unity. Then we have a specialization g_{e} : A → K_{e} which maps v ↦ ζ_{2e}, where ζ_{2e} = exp(πi/e) ∈ K_{e} is a primitive 2e-th root of unity. The corresponding specialized algebra will be denoted by K_{e}H. If P (ζ_{2e}) = 0 the algebra K_{e}H is not semisimple. Hence the specialization g_{e} induces a non-trivial homomorphism d_{e} between the Grothendieck groups of representations of KH and K_{e}H, which is called the Φ_{e}-modular decomposition map in [11]. We think of this map in terms of the corresponding decomposition matrix D_{e} defined as follows. Let χ be the character of an irreducible representation of KH, and g_{e}(χ) be the central function on K_{e}H defined by T_{w} ↦ g_{e}(χ(T_{w})) (w ∈ W). Then g_{e}(χ) is the character of the corresponding ‘specialized’ representation of K_{e}H (not necessarily irreducible). Hence we can write

g_{e}(χ) = ∑_{ψ} d_{χ,ψ}ψ,

where ψ runs over the irreducible characters of K_{e}H and the d_{χ,ψ} are non-negative integers. The matrix D_{e} := (d_{χ,ψ})_{χ,ψ} is the Φ_{e}-modular decomposition matrix of H. By [11], Proposition 5.4, the algebra K_{e}H is split (that is, every irreducible representation is absolutely irreducible) and the elementary divisors of D_{e} are all 1.

Let D_{e} have n rows corresponding to the irreducible characters χ_{1}, …, χ_{n} of KH, and l columns corresponding to the irreducible characters ψ_{1}, …, ψ_{l} of K_{e}H. Then matrix multiplication defines a ZZ-linear map ZZ^{n} → ZZ^{l}, x ↦ xD_{e} (where x is a row vector). Since the matrix D_{e} has full rank, the kernel of this map is a free ZZ-module of rank n − l. Let x = (x_{1}, …, x_{n}) be any element in this kernel. Then the corresponding linear combination of the specialized characters g_{e}(χ_{j}) is identically zero. Writing this out in terms of the above sum formula for χ(T_{w}) yields a relation

∑_{j=1}^{n} ∑_{i=1}^{k_{j}} x_{j}a_{ji}ζ_{2e}^{n_{j,i}} = 0,

where we have written k_{j} = k_{χ_{j}}, a_{ji} = a_{χ_{j},i} and n_{j,i} = n_{χ_{j},i} for all i, j. If we let x vary over a basis of the kernel, we obtain a system of n − l linear equations for the unknowns a_{j,i}.

It is now important to remark that, in order to obtain this system of equations, it is actually not necessary to know the exact entries of the decomposition matrix D_{e}. All we need to know is the kernel of the map x ↦ xD_{e}, and this does not change if we replace D_{e} by D_{e}D′ where D′ ∈ ZZ^{l×l} is any matrix with determinant ±1. On the level of characters of K_{e}H, this amounts to replacing the set of irreducible characters {ψ_{1}, …, ψ_{l}} by a set of central functions {β_{1}, …, β_{l}} on K_{e}H where the transition matrix between these two sets of central functions has integer entries and determinant ±1. Such a set of central functions on K_{e}H is usually called a basic set of characters. Thus, what we really need in order to obtain a linear system of equations as above is a decomposition matrix of HK with respect to some basic set of characters of K_{e}H.

5.3 Methods for finding Φ_{e}-modular relations.

We now describe two methods for finding vectors x ∈ ZZ^{n} in the kernel of the map x ↦ xD_{e} (where D_{e} is the Φ_{e}-modular decomposition matrix of KH as before).

(a) Induction from parabolic subalgebras. Let J ⊂ I be any proper subset and KH_{J} the corresponding parabolic sub-algebra of KH. We denote by ind_{J}^{I} the operation of induction of representations from KH_{J} to KH. Since this operation commutes with specialization (see [12], (1.5)), relations mod Φ_{2e} among the characters of KH_{J} ‘induce’ to relations among the characters of KH. Let χ′_{1}, …, χ′_{n′} be the irreducible characters of KH_{J}, and denote by m_{χ,χ′} the induce/restrict coefficients as in (5.1)(a). If x = (x_{1}, …, x_{n′}) ∈ ZZ^{n′} gives a relation

(∗)  ∑_{i} x_{i}g_{e}(χ′_{i}) = 0

then we can induce this to KH, use the commutation of induction and specialization, decompose into irreducibles, and obtain a relation for the characters of KH:

∑_{i=1}^{n′} ∑_{j=1}^{n} x_{i}m_{χ′_{i},χ_{j}}g_{e}(χ_{j}) = 0.

By an inductive hypothesis, we can assume that the character table of KH_{J} is already known. Then, of course, relations of the form (∗) can be computed directly using the specialized character table, without first trying to determine the decomposition matrix of KH_{J}. – In our applications and examples, we will consider all proper maximal parabolic sub-algebras of KH, compute the relations for their specialized character tables, and induce them to KH. In general, this method will not give a full basis for the kernel, nor will it give sufficiently many relations to determine the character values.

(b) Brauer trees. The minimal two-sided ideals in the specialized algebra K_{e}H are called blocks. In [11], we have associated with every block a numerical invariant, its defect, and we have shown that blocks of defect 1 are so-called Brauer tree algebras. In particular, the decomposition matrix of a block of defect 1 can be encoded in terms of the associated Brauer tree which, in the case of Iwahori-Hecke algebras, always is a straight line. Each such tree yields exactly one non-trivial Φ_{e}-modular relation. Suppose that {χ_{1}, …, χ_{a}} are the irreducible characters of KH belonging to a block of defect 1, lined up on the corresponding tree. Then we have the relation

∑_{i=1}^{a}(−1)^{i}g_{e}(χ_{i}) = 0.

For type E_{6}, E_{7}, E_{8} and the cases where Φ_{2e} divides the Poincaré polynomial to the first power only, all trees have been determined in [11], § 12, without knowing anything about the character values of KH. (Caution. In [11], the bijection between the irreducible characters of KH and of W was chosen to be compatible with the specialization v ↦ −1, while here we use v ↦ 1. In order to match up the two labellings, one simply has to interchange the two characters of degree 512 in type E_{7}, and the two characters in each of the two Galois conjugate pairs {(4096, 11), (4096, 12)} and {(4096, 26), (4096, 27)} for type E_{8}.).

5.4 Continuation of the examples in type E_{6}.

We return to the example in (4.5) and keep the notation introduced there. We have seen that, for the chosen good element w in the class A_{5} + A_{1}, at most three different powers of v are involved in the eigenvalues of the corresponding representing matrices ρ(T_{w}). Hence we have

χ(T_{w}) = a_{χ,1}v^{n_{χ,1}} + a_{χ,2}v^{n_{χ,2}} + a_{χ,3}v^{n_{χ,3}},

where the n_{χ,i} are known and a_{χ,i} are the unknown integers that we are looking for. (We use the convention that, if there is only one entry in the table of (4.5) we let a_{χ,1} = a_{χ,3} = 0, and if there are only two entries in that table we let a_{χ,2} = 0.) If we label the irreducible characters of KH by χ_{j} (1 ≤ j ≤ 25) in the order specified by the table in (4.5), and write a_{j,i} = a_{χ_{j},i} (1 ≤ i ≤ 3) etc., we have a 25×3-matrix of unknowns a_{j,i}.

Now we first apply the rules in (5.1) to eliminate some unknowns.

First we note that all characters of KH are rational. From the table in (4.5) we see that the eigenvalues of the two characters of degree 64 involve odd powers of v, hence the corresponding character values on T_{w} must be zero. Furthermore, the eigenvalues for the character of degree 81 even involve fractional powers of v, hence the values are also zero. In other words, we have a_{j,1} = a_{j,2} = a_{j,3} = 0 for j = 22, 23, 24, 26.

Now we specialize v ↦ 1. Then the values χ_{j}(w) of the corresponding specialized characters of W are known and given as follows: 1, 1, 0, -2, -2, -2, 2, 2, 1, 1, 1, 1, -1, -1, -1, -1, 0, 2, 0, -1, -1, 0, 0, 0, 0. The sum of the entries in the j-th row of our matrix of unknowns must be equal to the value at the j-th position in this list. Thus, we can eliminate the third entry in each row (say) by writing

a_{j,3} = χ_{j}(w) − a_{j,1} − a_{j,2} for j = 1, …, 25.

The next step is to use Curtis-Alvis duality. We have l(w) = 14. The characters with labels (10, 9) (j = 3), (20, 10) (j = 6), (60, 8) (j = 17), (80, 7) (j = 18) and (90, 8) (j = 19) are self-dual. From table (4.5) we see that, in each case, we have

χ_{j}(T_{w}) = a_{j,1}v^{12} + a_{j,2}v^{14} + a_{j,3}v^{16},

hence rule (5.1)(d) yields that we must have a_{j,1} = a_{j,3} for j = 3, 6, 17, 18, 19. Similarly, we see that a_{j,2} = a_{j′,2}, a_{j,1} = a_{j′,3} and a_{j,3} = a_{j′,1} if {χ_{j}, χ_{j′}} is a pair of dual characters. (This works even in the cases where the table in (4.5) contains only two entries because of our above convention about the unknowns a_{j,i}.)

Finally, we can compute the character values of the exterior powers of the reflection representations. They are given in Table 2 (where the i-th row corresponds to the (i − 1)-th exterior power). We shall now use the Brauer trees of [11],

Table 2: Example E_{6}

j pair (n, f) value a_{j,1} a_{j,2} a_{j,3}
1  (1, 0)  v^{28}  0  1  0
4  (6, 1)  −2v^{22} −2  0  0
7 (15, 5) v^{16} + v^{20}  1  0  1
6 (20, 10)  −2v^{14}  0 −2  0
8 (15, 17) v^{8} + v^{12}  1  0  1
5 (6, 25)  −2v^{6}  0  0  1
2 (1, 36)  1  0  1  0

Theorem 12.5, to obtain more relations, according to the rule in (5.3)(b). In the following, we simply denote an irreducible character of KH by its label (n, f). The first tree in [loc. cit.] for e = 5 yields the relation

(1, 0) − (24, 6) + (81, 10) − (64, 13) + (6, 25) ≡ 0 mod Φ_{10}.

We know already that the values of (1, 0), (6, 25), (81, 10) on T_{w} are v^{28}, −2v^{6}, 0 respectively. Let the value of (24, 6) at T_{w} be a_{1}v^{14} + a_{2}v^{16} + a_{3}c^{18} (cf. the table in (4.5)). If we now denote a primitive 10-th root of unity by v̄, we arrive at the following relation:

a_{1}v̄^{14} + a_{2}v̄^{16} + a_{3}v̄^{18} = v̄^{28} − 2v̄^{6} (equality in K_{5}).

We have Φ_{10} = v^{4} − v^{3} + v^{2} − v + 1, so we can choose 1, v̄, v̄^{2}, v̄^{3} as a Q I-basis of K_{5}. Reducing the exponents in the above equation now yields

−a_{1} + (a_{1} − 2 − a_{2})v̄ − a_{1}v̄^{2} + (a_{1} − a_{3} + 1)v̄^{3} = 0

and hence a_{1} = 0, a_{2} = −2, a_{3} = 1. Applying also rule (5.1)(d) about dual characters, we can deduce that

χ_{(24,6)}(T_{w}) = −2v^{16} + v^{18} and χ_{(24,12)}(T_{w}) = v^{10} − 2v^{12}.

The tree in [loc. cit.] for e = 8 yields the relation

(1, 0) − (30, 3) + (81, 6) − (81, 10) + (30, 15) − (1, 36) ≡ 0 mod Φ_{16}.

Let the value of (30, 3) at T_{w} be a_{1}v^{16} + a_{2}v^{18} + a_{3}c^{20}. Then the value of (30, 15), which is dual to (30, 3), is given by a_{1}v^{12} + a_{2}v^{10} + a_{3}v^{8} (see rule (5.1)(d)). If we now denote a primitive 16-th root of unity by v̄, we arrive at the following relation:

(1 + a_{1} + a_{3}) + 2a_{2}v̄^{2} + (1 + a_{1} + a_{3})v̄^{2} = 0.

Hence we have a_{2} = 0, 1 + a_{1} + a_{3} = 0. (Rule (5.1)(c) does not give any new information.) The tree in [loc. cit.] for e = 9 yields the relation

(1, 0) − (20, 2) + (64, 4) − (90, 8) + (64, 13) − (20, 20) + (1, 36) ≡ 0 mod Φ_{18}.

We know the values of (1, 0), (1, 36), (64, 4), (64, 13). Arguing similarly as before, we can deduce that the values of (20, 2), (20, 20) and (90, 8) at T_{w} are

2v^{20} − v^{22}, −v^{6} + 2v^{8}, −v^{12} + 2v^{14} − v^{16} respectively.

In order to solve the remaining problems, we shall now use relations among the Φ_{6}-modular reductions of the irreducible characters of KH. Using the method described in (5.3)(a), applied to the parabolic sub-algebras of type D_{5} and A_{5}, we obtain the following three relations modulo Φ_{12}.

(1)  (1, 0) + (6, 25) + (15, 4) − 2(15, 16) + (20, 20) + 2(24, 6)
−2(30, 15) + (60, 8) − 2(60, 5) + (60, 11) ≡ 0,
(2)  (1, 36) + (6, 25) − (15, 16) + (20, 2) + (24, 6) + (24, 12)
−(30, 3) − (30, 15) + (60, 8) − (60, 5) ≡ 0,
(3)  (6, 1) − (6, 25) − 2(15, 4) + 2(15, 16) − (24, 6) + (24, 12)
−(30, 3) + (30, 15) + 2(60, 5) − 2(60, 11) ≡ 0.

We subtract the second relation from the first, multiply the result by 2 and then add the third relation; we obtain:

2(1, 0) − 2(1, 36) + (6, 1) − (6, 25) − 2(20, 2) + 2(20, 20) + (24, 6)
−(24, 12) + (30, 3) − (30, 15) ≡ 0 mod Φ_{12}.

All character values at T_{w} are known in this relation except for (30, 3) and (30, 15). Inserting the known values yields that the value of (30, 3) − (30, 15) at T_{w} must be ≡ (2 − v^{2}) mod Φ_{12}. Above, we have already found that the values of (30, 3) and (30, 15) at T_{w} are of the form a_{1}v16+(−1−a_{1})v^{20} respectively a_{1}v^{12} +(−1−a_{1})v^{8}. Reducing modulo Φ_{12} and comparing with the relations that we just found yields that a_{1} = −1, and hence

χ_{(30,3)}(T_{w}) = −v^{16} and χ_{(30,15)}(T_{w}) = −v^{12}.

As above we check, using the table in (4.5) and rules (5.1)(b,c), that we can write χ_{(15,4)}(T_{w}) = av^{18} + (1 − a)v^{20}, χ_{(15,16)}(T_{w}) = (1 − a)v^{8} + av^{10}, χ_{(60,8)}(T_{w}) = bv^{12} − 2bv^{14} + bv^{16}, and χ_{(60,5)}(T_{w}) = cv^{14} + dv^{16} + (−1 − c − d)v^{18}, χ_{(60,11)}(T_{w}) = (−1 − c − d)v^{10} + dv^{12} + cv^{14} for some a, b, c, d ∈ ZZ. Inserting this into the first two relations above, we find the following linear equations:

a + c = 1, 2a + b + d = 3 from (1), a + c = 1, b + c + d = −2 from (2).

We conclude that c = 1, a = 0, b + d = −3, and hence that

χ_{(15,4)}(T_{w}) = v^{20} and χ_{(15,16)}(T_{w}) = v^{8}.

To proceed, we also induce relations for e = 4 from a parabolic sub-algebra of type D_{5}. Among others, we find the following relation between characters of KH:

(20, 2) − (60, 5) + (60, 11) − (20, 20) ≡ 0 mod Φ_{8}.

Arguing as above we find the single relation d = −2. Combining with the above equations yields that b = −1, and hence the values of (60, 8), (60, 5) and (60, 11) at T_{w} are

−v^{12} + 2v^{14} − v^{16}, v^{14} − 2v^{16}, −2v^{12} + v^{14} respectively.

Now we have determined all character values at T_{w} except those for (10, 9) and (80, 7). Here, we use the relation involving the generic degrees for KH. Solving the corresponding system of equations finally yields that

χ_{(10,9)}(T_{w}) = −v^{12} + 2v^{14} − v^{16} χ_{(80,7)}(T_{w}) = v^{12} + v^{16}.

Thus, we have re-computed the character table for KH of type E_{6}. Originally, this table was found in [12, 13] using different methods.

5.5 The character tables of KH for type E_{7}, E_{8}.

In (5.4), we have followed the rules (5.1)–(5.3) step by step in the example of type E_{6}. For types E_{7} and E_{8} the number of unknowns and relations in each step is considerably larger but, in principle, the same methods can be applied. Using an implementation of the above rules (5.1)–(5.3) in the computer algebra system GAP [23] and CHEVIE [14] we were thus able to compute the character tables for the Iwahori-Hecke algebras of types E_{7} and E_{8}. For type E_{7} this table has already been computed in [13], § 3.4, using different methods (and we thus have a check for these results). For type E_{8} this table was not known before.

In summary we can state that, in order to determine the character tables of the Iwahori-Hecke algebras of type E_{6}, E_{7}, E_{8}, we first use an inductive procedure to compute the values of the irreducible characters on basis elements T_{w} where w ∈ W has minimal length in a non-cuspidal class of W. If w ∈ W has minimal length in a cuspidal class, we basically use four types of ‘new’ critical information:

Table 3: Character values for type E_{8}

Cusp. class  Character value of χ_{4096,11}
8A_{1}  −4096v^{135}
2D_{4}(a_{1})  0
D_{4} + 4A_{1}  −60v^{65} + 10v^{69} − 84v^{73} + 64v^{75} + 6v^{85}
4A_{2}  64v^{90}
E_{8}(a_{8})  −64v^{45}
E_{7}(a_{4}) + A_{1}  −16v^{45} + 48v^{47} − 24v^{49}
2D_{4}  −4v^{45} + 24v^{47} − 44v^{49} + 40v^{51} − 8v^{53}
2A_{3} + 2A_{1}  0
D_{8}(a_{3})  0
D_{6} + 2A_{1}  v^{27} + 6v^{35} − 4v^{39} + v^{43}
2A_{4}  −4v^{54}
E_{8}(a_{6})  4v^{27}
E_{6}(a_{2}) + A_{2}  8v^{47} − 20v^{49} + 16v^{51} − 4v^{53}
E_{8}(a_{3})  0
A_{5} + A_{2} + A_{1} 2v^{47} − 9v^{49} + 20v^{51} − 22v^{53} + 10v^{55} − v^{57}
D_{8}(a_{1})  0
D_{8}  −v^{21}
A_{7} + A_{1}  0
E_{7} + A_{1}  −v^{15}
A_{8}  v^{30}
E_{8}(a_{4})  −v^{15}
E_{8}(a_{2})  0
D_{5}(a_{1}) + A_{3}  2v^{51} − 4v^{53} + 2v^{55}
E_{6} + A_{2}  0
E_{8}(a_{7})  0
E_{7}(a_{2}) + A_{1}  0
E_{8}(a_{1})  0
D_{8}(a_{2})  v^{27} − v^{29} + v^{31}
E_{8}(a_{5})  −v^{18}
E_{8}  v^{9}

(1) The ordinary character table of W plus induce/restrict matrices for parabolic subgroups. (This is used in (4.3), and in rules (5.1)(a,c), (5.3)(a).)

(2) Good elements in the cuspidal classes of W. (This is used in (4.3).)

(3) The generic degrees for the irreducible characters of KH. (This is used in rule (5.2)(a).)

(4) The Brauer trees for all Φ_{e}-blocks of KH of defect 1, in the cases where Φ_{2e} divides the Poincaré polynomial to first power. (This is used as in (5.3)(b).)

(Note that all the other rules, like (5.1)(c,f), use information which can be computed directly from basic combinatorial or group theoretical data about (W, I).) It is intriguing to speculate whether this is always sufficient to determine the character table for any given type of Iwahori-Hecke algebra. (This can be checked for the two-parameter algebra of type F_{4}, for example.)

For type E_{8}, we give in Table 3 the values for the character (4096, 11) on T_{w} for good elements w in the cuspidal classes in (3.9). (For type E_{7}, the values of one of the non-rational characters of degree 512 are given in [12].)

References

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[21] G. Pfeiffer, Character values of Iwahori-Hecke algebras of type B, to appear in Progress in Math., Birkhäuser.

[22] A. Ram, A Frobenius formula for the characters of the Hecke algebras, Invent. Math. 106 (1991), 461–488.

[23] M. Schönert et al., GAP – Groups, Algorithms, and Programming. Lehrstuhl D für Mathematik, RWTH Aachen, Germany, fourth ed., (1994).

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