# Representations of finite Coxeter groups

What is reference for complex irreducible representations of Hecke algebra of finite Coxeter groups (say generic case q =1)? I am interested in knowing its Wedderburn decomposition. So want explicit information regarding the number of irreducible representations (and parameterization, if any) with their multiplicities. I looked at Kazhdan-Lusztig's paper but not able to get required information.

For example: In case of symmetric group $S_n$, these representations are parametrized by all partitions of $n$ with multiplicities equal to number of standard tableaux.

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Please, clarify what kind of information you need and what you mean by "their multiplicities". – Victor Protsak Jul 14 '10 at 16:57
For $q=1$ the Hecke algebra by definition coincides with the group algebra of the corresponding Coxeter group $\mathbb{C}[W]$, so it is in no way "generic" case. For non-zero $q$ that isn't the root of unity, the algebra is semisimple and canonically isomorphic to $\mathbb{C}[W].$ Therefore, multiplicity=dimension is the same as for $W$. Is that what you were looking for? NB: "Generic" implies that $q$ is treated as a parameter and the Hecke algebra is considered over the ring $\mathbb{C}[q,q^{-1}]$ or its extension. – Victor Protsak Jul 16 '10 at 4:38
Oh ok I am really sorry then for wrong use of word 'generic'. Yeah as a first case I want to know about representations of C[W] for any coxeter group W. Would you please comment on that? I shall be greatly thankful for that. My apologies for my little knowledge in this area. – Pooja Jul 16 '10 at 13:16
@Victor: I don't understand why the iso. between CW and the Hecke algebra at non-zero q that isn't a root of unity should be canonical at all (though I believe that specific iso's have been written down). In fact it seems reasonable that there should be one such iso. for each homotopy class of paths from 1 to q (in the space of q for which the Hecke algebra is semisimple). – Sheikraisinrollbank Jan 31 '11 at 10:22

There are many relevant papers, but the most convenient book to consult is: MR1778802 (2002k:20017) 20C15 (20C08 20F55), Geck, Meinolf (F-LYON-GD); Pfeiffer,G¨otz (IRL-GLWY) Characters of finite Coxeter groups and Iwahori-Hecke algebras. London Mathematical Society Monographs. New Series, 21. The Clarendon Press, Oxford University Press, New York, 2000. xvi+446 pp.

See also their earlier paper: MR1250466 (94m:20018) 20C15, Geck, Meinolf (D-AACH-DM); Pfeiffer,G¨otz (D-AACH-DM), On the irreducible characters of Hecke algebras. Adv. Math. 102 (1993), no. 1, 79–94.

In the last chapter of my book Reflection Groups and Coxeter Groups (Cambridge, 1990) there is a brief summary of earlier work done on irreducible representations or character tables of finite Coxeter groups and their Iwahori-Hecke algebras. The 1979 Kazhdan-Lusztig paper was partly motivated by earlier papers of people like Iwahori and Curtis, but especially by Springer's theory of Weyl group representations on cohomology of flag varieties. The powerful "cell" construction by Kazhdan-Lusztig does not in general give explicitly the irreducible representations, however. Carter's 1985 book on characters of finite groups of Lie type discusses how all of this feeds into that kind of representation theory. Much of the progress has been due to Lusztig.

It's important to distinguish between finite crystallographic Coxeter groups (Weyl groups) and the remaining dihedral groups along with exceptions $H_3, H_4$. Even in the latter cases, much of the Weyl group theory has good analogues in spite of being outside the classical framework of groups of Lie type. In any case, whether the results in the literature are explicit enough for some purposes may be an open question. Certainly the case of symmetric groups and their Iwahori-Hecke algebras has been developed most concretely.

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Although I know several books treating representations of Iwahori-Hecke algebras from various points of view, I forgot about this one (as a weak excuse, I probably haven't read it). – Victor Protsak Jul 14 '10 at 16:55

More context is needed for this question. I am going to address the case of generic representations in the spherical case.

It is an old theorem of Lusztig that over the ring $\mathbb{C}[v,v^{-1}]$, the generic Hecke algebra $H_v$ with generators $T_i$ and relations

$$T_i T_j \ldots = T_j T_i \ldots \ (m_{ij} \text{ factors}), \qquad (T_i+v^2)(T_i-v^2)=0$$

is isomorphic to the group ring of the corresponding finite Coxeter group $W.$ Thus every representation of $W$ can be canonically deformed to a representation of $H_v.$ In the course of developing representation theory of reductive groups over a finite field, Lusztig developed quite a bit of machinery describing these representations (fake degrees, etc). This is described in his book

G. Lusztig, Characters of reductive groups over a finite field. Annals of Mathematics Studies, 107. Princeton University Press, Princeton, NJ, 1984

A more recent source, in a more general situation and with improved proofs, is

G. Lusztig, Hecke algebras with unequal parameters. CRM Monograph Series, 18. American Mathematical Society, Providence, RI, 2003

For specific non-zero $v$ that are not roots of unity, the same result holds. The case of roots of unity and of more general Hecke algebras (in particular, for Coxeter system of affine type) has also been studied.

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