weyl group representations

I am looking for some references about irreducible representations of the Weyl Group over simple Lie Groups, both classical and exceptional ones. In particular I want to know the dimensions and the character tables of the irreps.

I heard of those Mackey theory for a while (which should be helpful for the Weyl group over classical simple Lie Groups), but I failed to find some nice references for it. It will be very helpful if there is a nice account on the theory.

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Section 8.10 of James Humphreys' book "Reflection groups and Coxeter groups" lists a number of references for the representations of Weyl groups (and, more generally, finite reflection groups). –  Christopher Drupieski Apr 8 '11 at 21:10
You can find a lot of information in the book "Characters of finite Coxeter groups and Iwahori-Hecke algebras" by Geck and Pfeiffer. –  Victor Ostrik Apr 8 '11 at 22:43
Besides Section 8.10 of my book, realizations of the exceptional reflection groups in Sections 2.12-2.13 include references to relevant character tables in the Atlas of Finite Simple Groups. For types A, B/C, D, combinatorial treatments starting with symmetric groups are optimal and even give some closed formulas for dimensions, but for other types of rank >2 it's hard to construct explicit matrix representations. Anyway, these approaches require case-by-case treatment. Springer theory unifies Weyl group representations but doesn't give more information about characters. –  Jim Humphreys Apr 9 '11 at 12:30

2 Answers

Let me add a useful reference book, probably no longer in print but found in many libraries: R.W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Wiley-Interscience, 1985. The book includes a lot of information about the irreducible representations of Weyl groups (though not with complete proofs) in the context of characters of finite groups of Lie type, unipotent classes in simple algebraic groups, Springer correspondence, etc. Chapters 11 and 13 involve the Weyl groups most heavily. (Though the book by Chriss and Ginzburg goes much deeper into the geometry of the Springer correspondence, the treatment there is mostly limited to the case of symmetric groups which is often more straightforward than the general case; for this Carter gives a helpful overview.)

For example, the $25 \times 25$ character table of the Weyl group of type $F_4$ originally worked out by T. Kondo in a 1965 journal article is displayed on page 413 of the book, while the role of these characters in the Springer correspondence is summarized on page 428 (involving the previous study of unipotent classes for the simple algebraic group of type $F_4$). Like all character tables of Weyl groups, this one has entries in $\mathbb{Z}$. It's not at all easy in a case like $F_4$ to write down explicit integral matrices affording the irreducible representations, but fortunately the characters alone are sufficient for some applications like those developed by Carter. One caveat is that notation for conjugacy classes and characters differs in various sources.

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Thanks a lot. The book is very helpful to me. I have some knowledge on the Springer Correspondence, but I never find a book which writes down the dimemsions and character tables clearly. –  wky Apr 10 '11 at 21:19

I'm not sure I know what you mean by Mackey theory or how it relates to the representation theory of Weyl groups. I guess you could mean Mackey's approach to the representation theory of semidirect products and group extensions, which would certainly be helpful for a case-by-case approach to the problem. For a more unified (and geometric) approach, there's Springer theory (a.k.a. the Springer correspondence). The prerequisites for understanding this theory are somewhat sophisticated; for a good entry point, you could try section 3.6 in Chriss and Ginzburg, Representation Theory and Complex Geometry.

If you just want to get character tables and explicit numerical data, then using something like GAP would probably be your best bet.

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