2003/02/12 by Susumu Ariki, Ariki, Susumu
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.QA #math.RT
paper · pdf · doi:10.48550/arxiv.math/0302136
57 pages, Appendix (errata to my book) expanded, (case 5a) corrected
openalex publication_date 2003/02/12 · arxiv created 2004/07/26 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let W be a finite Weyl group of classical type which may not be irreducible, F an algebraically closed field, q an invertible element of F. We denote by \mathcal HW(q) the associated Hecke algebra. If q=1 then it is FW and we know the representation type. Thus, we assume that q≠ 1. Let PW(x) be the Poincare polynomial of W. It is well-known that \mathcal HW(q) is semisimple if and only if x-q does not divide PW(x). We show that the similar results hold for finiteness, tameness and wildness. In other words, the Poincare polynomial governs the representation type of \mathcal HW(q) completely. Note that the finiteness result was already given in the author's previous papers, some of which were written with Andrew Mathas. The proof uses the Fock space theory, which was developed for proving the LLT conjecture (see AMS Univ. Lec. Ser. 26), the Specht module theory, which was developed by Dipper, James and Murphy in this case, and results from the theory of finite dimensional algebras.