Invariants for representations of Weyl groups, two-sided cells, and modular representations of Iwahori-Hecke algebras
type: Item
instance of: scholarly article
zbMATH Open document ID: 1005.20031
author: Akihiko Gyoja
author: Kyo Nishiyama
author: Kenji Taniguchi
publication date: 2003-02-19T00:00:00Z
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20C08
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 1642284
zbMATH Keywords: Young diagrams
zbMATH Keywords: Schur functions
zbMATH Keywords: \(q\)-series identities
zbMATH Keywords: symmetric groups
zbMATH Keywords: Weyl groups
zbMATH Keywords: reflection groups
zbMATH Keywords: hook lengths
zbMATH Keywords: Kawanaka invariants
zbMATH Keywords: Littlewood-Richardson coefficients
zbMATH Keywords: combinatorics of partitions
zbMATH Keywords: Hecke algebras
zbMATH Keywords: Kazhdan-Lusztig two-sided cells
MaRDI profile type: Publication
title: Invariants for representations of Weyl groups, two-sided cells, and modular representations of Iwahori-Hecke algebras
review text: Let \(\lambda\) be the shape of a Young diagram with corresponding Schur function \(s_\lambda(x)\) in infinite variables, and denote by \(h(v)\) the hook length of \(\lambda\) at \(v\). In [Osaka J. Math. 36, No. 1, 157-176 (1999; Zbl 0915.05008)], \textit{N. Kawanaka} obtained a \(q\)-series identity for \(\sum_\lambda I_\lambda(q)s_\lambda(x)\), where the sum is over all Young diagrams \(\lambda\) and NEWLINE\[NEWLINEI_\lambda(q)=\prod_{v\in\lambda}\tfrac{1+q^{h(v)}}{1-q^{h(v)}}.NEWLINE\]NEWLINE Kawanaka used this identity to find an expression for \(I_\lambda(q)\) in terms of the characters of the symmetric group and the representation of the group by permutation matrices.NEWLINENEWLINENEWLINEDefinition 1.1 of this paper defines a \(q\)-series \(I_W(\chi;q)\) for a Weyl group \(W\) acting on a complex vector space as a reflection group, where \(\chi\) is a character of a finite-dimensional representation \(\pi\) of \(W\). The authors call this the Kawanaka invariant of \(\pi\); it generalizes the aforementioned expression for \(I_\lambda(q)\) to the case of an arbitrary Weyl group.NEWLINENEWLINENEWLINEThe authors establish, in Theorem 2.1, an expression for the Kawanaka invariant in type \(B\) in terms of hook lengths. They conjecture a more complicated closed formula for the Kawanaka invariants in type \(D\) in Conjecture 3.2, and they establish in Theorem 2.2 an expression for these invariants in terms of Littlewood-Richardson coefficients. In Theorem 3.5, a recursive formula for the type \(D\) invariants is given in terms of the combinatorics of partitions. The exceptional and dihedral cases of the invariants have also been explicitly calculated (\S 2.4).NEWLINENEWLINENEWLINEIn \S 4, the authors define an equivalence relation on the characters of the Weyl group (or Hecke algebra). The resulting equivalence classes, which give a more refined partition than that induced by the usual Kazhdan-Lusztig two-sided cells, are called refined two-sided cells. The authors…
reviewed by: R. M. Green
description: scientific article; zbMATH DE number 1642284