Robinson--Schensted--Knuth insertion and characters of cyclotomic Hecke algebras
type: Item
instance of: scholarly article
title: Robinson--Schensted--Knuth insertion and characters of cyclotomic Hecke algebras
zbMATH Open document ID: 1016.05071
author: Andy Cantrell
author: Tom Halverson
author: Brian J. Miller
published in: Journal of Combinatorial Theory. Series A
publication date: 2003-03-26T00:00:00Z
review text: The cyclotomic Hecke algebras \(H_{n,r}=H_n(u_1,\ldots,u_r;q)\) are defined as Iwahori-Hecke algebras of the complex reflection groups \(G_{n,r}=S_n\wr ({\mathbb{Z}}/r{\mathbb{Z}})^n\). If \(\zeta\) is a primitive complex \(r\)th root of unity, then when \(q\to 1\) and \(u_i\to\zeta^i\), the algebra \(H_{n,r}\) specializes to the group algebra \({\mathbb{C}}[G_{n,r}]\). The irreducible representations of \(H_{n,r}\) are described in \textit{S. Ariki} and \textit{K. Koike} [Adv. Math. 106, 216-243 (1994; Zbl 0840.20007)] and are indexed by \(r\)-tuples of partitions with a total of \(n\) boxes, called \(r\)-partitions. In \textit{T. Shoji} [J. Algebra 226, 818-856 (2000; Zbl 0959.20007)] a symmetric function \(q_{\mu}\) was related with the \(r\)-partition \(\mu\) satisfying the condition \[ q_{\mu}=\sum_{\lambda}\chi_q^{\lambda}(a_{\mu})s_{\lambda}, \] where \(s_{\lambda}\) is the Schur function associated with the \(r\)-partition \(\lambda\) and \(\chi_q^{\lambda}(a_{\mu})\) is the irreducible \(H_{n,r}\)-character associated with \(\lambda\) and evaluated at an element \(a_{\mu}\). The symmetric function \(q_{\mu}\) is a deformation of the power sum symmetric function and the above formula is analogous to the Frobenius formula for the characters of \(S_n\). In the paper under review the authors derive the formula \[ q_{\mu}=\sum_{\lambda}\left(\sum_{Q_{\lambda}}\text{wt}_{\mu}(Q_{\lambda})\right)s_{\lambda}, \] where \(Q_{\lambda}\) ranges over the set of ``standard tableaux'' of shape \(\lambda\) and the weight \(\text{wt}_{\mu}\) depends on the parameters \(q\) and \(u_i\) and can be computed combinatorially. Comparing the coefficients of \(s_{\lambda}\), the authors obtain an expression which computes the irreducible \(H_{n,r}\)-characters as a sum over standard tableaux. For \(q=1\) and \(u_i=\zeta^i\) the formula gives a character formula for the complex reflection group \(G_{n,r}\). For the proof, the authors apply ideas from \textit{A. Ram} [in: Sagan, Bruc…
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 20C08
zbMATH DE Number: 1888397
zbMATH Keywords: Robinson-Schensted-Knuth insertion
zbMATH Keywords: cyclotomic Hecke algebras
zbMATH Keywords: characters
zbMATH Keywords: symmetric functions
zbMATH Keywords: partitions
zbMATH Keywords: tableaux
MaRDI profile type: Publication
full work available at URL: jcta.2002.3252
OpenAlex ID: W2056657952
cites work: A Hecke algebra of \((\mathbb{Z}/r\mathbb{Z})\wr{\mathfrak S}_ n\) and construction of its irreducible representations
cites work: Murnaghan-Nakayama Rules for Characters of Iwahori-Hecke Algebras of the Complex Reflection Groups G(r, p, n)
cites work: Schur-Weyl reciprocity for Ariki-Koike algebras
cites work: A Frobenius formula for the characters of Ariki-Koike algebras
cites work: Q3947818
cites work: A Frobenius formula for the characters of the Hecke algebras
cites work: Q4400937
cites work: A recursive rule for Kazhdan-Lusztig characters
cites work: Q4001520
DOI: 10.1006/JCTA.2002.3252
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description: scientific article; zbMATH DE number 1888397