Robinson--Schensted--Knuth insertion and characters of cyclotomic Hecke algebras
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, Bruce E. (ed.) et al., Mathematical essays in honor of Gian-Carlo Rota's 65th birthday. Boston, MA: Birkhäuser. Prog. Math. 161, 335-342 (1998; Zbl 0905.05082)] and use a version for \(r\)-partitions of the Robinson-Schensted-Knuth insertion.
scientific article; zbMATH DE number 1888397
scientific article