Cyclic characters of symmetric groups
type: Item
instance of: scholarly article
title: Cyclic characters of symmetric groups
zbMATH Open document ID: 0979.20017
DOI: 10.1023/A:1026592027019
author: Armin Jöllenbeck
author: Manfred Schocker
published in: Journal of Algebraic Combinatorics
publication date: 2002-02-12T00:00:00Z
review text: Let \(S_n\) denote the symmetric group of degree \(n\) and \(C\) denote a subgroup of \(S_n\) generated by a cycle \(\sigma\) of order \(n\). Then \(\psi_i\colon C\to\mathbb{C}^*\) defined by \(\psi_i(\sigma)=\xi^i\), where \(\xi\) is a primitive \(n\)-th root of unity in the complex field \(\mathbb{C}\), is an irreducible ordinary character of \(C\). It is well-known that there is a one-to-one correspondence between the complex irreducible characters of \(S_n\) and the partitions of \(n\). Now, if \(p\) is a partition of \(n\), then the irreducible character of \(S_n\) corresponding to \(p\) is denoted by \(\xi^p\). In the paper under review the authors prove that the multiplicity \((\psi^{S_n}_i,\xi^p)\) is equal to the number of standard Young tableaux of shape \(p\) and its major index is congruent to \(i\) modulo \(n\). This fact was already proved by \textit{W. Kraśkiewicz} and \textit{J. Weyman} [Bayreuther Math. Schr. 63, 265-284 (2001; Zbl 1037.20012)]. In the present paper the authors give a new proof.
reviewed by: M. R. Darafsheh
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 05E05
zbMATH DE Number: 1553310
zbMATH Keywords: symmetric groups
zbMATH Keywords: irreducible ordinary characters
zbMATH Keywords: partitions
zbMATH Keywords: multiplicities
zbMATH Keywords: standard Young tableaux
MaRDI profile type: Publication
cites work: Q5785190
cites work: Q3491770
cites work: Q4141936
cites work: Q4267501
cites work: Lie elements in the tensor algebra
cites work: Noncommutative cyclic characters of symmetric groups
cites work: Q3134850
cites work: A Mackey formula in the group of a Coxeter group. With an appendix by J. Tits: Two properties of Coxeter complexes
cites work: Die linearen Beziehungen zwischen höheren Kommutatoren
cites work: On the eigenvalues of representations of reflection groups and wreath products
cites work: Über Invarianten in Lieschen Ringen
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description: scientific article; zbMATH DE number 1553310