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Funktsional. Anal. i Prilozhen., 2017, Volume 51, Issue 1, Pages 28–39 (Mi faa3258)  

This article is cited in 1 scientific paper (total in 1 paper)

On the Relationship between Combinatorial Functions and Representation Theory

A. M. Vershikabc, N. V. Tsilevichab

a St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
b Saint Petersburg State University
c Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow

Abstract: The paper is devoted to the study of well-known combinatorial functions on the symmetric group $\mathfrak{S}_n$—the major index $\operatorname{maj}$, the descent number $\operatorname{des}$, and the inversion number $\operatorname{inv}$—from the representation-theoretic point of view. We show that these functions generate the same ideal in the group algebra $\mathbb{C}[\mathfrak{S}_n]$, and the restriction of the left regular representation of the group $\mathfrak{S}_n$ to this ideal is isomorphic to its representation in the space of $n\times n$ skew-symmetric matrices. This allows us to obtain formulas for the functions $\operatorname{maj}$, $\operatorname{des}$, and $\operatorname{inv}$ in terms of matrices of an exceptionally simple form. These formulas are applied to find the spectra of the elements under study in the regular representation, as well as derive a series of identities relating these functions to one another and to the number $\operatorname{fix}$ of fixed points.

Keywords: major index, descent number, inversion number, representations of the symmetric group, skew-symmetric matrices, dual complexity

Funding Agency Grant Number
Russian Science Foundation 14-50-00150


DOI: https://doi.org/10.4213/faa3258

Full text: PDF file (224 kB)
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English version:
Functional Analysis and Its Applications, 2017, 51:1, 22–31

Bibliographic databases:

UDC: 517.986.6, 519.12
Received: 14.12.2016
Accepted:24.01.2017

Citation: A. M. Vershik, N. V. Tsilevich, “On the Relationship between Combinatorial Functions and Representation Theory”, Funktsional. Anal. i Prilozhen., 51:1 (2017), 28–39; Funct. Anal. Appl., 51:1 (2017), 22–31

Citation in format AMSBIB
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\by A.~M.~Vershik, N.~V.~Tsilevich
\paper On the Relationship between Combinatorial Functions and Representation Theory
\jour Funktsional. Anal. i Prilozhen.
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\vol 51
\issue 1
\pages 28--39
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\yr 2017
\vol 51
\issue 1
\pages 22--31
\crossref{https://doi.org/10.1007/s10688-017-0165-4}
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    This publication is cited in the following articles:
    1. J. Math. Sci. (N. Y.), 232:2 (2018), 170–176  mathnet  crossref
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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