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 Funktsional. Anal. i Prilozhen.: Year: Volume: Issue: Page: Find

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

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

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English version:
Functional Analysis and Its Applications, 2017, 51:1, 22–31

Bibliographic databases:

UDC: 517.986.6, 519.12
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
\Bibitem{VerTsi17} \by A.~M.~Vershik, N.~V.~Tsilevich \paper On the Relationship between Combinatorial Functions and Representation Theory \jour Funktsional. Anal. i Prilozhen. \yr 2017 \vol 51 \issue 1 \pages 28--39 \mathnet{http://mi.mathnet.ru/faa3258} \crossref{https://doi.org/10.4213/faa3258} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3647780} \elib{http://elibrary.ru/item.asp?id=28169173} \transl \jour Funct. Anal. Appl. \yr 2017 \vol 51 \issue 1 \pages 22--31 \crossref{https://doi.org/10.1007/s10688-017-0165-4} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000396373700002} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85015421144} 

• http://mi.mathnet.ru/eng/faa3258
• https://doi.org/10.4213/faa3258
• http://mi.mathnet.ru/eng/faa/v51/i1/p28

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This publication is cited in the following articles:
1. J. Math. Sci. (N. Y.), 232:2 (2018), 170–176
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