On the relationship between combinatorial functions and representation theory

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On the relationship between combinatorial functions and representation theory. / Vershik, A. M.; Tsilevich, N. V.

In: Functional Analysis and its Applications, Vol. 51, No. 1, 01.01.2017, p. 22-31.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{1c3ab68ef8484215814a48d7551e4a16,

title = "On the relationship between combinatorial functions and representation theory",

abstract = "The paper is devoted to the study of well-known combinatorial functions on the symmetric group Sn—the major index maj, the descent number des, and the inversion number inv—from the representation-theoretic point of view. We show that these functions generate the same ideal in the group algebra C[Sn], and the restriction of the left regular representation of the group Sn to this ideal is isomorphic to its representation in the space of n×n skew-symmetric matrices. This allows us to obtain formulas for the functions maj, des, and 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 fix of fixed points.",

keywords = "descent number, dual complexity, inversion number, major index, representations of the symmetric group, skew-symmetric matrices",

author = "Vershik, {A. M.} and Tsilevich, {N. V.}",

year = "2017",

month = jan,

day = "1",

doi = "10.1007/s10688-017-0165-4",

language = "English",

volume = "51",

pages = "22--31",

journal = "Functional Analysis and its Applications",

issn = "0016-2663",

publisher = "Springer Nature",

number = "1",

}

RIS

TY - JOUR

T1 - On the relationship between combinatorial functions and representation theory

AU - Vershik, A. M.

AU - Tsilevich, N. V.

PY - 2017/1/1

Y1 - 2017/1/1

N2 - The paper is devoted to the study of well-known combinatorial functions on the symmetric group Sn—the major index maj, the descent number des, and the inversion number inv—from the representation-theoretic point of view. We show that these functions generate the same ideal in the group algebra C[Sn], and the restriction of the left regular representation of the group Sn to this ideal is isomorphic to its representation in the space of n×n skew-symmetric matrices. This allows us to obtain formulas for the functions maj, des, and 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 fix of fixed points.

AB - The paper is devoted to the study of well-known combinatorial functions on the symmetric group Sn—the major index maj, the descent number des, and the inversion number inv—from the representation-theoretic point of view. We show that these functions generate the same ideal in the group algebra C[Sn], and the restriction of the left regular representation of the group Sn to this ideal is isomorphic to its representation in the space of n×n skew-symmetric matrices. This allows us to obtain formulas for the functions maj, des, and 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 fix of fixed points.

KW - descent number

KW - dual complexity

KW - inversion number

KW - major index

KW - representations of the symmetric group

KW - skew-symmetric matrices

UR - http://www.scopus.com/inward/record.url?scp=85015421144&partnerID=8YFLogxK

U2 - 10.1007/s10688-017-0165-4

DO - 10.1007/s10688-017-0165-4

M3 - Article

AN - SCOPUS:85015421144

VL - 51

SP - 22

EP - 31

JO - Functional Analysis and its Applications

JF - Functional Analysis and its Applications

SN - 0016-2663

IS - 1

ER -

ID: 49789523