Asymptotic behavior of \(S_n\) degrees associated with rational functions.
type: Item
instance of: scholarly article
title: Asymptotic behavior of \(S_n\) degrees associated with rational functions.
zbMATH Open document ID: 1343.20018
author: Stefan Catoiu
author: Allan Berele
published in: Israel Journal of Mathematics
publication date: 2015-11-16T00:00:00Z
review text: The paper under review is motivated by the result of \textit{A. Berele} and \textit{A. Regev}, [in Trans. Am. Math. Soc. 360, No. 10, 5155-5172 (2008; Zbl 1161.16016)] that over a field of characteristic 0 the codimension sequence \(c_n(A)\) of a finitely generated unitary PI-algebra \(A\) is asymptotically equal to \(\alpha\cdot n^g\cdot\ell^n\), where \(\alpha\) is a non-negative real, \(g\in\mathbb Z/2\), and \(\ell\) is a non-negative integer. (The general result for arbitrary unitary PI-algebras was established by the first author [in Adv. Appl. Math. 41, No. 1, 52-75 (2008; Zbl 1145.05052); corrigendum 43, No. 2, 199-200 (2009)].) Now the authors consider sequences of characters \(\chi_n=\sum_{\lambda\vdash n}m_\lambda\chi_\lambda\) of the symmetric groups \(S_n\) which share the main properties of the cocharacter sequences of finitely generated unitary PI-algebras: (i) All \(\lambda\)'s are partitions in not more than \(k\) parts; (ii) The associated symmetric function \(P(t_1,\ldots,t_k)=\sum m_\lambda S_\lambda(t_1,\ldots,t_k)\) (where \(S_\lambda(t_1,\ldots,t_k)\) is the Schur function corresponding to the partition \(\lambda\)) is a nice rational function, i.e., it can be written as a fraction of polynomials with denominator a product of terms of the form \((1-u)\), where \(u\) is a monic monomial; (iii) The symmetric function \(P(t_1,\ldots,t_k)\) is Young derived, i.e., both \(P(t_1,\ldots,t_k)\) and \(\prod_i(1-t_i)P(t_1,\ldots,t_k)\) can be expanded into formal series of Schur functions with non-negative integer coefficients. The main result of the paper states that in this case the sequence \(c_n=\sum_{\lambda\vdash n}m_\lambda\deg(\chi_\lambda)\) has asymptotic behaviour \(c_n\simeq\alpha\cdot n^g\cdot\ell^n\), where \(g\in\mathbb Z/2\), and \(\ell\) is algebraic over \(\mathbb Q\). If one replaces the condition to be Young derived with the weaker condition that each \(m_\lambda\) is a non-negative integer, then there exists a modulus \(d\) such tha…
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 16P90
Mathematics Subject Classification ID: 16R10
zbMATH DE Number: 6509132
zbMATH Keywords: representations of symmetric groups
zbMATH Keywords: degrees of representations
zbMATH Keywords: algebras with polynomial identity
zbMATH Keywords: codimensions of PI-algebras
zbMATH Keywords: nice rational functions
zbMATH Keywords: asymptotics of codimensions
zbMATH Keywords: cocharacter sequences
zbMATH Keywords: codimension sequences
zbMATH Keywords: symmetric functions
MaRDI profile type: Publication
full work available at URL: s11856-015-1202-8
OpenAlex ID: W2464139685
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DOI: 10.1007/S11856-015-1202-8
description: scientific article; zbMATH DE number 6509132