A new tableau representation for supersymmetric Schur functions
type: Item
instance of: scholarly article
title: A new tableau representation for supersymmetric Schur functions
zbMATH Open document ID: 0840.20008
author: Curtis Greene
author: I. P. Goulden
published in: Journal of Algebra
publication date: 1994-12-19T00:00:00Z
review text: The so-called supersymmetric skew Schur functions \(s_{\lambda/ \mu}\) in variables \(X_m=(x_1, x_2, \dots, x_m)\) and \(Y_n=(y_1, y_2, \dots, y_n)\) is related to the corresponding symmetric function in \(X_m\cup Y_n\) by the involutory automorphism on the ring of symmetric polynomials in \(Y_n\): \[ s_{\lambda/ \mu} (X_m/ Y_n)=\sum_{\mu\subseteq \nu\subseteq \lambda} s_{\nu/ \mu} (X_m) s_{\lambda'/ \nu'} (Y_n), \] have combinatorial interest in their own right, beyond their wide application, for example in the character theory of Lie superalgebras [\textit{V. G. Kac}, Adv. Math. 26, 8-96 (1977; Zbl 0366.17012)], in the formalism of the KP and derived integrable hierarchies, and related algebraic-geometrical contexts. The above formula for the case \(\mu=\emptyset\) was given by \textit{P. H. Dondi} and \textit{P. D. Jarvis} [Z. Phys. C 4, 201-205 (1980), J. Phys. A 14, 547-563 (1981; Zbl 0449.17002)], studied by \textit{A. B. Balantekin} and \textit{I. Bars} [in J. Math. Phys. 22, 1149-1162, 1810-1818 (1981; Zbl 0469.22017, Zbl 0547.22014)], formulated for the skew case by \textit{A. Berele} and \textit{A. Regev} [Adv. Math. 64, 118-175 (1987; Zbl 0617.17002)]\ and further developed by \textit{R. C. King} [Ars Comb. 16A, 269-287 (1983; Zbl 0547.22016)]. The authors present a new tableau description of such functions, and consequently derive and shed light on several other combinatorial characterizations and related (super)symmetric functions. Let \({\mathcal T}_{\lambda/\mu} (A)\) be the set of skew tableaux of shape \(\lambda/\mu\) with entries in \(A\subseteq \mathbb{Z}\), \(T(\alpha)\) the entry of cell \(\alpha\) belonging to some \(T\in {\mathcal T}_{\lambda, \mu} (A)\), and \(C(\alpha)\) the content of that cell (where \(C(i,j)=j-i)\). Then the major theorem of the paper reads \[ s_{\lambda/\mu} (X/Y)=\sum_{T\in {\mathcal T}_{\lambda/ \mu} (\mathbb{Z})} \prod_{\alpha\in \lambda/\mu} (x_{T (\alpha)}+ y_{T(\alpha)+ C(\alpha)}) \] for supersymmetric skew Schur …
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 17A70
zbMATH DE Number: 703243
zbMATH Keywords: supersymmetric skew Schur functions
zbMATH Keywords: ring of symmetric polynomials
zbMATH Keywords: skew tableaux
zbMATH Keywords: Bender-Knuth algorithm
zbMATH Keywords: supersymmetric functions
reviewed by: Peter D Jarvis
MaRDI profile type: Publication
full work available at URL: jabr.1994.1361
OpenAlex ID: W2032817839
DOI: 10.1006/JABR.1994.1361
Recommended article: Factorial supersymmetric skew Schur functions and ninth variation determinantal identities
Recommended article: Set superpartitions and superspace duality modules
Recommended article: Interpolation Analogs of Schur Q-Functions
Recommended article: Lattice Paths and a Sergeev-Pragacz Formula for Skew Supersymmetric Functions
Recommended article: A Murnaghan Nakayama rule for Schur functions in superspace
Recommended article: Q4437895
Recommended article: Q5377017
Recommended article: Combinatorial relations on skew Schur and skew stable Grothendieck polynomials
Recommended article: Robinson-Schensted algorithms for skew tableaux
Recommended article: A determinantal formula for supersymmetric Schur polynomials
description: scientific article; zbMATH DE number 703243