Generalized hook and content numbers identities
type: Item
instance of: scholarly article
title: Generalized hook and content numbers identities
zbMATH Open document ID: 0965.05096
author: Amitai Regev
published in: European Journal of Combinatorics
publication date: 2001-07-24T00:00:00Z
review text: For any cell \(x=(i,j)\) in a skew diagram \(\lambda/\eta\) one defines the hook pair \((\text{arm}(x),\text{leg}(x))=(\lambda_i-j,\lambda'_j-i)\) and the content pair \((j-1,i-1)\). Thus, the diagram determines the multisets of hook pairs and content pairs. The author proves various identities between these multisets. In particular, these identities imply new ``hook'' formulas for the dimensions of the irreducible \(\text{GL}(n)\)-representations. In the more general situation of the Jack symmetric functions \(P_{\lambda}^{(\alpha)}\), these identities give new ``\(\alpha\)-hook'' formulas for \(P_{\lambda}^{(\alpha)}(1,\ldots,1)\). An important feature of the paper is that beyond some standard definitions, the main results and their proofs are completely elementary and no previous knowledge of any kind is assumed on the part of the reader. The proofs are inductive. Recently, bijective proofs of the main results of the present paper and of [\textit{A. Regev}, Eur. J. Comb. 21, No. 7, 959-966 (2000; Zbl 0962.05062)] have been given in [\textit{C. Krattenthaler}, Electron. J. Comb. 7, No. 1, Research paper R27 (2000; Zbl 0939.05011)].
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 05E05
zbMATH DE Number: 1538576
zbMATH Keywords: skew diagrams
zbMATH Keywords: Schur functions
zbMATH Keywords: Jack symmetric functions
zbMATH Keywords: multiset identities
zbMATH Keywords: hook formulas
MaRDI profile type: Publication
full work available at URL: eujc.2000.0395
OpenAlex ID: W2000510453
cites work: On hooks of Young diagrams
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cites work: Q4236280
DOI: 10.1006/EUJC.2000.0395
description: scientific article; zbMATH DE number 1538576