The combinatorics of a class of representation functions
type: Item
instance of: scholarly article
title: The combinatorics of a class of representation functions
zbMATH Open document ID: 0943.05086
author: William Y. C. Chen
author: J. D. Louck
published in: Advances in Mathematics
publication date: 2000-08-28T00:00:00Z
review text: The authors study a class of polynomials that arise from the irreducible representations of the unitary group \(U(n)\). These polynomials have been studied extensively in the physics literature where they are formulated in terms of the boson calculus. This paper serves as an introduction for mathematicians and describes what is needed for a rigorous mathematical treatment. They begin this program by proving the following result. Let \(\alpha\) and \(\beta\) be compositions of \(m\) into \(n\) nonnegative integers. Define the homogeneous polynomial of degree \(m\) in \(z_{ij}\), \(1 \leq i,j \leq n\), by \[ L_{\alpha,\beta}(Z) = \sum_{A\in M(\alpha,\beta)}\prod_{i,j=1}^n z_{ij}^{a_{ij}}/a_{ij}!, \] where \(M(\alpha, \beta)\) is the set of \(n \times n\) nonnegative integral matrices with row sums given by \(\alpha\) and column sums given by \(\beta\). Let \(\alpha!\) be the product of the factorials of the components of \(\alpha\), and let \(D_{\alpha,\beta}(Z) = \sqrt{\alpha! \beta!} L_{\alpha,\beta}(Z)\). Let \(D(Z)\) be the matrix indexed by the compositions of \(m\) with entries \(D_{\alpha,\beta}(Z)\). The authors prove that \(D(XY) = D(X) D(Y)\) and relate this result to MacMahon's master theorem. They also discuss specializations which include the generalized beta function of Gelfand and Graev, Jacobi polynomials, and Clebsch-Gordon coefficients.
reviewed by: David M. Bressoud
Mathematics Subject Classification ID: 05E15
Mathematics Subject Classification ID: 20C35
zbMATH DE Number: 1268550
zbMATH Keywords: irreducible representations
zbMATH Keywords: unitary group
zbMATH Keywords: beta function
zbMATH Keywords: Jacobi polynomials
zbMATH Keywords: Clebsch-Gordon coefficients
MaRDI profile type: Publication
full work available at URL: aima.1998.1769
OpenAlex ID: W2037111177
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DOI: 10.1006/AIMA.1998.1769
description: scientific article; zbMATH DE number 1268550