The universal form of the Littlewood-Richardson rule
type: Item
instance of: scholarly article
title: The universal form of the Littlewood-Richardson rule
zbMATH Open document ID: 0659.20035
DOI: 10.1016/0001-8708(88)90007-2
author: Giandomenico Boffi
published in: Advances in Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let F be a free module of rank n over any commutative ring R and let \(L_{\lambda}F\) be the GL(F)-irreducible homogeneous polynomial representations parametrized by the partition \(\lambda =(\lambda_ 1,\lambda_ 2,...)\) of n. Given any two partitions \(\lambda\), \(\mu\) of n, one has \(L_{\lambda}F\otimes L_{\mu}F\cong \sum c(\lambda,\mu;\nu)L_{\nu}F\), \(c(\lambda,\mu;\nu)\in {\mathbb{N}}\), in the case when \(R={\mathbb{C}}\), where the coefficients \(c(\lambda,\mu;\nu)\) can be described by the Littlewood-Richardson rule. In the present paper, the author proves that there exists a universal filtration for \(L_{\lambda}F\otimes L_{\mu}F\) such that its associated graded module is isomorphic to \(\sum c(\lambda,\mu;\nu)L_{\nu}F\). Furthermore, he gives the coefficients \(c(\lambda,\mu;\nu)\) an interpretation in his description of the associated graded module \(\sum L_{\nu}F\otimes C_{\nu}\), where \(C_{\nu}\) is a suitable universally free R-module of \(U^+(\lambda_ 1,R)\)-invariants, \(U^+(\lambda_ 1,R)\) denotes the upper-unitriangular matrices of order \(\lambda_ 1\) with entries in R.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 20G20
zbMATH DE Number: 4077556
zbMATH Keywords: free module
zbMATH Keywords: irreducible homogeneous polynomial representations
zbMATH Keywords: partition
zbMATH Keywords: Littlewood-Richardson rule
zbMATH Keywords: universal filtration
zbMATH Keywords: graded module
zbMATH Keywords: invariants
MaRDI profile type: Publication
full work available at URL: 0001-8708(88)90007-2
OpenAlex ID: W2074816468
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description: scientific article; zbMATH DE number 4077556