Tensor product varieties and crystals: GL case
type: Item
instance of: scholarly article
zbMATH Open document ID: 0990.20030
DOI: 10.1090/S0002-9947-01-02899-9
author: Anton Malkin
published in: Transactions of the American Mathematical Society
publication date: 2001-12-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0103026
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 20G20
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 14A10
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 17B67
zbMATH DE Number: 1680730
zbMATH Keywords: tensor product varieties
zbMATH Keywords: crystals
zbMATH Keywords: canonical bases
zbMATH Keywords: Littlewood-Richardson coefficients
zbMATH Keywords: nilpotent orbits
zbMATH Keywords: Spaltenstein varieties
zbMATH Keywords: irreducible components
zbMATH Keywords: simple modules
zbMATH Keywords: Kac-Moody algebras
MaRDI profile type: Publication
arXiv ID: math/0103026
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cites work: Tensor product varieties and crystals: The \(ADE\) case.
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title: Tensor product varieties and crystals: GL case
review text: Let \(\lambda\) be a partition of an integer \(k\) into \(N\) parts, and let \(\mu^1,\mu^2,\dots,\mu^l\) be partitions of integers \(k^1,k^2,\dots,k^l\) into \(N\) parts each, where \(k^1+k^2+\cdots+k^l=k\). Let \(S_l((\mu^1,\mu^2,\dots,\mu^l),\lambda)\) be the variety of all \(l\)-step partial flags in \(\mathbb{C}^k\) with dimensions of the subfactors given by \(k^1,k^2,\dots,k^l\) and such that \(t\) preserves each subspace in the flag, and when restricted to the subfactors it defines operators with the Jordan forms given by \(\mu^1,\mu^2,\dots,\mu^l\). This variety is known as the Spaltenstein variety; Spaltenstein proved that it has pure dimension.NEWLINENEWLINENEWLINELet \(C_l((\mu^1,\mu^2,\dots,\mu^l),\lambda)\) be the set of irreducible components of \(S_l((\mu^1,\mu^2,\dots,\mu^l),\lambda)\). For any partition \(\nu\) of \(k\) into \(N\) parts, let \(L_\nu\) denote the simple \(\text{GL}_N(\mathbb{C})\) representation over \(\mathbb{C}\) with highest weight \(\nu\), and let \(\rho_\nu\) denote the simple \(\mathbb{C} S_k\)-module corresponding to \(\nu\). It follows from \textit{P. Hall} [Proc. 4th Can. Math. Congr. Banff 1957, 147-159 (1959; Zbl 0122.03403)] that: NEWLINE\[NEWLINE|C_l((\mu^1,\mu^2,\dots,\mu^l),\lambda)|=\dim_\mathbb{C}\Hom_{\text{GL}_N(\mathbb{C})}(L_{\mu^1}\otimes\cdots\otimes L_{\mu^l},L_\lambda),NEWLINE\]NEWLINE and that NEWLINE\[NEWLINE|C_l((\mu^1,\mu^2,\dots,\mu^l),\lambda)|=\dim_\mathbb{C}\Hom_{S_{k^1}\times\cdots\times S_{k^l}}(\rho_{\mu^1}\otimes\cdots\otimes\rho_{\mu^l},\text{res}^{S_{k^1}\times\cdots\times S_{k^l}}_{S_k}\rho_\lambda).NEWLINE\]NEWLINE Schur-Weyl duality implies the equality of the right hand sides, and either equality can be derived combinatorially, but it is interesting to try to interpret these two formulas geometrically, and this is the aim of the author of the paper under review. Some results in this direction have already been achieved -- see, for example, \textit{W. Borho} and \textit{R. Macpherson} [Astéris…
reviewed by: Bethany R. Marsh
description: scientific article; zbMATH DE number 1680730