A generalization of Hall polynomials to ADE case
type: Item
instance of: scholarly article
zbMATH Open document ID: 1026.17008
DOI: 10.1155/S1073792801000563
author: Anton Malkin
publication date: 2003-02-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0103212
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 1719041
zbMATH Keywords: Hall polynomials
zbMATH Keywords: representations of formal power series
zbMATH Keywords: partition of integers
MaRDI profile type: Publication
arXiv ID: math/0103212
OpenAlex ID: W1845187161
published in: IMRN. International Mathematics Research Notices
title: A generalization of Hall polynomials to ADE case
review text: For a representation \(\pi\) of \(kt\) in a finite-dimensional \(k\)-linear space \(D\), one associates a partition \(\lambda\) with \(\lambda_i=\dim(\text{Im }\pi(t^{i-1}))/\dim(\text{Im }\pi(t^i))\), called the type of \(\pi\). Let \({\mathfrak N}^{\lambda}_{\mu^1\cdots\mu^n}(k)\) be the set of all pairs \((\pi,\underline{D})\) consisting of a representation \(\pi\) of \(kt\) of type \(\lambda\) and \(n\)-step filtration \(\underline{D}=(\{0\}=D^0\subset D^1\subset\cdots\subset D^n=k^{|\lambda|})\) of \(k^{|\lambda|}\) by subrepresentations of \(\pi\) such that \(\pi|_{D^a/D^{a-1}}\) has type \(\mu^a\) for \(a=1,\ldots,n\). If \(k\) is algebraically closed, then \({\mathfrak N}^{\lambda}_{\mu^1\cdots\mu^n}(k)\) is a quasi-projective variety over \(k\). The set \({\mathfrak N}^{\lambda}_{\mu^1\cdots\mu^n}({\mathbb F}_{p^l})\) can be identified with the set of \({\mathbb F}_{p^l}\)-rational points of the variety \({\mathfrak N}^{\lambda}_{\mu^1\cdots\mu^n}(\bar{\mathbb F}_{p^l})\). This is a classical observation of \textit{E. Steinitz} [Zur Theorie der Abelschen Gruppen, Deutsche Math.-Ver. \(9_1\), 80-85 (1901; JFM 32.0149.02)] and \textit{P. Hall} [Proc. 4th Can. Math. Congr. Banff, 1957, 147-159 (1959; Zbl 0122.03403)] that the sets \({\mathfrak N}^{\lambda}_{\mu^1\cdots\mu^n}(k)\) have relation to the theory of representations of \(GL(N,{\mathbb C})\) and \(S_N\). In particular, the number of irreducible components of \({\mathfrak N}^{\lambda}_{\mu^1\cdots\mu^n}({\mathbb C})\) is equal to \(c^{\lambda}_{\mu^1\cdots \mu^n}=\dim_{\mathbb C}\Hom_{GL(N,{\mathbb C})} (L_{\lambda},L_{\mu^1}\otimes\cdots\otimes L_{\mu^n})\). NEWLINENEWLINENEWLINEIn the paper under review the author considers other varieties relevant to tensor product multiplicities, for reductive groups other than \(GL(N)\). These varieties were independently described by \textit{H. Nakajima} [Invent. Math. 146, 399-449 (2001; Zbl 1023.17008), see also math.QA/0103008], \textit{M. Varagnolo} a…
reviewed by: Vesselin Drensky
description: scientific article; zbMATH DE number 1719041