On the Littlewood-Richardson polynomials
type: Item
instance of: scholarly article
title: On the Littlewood-Richardson polynomials
zbMATH Open document ID: 1018.16012
DOI: 10.1016/S0021-8693(02)00125-4
author: Jerzy Weyman
author: Harm Derksen
published in: Journal of Algebra
publication date: 2003-02-12T00:00:00Z
review text: The authors continue their study of semi-invariants of quivers. Let \(Q\) be a quiver, \(\alpha\) a dimension vector, \(\text{GL}(\alpha)\) the corresponding product of general linear groups. Write \(SI(Q,\alpha)\) for the algebra of polynomial invariants of the derived subgroup \(\text{SL}(\alpha)\) of \(\text{GL}(\alpha)\), acting on the space of \(\alpha\)-dimensional representations of \(Q\) over an algebraically closed field. Under the assumption that \(Q\) has no oriented cycles, a spanning set of \(SI(Q,\alpha)\) was given by \textit{H. Derksen} and \textit{J. Weyman} [J. Am. Math. Soc. 13, No. 3, 467-479 (2000; Zbl 0993.16011)]. Here Schofield's double quiver construction is applied, in order to derive from this generators of \(SI(Q,\alpha)\) for arbitrary quivers, recovering this way results of \textit{L. Le Bruyn} and \textit{C. Procesi} [Trans. Am. Math. Soc. 317, No. 2, 585-598 (1990; Zbl 0693.16018)] and \textit{S. Donkin} [Comment. Math. Helv. 69, No. 1, 137-141 (1994; Zbl 0816.16015)] on generating \(\text{GL}(\alpha)\)-invariants, and results of \textit{A. Schofield} and \textit{M. Van den Bergh} [Indag. Math., New Ser. 12, No. 1, 125-138 (2001; Zbl 1004.16012)] and \textit{M. Domokos} and \textit{A. N. Zubkov} [Transform. Groups 6, No. 1, 9-24 (2001; Zbl 0984.16023)] on generating semi-invariants. The second part of the paper concentrates on the subalgebra \(SI(Q,\alpha,\sigma)\) spanned by those relative \(\text{GL}(\alpha)\)-invariants whose weight is a scalar multiple of some fixed weight \(\sigma\) (where \(Q\) has no oriented cycles). This is a graded algebra, and its projective spectrum provides a moduli space for certain representations of \(Q\). The authors prove that for all sufficiently large \(n\), the algebra \(SI(Q,\alpha,\sigma)\) has a homogeneous system of parameters consisting of elements of degree \(n\). The Poincaré series of \(SI(Q,\alpha,\sigma)\) can be written as a rational function \(P(t)/(1-t)^d\), where \(d\) is the Krull d…
reviewed by: Matyas Domokos
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16R30
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 14L24
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 16W22
zbMATH DE Number: 1868002
zbMATH Keywords: semi-invariants
zbMATH Keywords: quivers
zbMATH Keywords: Littlewood-Richardson polynomials
zbMATH Keywords: systems of parameters
zbMATH Keywords: Poincaré series
MaRDI profile type: Publication
cites work: Singularités rationnelles et quotients par les groupes réductifs. (Rational singularities and quotients by reductive groups)
cites work: Polynomial invariants of representations of quivers
cites work: Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients
cites work: Semi-invariants of quivers as determinants
cites work: A characterization of rational singularities in terms of injectivity of Frobenius maps
cites work: MODULI OF REPRESENTATIONS OF FINITE DIMENSIONAL ALGEBRAS
cites work: Semisimple Representations of Quivers
cites work: General Representations of Quivers
cites work: Semi-Invariants of Quivers
cites work: Semi-invariants of quivers for arbitrary dimension vectors
cites work: F-rational rings have rational singularities
description: scientific article; zbMATH DE number 1868002