Free resolutions of generic symmetric matrices
type: Item
instance of: scholarly article
title: Free resolutions of generic symmetric matrices
zbMATH Open document ID: 0840.13001
DOI: 10.1016/0024-3795(93)00355-4
author: Helmer Aslaksen
author: Eng-Chye Tan
author: Chen-Bo Zhu
published in: Linear Algebra and its Applications
publication date: 1996-07-01T00:00:00Z
review text: For a field \(k\) of characteristic zero let \(G = O(m,k)\) and \(V = U^n\), where \(U\) is the standard module of \(G\). \(G\) acts on \(k[V]\), the space of polynomial functions on \(V\). We can identify \(k[V]\) with the polynomial ring \(k[X]\), where \(X\) is an \(m \times n\) matrix of indeterminates. If \(P\) is an \(n \times n\) symmetric matrix of indeterminates, then \(k[V]^G \simeq k[X^tX] \simeq k[P]/I_{m + 1}\), where \(I_{m + 1} \subset k[P]\) is the ideal generated by the \((m + 1) \times (m + 1)\) minors of \(P\). The authors give an elementary construction of a finite free resolution of \(k[P]/I_{m + 1}\) in the case \(m = n - 2\). The theorem was earlier known [cf. \textit{S. Goto} and \textit{S. Tachibana}, J. Math. Kyoto Univ. 17, 51-54 (1977; Zbl 0375.13004) and \textit{T. Józefiak} [Commentarii Math. Helvet. 53, 595-607 (1978; Zbl 0398.13009)]. However, the present authors' approach is elementary.
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 826706
zbMATH Keywords: polynomial ring
reviewed by: Witold Więsław
MaRDI profile type: Publication
full work available at URL: 0024-3795(93)00355-4
OpenAlex ID: W2031603390
cites work: A complex associated with a symmetric matrix
cites work: Q5648459
cites work: Q3331451
cites work: Ideals generated by minors of a symmetric matrix
description: scientific article; zbMATH DE number 826706