Presentations of Schur complexes attached to row convex shapes
type: Item
instance of: scholarly article
title: Presentations of Schur complexes attached to row convex shapes
zbMATH Open document ID: 0930.13012
author: M. Klucznik
published in: Advances in Mathematics
publication date: 1999-10-21T00:00:00Z
review text: This paper is devoted to the study of a special class of the Schur complexes originally introduced by \textit{K. Akin, D. A. Buchsbaum} and \textit{J. Weyman} [Adv. Math. 44, 207-278 (1982; Zbl 0497.15020)]. If \((\lambda/ \mu)\) is a shape, i.e., \(\lambda = (\lambda_1 \ldots \lambda_n)\) is a partition (\(\lambda_i \in {\mathbb Z}^+\), \(\lambda_i \geq \lambda_{i + 1}\)) and \(\mu = (\mu_1 \ldots \mu_n)\) a composition (\(\mu_i \in {\mathbb Z}^+\), \(\lambda_i \geq \mu_i\)) then there is associated a shape matrix (\(a_{ij} =1\) if \(\mu_i < j \leq \lambda_i\) and \(a_{ij} =0\) otherwise) and to this a Schur complex generalizing the Koszul complex. If the shape has the stronger property that \(\lambda_i > \mu_i\) for all \(i\), then the shape is said to be row convex. The author proves that any complex associated to such a shape is the homomorphic image of one of the form \[ \Lambda_{\lambda /\mu} \phi = \Lambda_{\lambda_1 - \mu_1} \phi \otimes \cdots \otimes \Lambda_{\lambda_n - \mu_n} \phi \] and that the kernel is the homomorphic image of a complex of the form \[ \sum_{\substack{ 1 \leq i \leq j \leq n\\t > \max\{0, \mu_i - \mu_j\}}} \Lambda_{\lambda_1 - \mu_1} \phi \otimes \cdots \otimes \Lambda_{\lambda_i - \mu_i +t} \phi \otimes \cdots \otimes \Lambda_{\lambda_j - \mu_j -t} \phi \otimes \cdots \otimes \Lambda_{\lambda_n - \mu_n}\phi. \]
reviewed by: Keith Johnson
Mathematics Subject Classification ID: 13D25
zbMATH DE Number: 1211560
zbMATH Keywords: Schur complex
zbMATH Keywords: Koszul complex
zbMATH Keywords: row convex shape
MaRDI profile type: Publication
full work available at URL: aima.1997.1708
OpenAlex ID: W2015198224
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DOI: 10.1006/AIMA.1997.1708
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description: scientific article; zbMATH DE number 1211560