Presentations of Schur complexes attached to row convex shapes
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 \[ \sum_{\substack{ 1 \leq i \leq j \leq n\\t > \max\{0, \mu_i - \mu_j\}}} \Lambda_{\lambda_1 - \mu_1} \phi \otimes \cdots \otimes \Lambda_{\lambda_i - \mu_i +t} \phi \otimes \cdots \otimes \Lambda_{\lambda_j - \mu_j -t} \phi \otimes \cdots \otimes \Lambda_{\lambda_n - \mu_n}\phi. \]
scientific article; zbMATH DE number 1211560
scientific article