Natural transformations of tensor-products of representation-functors. I: Combinatorial preliminaries
type: Item
instance of: scholarly article
title: Natural transformations of tensor-products of representation-functors. I: Combinatorial preliminaries
zbMATH Open document ID: 0549.18001
author: Jacob Towber
published in: Pacific Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: A level 1 finite set is a finite set and Fin-1-Sets denotes the category consisting of level 1 finite sets and bijections. If \(n>1\), a level n finite set is a finite set of pairwise disjoint non-empty level n-1 finite sets; a level n morphism between two level n finite sets \({\mathcal D}_ 1,{\mathcal D}_ 2\) is defined to consist of a bijection \(\iota\) : \({\mathcal D}_ 1\to {\mathcal D}_ 2\), together with the assignment to each \(\Delta\in {\mathcal D}_ 1\) of a level n-1 morphism \(\iota_{\Delta}\) from \(\Delta\) to \(\iota\) (\(\Delta)\). Fin-n-sets is the category consisting of level n finite sets and level n morphisms. The author uses this concept to the structural study of the A-module \(NatTsf_ A(\alpha_ 1\times...\times\alpha_ m,\beta_ 1\times...\times\beta_ n)\) which consists of all natural transformations from the exterior product functor \(\wedge_ A^{\alpha_ 1}\otimes...\otimes\wedge_ A^{\alpha_ m}: Mod_ A\to Mod_ A, E\mapsto\wedge_ A^{\alpha_ 1}E\otimes...\otimes\wedge_ A^{\alpha_ m}E\) into the similar functor \(\wedge_ A^{\beta_ 1}\otimes...\otimes\wedge_ A^{\beta_ n},\) where \(\alpha_ 1,...,\alpha_ m\), \(\beta_ 1,...,\beta_ n\) are partitions and A is a commutative ring.
reviewed by: Ladislav Bican
Mathematics Subject Classification ID: 18A23
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 18B99
Mathematics Subject Classification ID: 05A17
zbMATH DE Number: 3875500
zbMATH Keywords: tensor product
zbMATH Keywords: Fin-n-sets
zbMATH Keywords: natural transformations
zbMATH Keywords: exterior product
zbMATH Keywords: partitions
MaRDI profile type: Publication
full work available at URL: pjm.1983.108.465
OpenAlex ID: W2001783636
DOI: 10.2140/PJM.1983.108.465
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description: scientific article; zbMATH DE number 3875500