Natural transformations of tensor-products of representation-functors. I: Combinatorial preliminaries
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.
scientific article; zbMATH DE number 3875500