Brauer-Fitting Correspondence on Tensor Algebra

We show that Brauer-Fitting correspondence as well as the notion of pseudoblocks of endomorphism algebras are both compatible with the external tensor product of modules and algebras. Mathematics Subject Classification: 20C20


Introduction
The tensor product of algebras and modules has been a subject for investigation by many authors in the past few decades.Külshammer [9] studied the tensor product of algebra and modules over an algebraically closed field.His results were generalized by M. Harris [2] who considered the tensor product of lattices and modules over local rings.Alghamdi and Khammash [1] investigated the tensor product of G-algebras as well as tensor product of the Brauer homomorphisms.Khammash [7] studied the points, pointed groups and the notion of the defect group in tensor product of G-algebras.W. Huang [3] analyzed the connection between block algebras and interior G-algebras in connection with the tensor product.The concept of pseudo-blocks for endomorphism algebras was introduced in [8] where it was also shown (Theorem2) that the distribution of the indecomposable summand of a module among the pseudo-blocks is compatible with the (Brauer) block distribution of the corresponding (under Brauer-Fitting correspondence (see section1)) irreducible representations for the endomorphism algebra of that module.In this paper we relate the Brauer-Fitting correspondence as well as the notion of pseudo-blocks to the external tensor product of modules and algebras.
First we show that the Brauer-Fitting correspondence is compatible with the external tensoring of modules and algebras in the following sense denotes the set of isomorphism classes of indecomposable direct is the set of isomorphism classes of irreducible representations of . A proof of theorem1 will be given in section 2. We also show that such tensor operation is compatible with the pseudoblock distribution for the members of ) (X

Inds
. Namely if we use the notation ) ( X psd  to denote the pseudo-block linkage relation defined on ) (X Inds (see section 4), then we have Theorem2 can also be applied in counting the pseudo-blocks of Y X  .In fact if we write  for the set of conjugacy classes for the relation Brauer-Fitting correspondence on tensor algebra 897

Preliminaries
We start by describing the Brauer-Fitting correspondence.Let  be a finite dimensional k-algebra where F is an algebraically closed field of characteristic 0  p .Let Y be a finite dimensional  -module and write ; the endomorphism algebra of Y.An early and direct (and in fact vital) connection between the representation theory of the endomorphism algebra and representation theory of  is provided by the Brauer-Fitting theorem (see [10] -module and the Brauer-Fitting correspondence is given by the following bijection where r denotes the radical.The interplay between properties of the modules i i Y S ., involved in the correspondence (1) can be found in the references [4], [5], [6] and [7].

Connection with Tensor Product
We now relate the Brauer-Fitting correspondence to the tensor product of modules and algebras over a field F .Suppose that as F -algebras.

Proof of Theorem1:
From the description of the Brauer-Fitting correspondence in (1), it follows that the Brauer-Fitting correspondent of

Blocks of Tensor Algebra
Here we recall some facts about the blocks and projective indecomposable modules for the tensor algebras and their block distribution which will be needed latter.

Lemma2.1:
(1) Every projective indecomposable -module has the form where P is projective indecomposable if and only if .

Connection with Pseudoblocks
The notion of pseudoblocks was introduced in [7]


as follows: dimensional algebras, and i