On the jordan form of the tensor product over fields of prime characteristic

Abstract

L-et 4. B denote the companion matrice\ of the polynomials rm. u" ovcr a field r ofprime order r, and let i. p be 11011-~efo elements oidn extension field K of F. The Jordan torm of the tensor product ( I ! t A ) @ (pi t B) of in\ertibleJordan matrices ovcr K is determmed vla an equivalent study oithe niipotent transformation 5 of rn x n matricer X over F, where (X)S = A T X I-XB. Ubing mod~de-theoretic concepts a Jordan bas18 for S 18 spccificd recursively ~n terms of the representations of m and n in the scale of p, znd rednction formulae for the ?!em-ntnrv divisor? of S 2 r e estah!ished.

On Jordan bases for the tensor product and Kronecker sum and their elementary divisors over fields of prime characteristic

“…, r 0 }. See p. 354 of [11] for the proof of Lemma 4. Notice that ¼ (m, n) as in Lemma 4 is non-increasing if and only if lðm, nÞ ¼ ðl 1 , l 2 , .…”

“…It is convenient to concentrate exclusively on S(m, n) which we regard as a linear mapping of V ¼ Vðm, nÞ ¼ m F n since c is a basis of V and (1.5) is a matrix over the prime field F. We recall some terminology from [11]. Write V r ¼ hEði, jÞ : i þ j ¼ r þ 1i ð 3:1Þ…”

Generators for decompositions of tensor products of modules

“…, V q } be a set of representatives of these isomorphism classes with dim V i = i. Many authors have investigated the decomposition of the KG-module V n ⊗ V m , where n ≥ m, into a direct sum of indecomposable KG-modules-for example, in order of publication, see [3][4][5][6][7][8][9][10]. From the works of these authors, it is well-known that V n ⊗ V m decomposes into a direct sum of m indecomposable KG-modules, but that the dimensions of the components depend on the characteristic p. However, in the special case of p ≥ n + m − 1 that we consider in this paper,…”

Decomposing tensor products and exterior and symmetric squares

“…In describing Norman's algorithm, we follow Hou's account [3, Theorem 2.2]. In the notation of [6] and [3], iðmÞ is the identity permutation of f1; . .…”

“…Many authors have investigated the decomposition of V m n V n into a direct sum of indecomposable KG-modules. See [2], [3], [5], [6], [7], and [8].…”