Decomposing tensor products and exterior and symmetric squares

Abstract

Abstract. We describe a new algorithm for decomposing tensor products of indecomposable KG-modules into a direct sum of indecomposable KG-modules when K is a field of finite characteristic p and G a cyclic group of order q ¼ p t . We use this algorithm to extend reciprocity results of Gow and La¤ey relating the exterior and symmetric squares of indecomposable modules when p is odd.

On a Question of Glasby, Praeger, and Xia

“…Recall that for a sequence s and an integer k, s k denotes the kth term of the sequence s. The following result was proved in [1]. Therefore, m n p satisfies the following conditions in the corresponding cases: …”

“…In [1], we gave a recursive definition of the combinatorial object s p m n , where m ≤ n, and proved that…”

“…Let V 1 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 m ⊗ V n , where m ≤ n, into a direct sum of indecomposable KG-modules-for example, in order of publication, see [5], [10], [7], [9], [8], [6], and [1]. From the works of these authors, it is well known that V m ⊗ V n decomposes into a direct sum V 1 ⊕ · · · ⊕ V m of m indecomposable KG-modules with 1 ≥ · · · ≥ m > 0, but that the dimensions i of the components depend on the characteristic p. Following [3], we define the Jordan Partition m n p of mn by m n p = 1 m We say that m n p is standard iff i = m + n − 2i + 1 for 1 ≤ i ≤ m. Clearly, 1 n p is standard for every positive integer n and every prime p.…”

A New Algorithm for Decomposing Modular Tensor Products

“…(1.1) [2] We now explain the connection of λ(r, s, p) to the modular representation of a cyclic group G of order q = p α where r ≤ s ≤ q. For a field F of characteristic p, it is well known that there are exactly q isomorphism classes of indecomposable FG-modules and that such modules are cyclic and uniserial [1, pages 24-25].…”

Generators for decompositions of tensor products of modules associated with standard Jordan partitions

“…, 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 m ⊗ V n , where m ≤ n, into a direct sum of indecomposable KG-modules-for example, in order of publication, see [9], [15], [11], [12], [14], [10], [13], and [3]. From the works of these authors, it is well-known that V m ⊗V n decomposes into a direct sum V λ1 ⊕V λ2 · · ·⊕V λm of m indecomposable KG-modules where λ 1 ≥ λ 2 ≥ · · · ≥ λ m > 0, but that the dimensions λ i of the components depend on the characteristic p. Now λ(m, n, p) = (λ 1 , λ 2 , .…”