Generators for Tensor Product Components

TL;DR

This work develops an explicit, recursive method to identify generators for each indecomposable component in the tensor product $V_r\otimes V_s$ of indecomposable modules over a cyclic $p$-group in characteristic $p$. Building on Renaud’s decomposition and the $J(r,s)$-matrix framework, the authors express each component’s generator through a carefully chosen $F$-basis and lift it via binomial-entry matrices, with the lifting guided by the action of $(g-1)$. A key innovation is the use of explicit inverses of binomial matrices (via Nordenstam–Young and Roberts results) to perform the liftings, enabling a precise construction of $y_\ell\in D_{r+s-\ell}$ satisfying $(g-1)^{\lambda_\ell-1}(y_\ell) = x_{r+s+1-\ell-\lambda_\ell}$. The approach treats both generic and degenerate cases ($m_{n-1}>0$ and $m_{n-1}=0$), yielding a complete, explicit generator description for all indecomposable tensor components with direct applicability to computational and theoretical decomposition problems in modular representation theory.

Abstract

Let $p$ be a prime number, $F$ a field of characteristic $p$, and $G$ a cyclic group of order $q =p^a$ for some positive integer $a$. Under these circumstances every indecomposable $F G$-module is cyclic. For indecomposable $F G$-modules $U$ and $W$, we present a new recursive method for identifying a generator for each of the indecomposable components of $U \otimes W$ in terms of a particular $F$-basis of $U \otimes W$.

Theorems & Definitions (67)

• Lemma 1
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• Lemma 2
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• Theorem 1
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• Corollary 1
• Lemma 3
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• Lemma 4