On а question concerning the tensor product of modules

A. V. Konygin

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 1, pp. 103-109

Résumé

Assume that $G$ is a group, $K$ is an algebraically closed field, and $V_1$ and $V_2$ are $KG$-modules. The following question is considered: under what constraints on $G$, $K$, $V_1$, and $V_2$ does $V_1 \otimes V_2 \cong V_1 \otimes I$ hold, where $I$ is the trivial $KG$-module (of dimension $\dim(V_2)$)? Earlier, when considering a problem of P. Cameron on finite primitive permutation groups, the author obtained and used some results on this question. This work continues the study of the question. The following results were obtained. 1. Assume that $G$ is a nontrivial connected reductive algebraic group, and $V_1$ and $V_2$ are faithful semisimple $KG$-modules. Then $V_1 \otimes V_2 \ncong V_1 \otimes I$. 2. Assume that $G$ is a nontrivial finite group, $\mathrm{char} (K) = 0$, $V_1$ is a $KG$-module, and $V_2$ is a faithful $KG$-module. Then $V_1 \otimes V_2 \cong V_1 \otimes I $ if and only if $V_1$ is the direct sum of $\frac {\dim (V_1)} {|G|}$ regular $KG$-modules. In addition, we consider the question of the possibility that $V_1 \otimes V_2 \cong V_1 \otimes I$ in the case where $G = SL_2(p^n)$, $V_1$ and $V_2$ are simple $KG$-modules, and $\mathrm{char}(K) = p$.

Détail

Keywords: finite group, group representation, tensor product of modules.
Mots-clés : algebraic group