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 $\mathrm{char}(K) = p$.
Keywords: finite group, algebraic group, group representation, tensor product of modules
Received November 22, 2020
Revised December 30, 2020
Accepted January 11, 2021
Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 20-01-00456).
Anton Vladimirovich Konygin, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia e-mail: konygin@imm.uran.ru
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Cite this article as: A.V. Konygin. On a question concerning the tensor product of modules, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 1, pp. 103–109.
