It is well known that all irreducible representations of Chevalley groups over infinite fields and modular representations in nice characteristics of fields of definition are exhausted by subrepresentations of tensor products of their natural representations. We consider two specific subrepresentations of this kind and use them to answer two questions on the number of generating involutions of some matrix groups. For an integral domain $D$ of characteristic different from 2, we establish the irreducibility of the symmetric and external squares of the natural representation of the group $SL_n(D)$ and find their kernels (Theorem 1). Denote by $n(G)$ (by $n_c(G)$) the minimum number of generating (and also conjugate, respectively) involutions of $G$ whose product is 1. Problems on finding the numbers $n(G)$ and $n_c(G)$ for finite simple groups are written by the author in the Kourovka Notebook (Question 14.69). Based on Theorem 1 and L.L. Scott's inequality, we prove the following result. Let $G$ be $SL_3(D)$ or $SL_6(D)$, where $D$ is an integral domain of characteristic different from 2. Then $n(G)>5$ and, in particular, $G$ is not generated by three involutions two of which commute; moreover, if $D$ is the ring of integers or a finite field (of odd order), then $n(G)=n_c(G)=6$ (Theorem 2).
Keywords: special linear group over the integral domain, tensor representations, generating sets of involutions
Received May 10, 2020
Revised July 6, 2020
Accepted July 20, 2020
Funding Agency: This work was supported by the Krasnoyarsk Mathematical Center, which is financed by the Ministry of Science and Higher Education of the Russian Federation within the project for the establishment and development of regional centers for mathematical research and education (agreement no. 075-02-2020-1534/1), and by the Russian Foundation for Basic Research (project no. 19-01-00566).
Yakov Nifant’evich Nuzhin, Dr. Phys.-Math. Sci., Prof., Siberian Federal University, pr. Svobodnyi 79, Krasnoyarsk, 660041 Russia, e-mail: nuzhin2008@rambler.ru
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Cite this article as: Ya.N. Nuzhin. Tensor representations and generating sets of involutions of some matrix groups, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 3, pp. 133–141.
