Expressing Matrices Into Products of Commutators of Involutions, Skew-Involutions, Finite Order and Skew Finite Order Matrices

Ivan Italo Gonzales Gargate; Michael Santos Gonzales Gargate

paper

Expressing Matrices Into Products of Commutators of Involutions, Skew-Involutions, Finite Order and Skew Finite Order Matrices

arXiv:2007.12305

Abstract

Let $R$ be an associative ring with unity $1$ and consider that $2, k$ and $2 k \in N$ are invertible in $R$ . For $m \geq 1$ denote by $U T_{n} (m, R)$ and $U T_{\infty} (m, R)$ , the subgroups of $U T_{n} (R)$ and $U T_{\infty} (R)$ respectively, which have zero entries on the first $m - 1$ super diagonals. We show that every element on the groups $U T_{n} (m, R)$ and $U T_{\infty} (m, R)$ can be expressed as a product of two commutators of involutions and also, can be expressed as a product of two commutators of skew-involutions and involutions in $U T_{\infty} (m, R)$ . Similarly, denote by $U T_{\infty}^{(s)} (R)$ the group of upper triangular infinite matrices whose diagonal entries are $s$ th roots of $1$ . We show that every element of the groups $U T_{n} (\infty, R)$ and $U T_{\infty} (m, R)$ can be expressed as a product of $4 k - 6$ commutators all depending of powers of elements in $U T_{\infty}^{(k)} (m, R)$ of order $k$ and, also, can be expressed as a product of $8 k - 6$ commutators of skew finite matrices of order $2 k$ and matrices of order $2 k$ in $U T_{\infty}^{(2 k)} (m, R)$ . If $R$ is the complex field or the real number field we prove that, in $S L_{n} (R)$ and in the subgroup $S L_{V K} (\infty, R)$ of the Vershik-Kerov group over $R$ , each element in these groups can be decomposed into a product of commutators of elements as described above.

15 pages