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

TL;DR

This paper studies how elements in certain upper triangular matrix groups over rings can be expressed as products of commutators of involutions, skew-involutions, and finite order matrices, providing explicit decompositions and extending to infinite matrices.

Contribution

It introduces new methods to decompose elements of upper triangular matrix groups into products of commutators of involutions and finite order matrices, including infinite matrices, with explicit bounds.

Findings

Every element in specific upper triangular matrix groups can be expressed as a product of two commutators of involutions.

Elements can also be written as products of commutators of skew-involutions and involutions.

Explicit bounds on the number of commutators needed for elements in infinite matrix groups are provided.

Abstract

Let $R$ be an associative ring with unity $1$ and consider that $2,k$ and $2k\in \mathbb{N}$ are invertible in $R$. For $m\geq 1$ denote by $UT_n(m,R)$ and $UT_{\infty}(m,R)$, the subgroups of $UT_n(R)$ and $UT_{\infty}(R)$ respectively, which have zero entries on the first $m-1$ super diagonals. We show that every element on the groups $UT_n(m,R)$ and $UT_{\infty}(m,R)$ 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 $UT_{\infty}(m,R)$. Similarly, denote by $UT^{(s)}_{\infty}(R)$ the group of upper triangular infinite matrices whose diagonal entries are $s$th roots of $1$. We show that every element of the groups $UT_n(\infty,R)$ and $UT_{\infty}(m,R)$ can be expressed as a product of $4k-6$ commutators all depending of powers of elements in $UT^{(k)}_{\infty}(m,R)$ of order $k$ and, also, can be expressed as a product of $8k-6$ commutators of skew finite matrices of order $2k$ and matrices of order $2k$ in $UT^{(2k)}_{\infty}(m,R)$. If $R$ is the complex field or the real number field we prove that, in $SL_n(R)$ and in the subgroup $SL_{VK}(\infty,R)$ 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.