论文标题
将矩阵表达成互动，偏斜措施，有限订单和偏斜有限订单矩阵的矩阵
Expressing Matrices Into Products of Commutators of Involutions, Skew-Involutions, Finite Order and Skew Finite Order Matrices
论文作者
论文摘要
让$ r $是一个具有Unity $ 1 $的联想戒指，并认为$ 2，K $和$ 2K \ in \ Mathbb {n} $在$ r $中是可逆的。对于$ m \ geq 1 $ $ ut_n（m，r）$和$ ut _ {\ infty}（m，r）$，分别为$ ut_n（r）$和$ ut _ {\ infty}（r）$的子组，在第一个$ M-1 $ $ super uigonals中为零入口。我们表明，组上的每个元素$ ut_n（m，r）$和$ ut _ {\ infty}（m，r）$可以表示为两个换向者的产物，而且也可以表示为两个偏斜率和参与的换向器的产物。同样，用$ ut^{（s）} _ {\ infty}（r）$表示上层三角形无限矩阵的组，其对角线条目为$ 1 $ 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 $8k-6$偏斜的有限矩阵订单$ 2K $和订单矩阵的换向器$ 2K $ in $ ut^{（2k）} _ {\ infty}（m，r）$。如果$ r $是复杂的字段或实际数字字段，我们证明，在$ sl_n（r）$中，在子组$ sl_ {vk}（\ infty，r）$中，Vershik-kerov组的$ r $超过$ r $，这些组中的每个元素都可以将这些元素分解为如上所述的元素的元素的产品。
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.
