On groups elementarily equivalent to a group of triangular matrices $T_n(R)$

三角矩阵的定律

Laws for triangular matrices

该研究探讨域上及含0和1的加法幂等半环上三角矩阵幺半群的恒等式，发现部分此类恒等式无有限公理化的新情况。

Conjugacy classes of centralizers in the group of upper triangular matrices

Let G be a group. Two elements x and y in G are said to be in the same z-class if their centralizers in G are conjugate within G. In this paper, we prove that the number of z-classes in the group of upper triangular matrices is infinite provided that the field is infinite and size of the matrices is at least 6, and finite otherwise.

The finite basis problem for the monoid of 2 by 2 upper triangular tropical matrices

For each positive $n$, let $u_n = v_n$ denote the identity obtained from the Adjan identity $(xy) (yx) (xy) (xy) (yx) = (xy) (yx) (yx) (xy) (yx)$ by substituting $(xy) \rightarrow (x_1 x_2 \dots x_n)$ and $(yx) \rightarrow (x_n \dots x_2 x_1)$. We show that every monoid which satisfies $u_n = v_n$ for each positive $n$ and generates the variety containing the bicyclic monoid is nonfinitely based. This implies that the monoid of 2 by 2 upper triangular tropical matrices over the tropical semiring is nonfinitely based.

三角布尔矩阵的恒等式

Identities of triangular Boolean matrices

本文给出上三角布尔矩阵半环中恒等式的组合刻画，并应用于复杂度、有限公理化及可识别语言类的研究。

A note on identities in plactic monoids and monoids of upper-triangular tropical matrices

This paper uses the combinatorics of Young tableaux to prove the plactic monoid of infinite rank does not satisfy a non-trivial identity, by showing that the plactic monoid of rank $n$ cannot satisfy a non-trivial identity of length less than or equal to $n$. A new identity is then proven to hold for the monoid of $n \times n$ upper-triangular tropical matrices. Finally, a straightforward embedding is exhibited of the plactic monoid of rank $3$ into the direct product of two copies of the monoid of $3\times 3$ upper-triangular tropical matrices, giving a new proof that the plactic monoid of ra…

On double cosets of groups $GL(n)$ with respect to subgroups of block strictly triangular matrices

We parametrize the space of double cosets of the group $GL(n,\Bbbk)$ with respect to two subgroups $T_-$, $T_+$ of block strictly triangular matrices. In Addendum, we consider the quasi-regular representation of $GL(n,\Bbb{C})$ in $L^2$ on $T_- \ GL(n,\Bbb{C})$, observe that it admits an additional group of symmetries, find the joint spectrum, and observe that it is multiplicity free.