Applications

Abstract

This section is devoted to a very concrete problem. As usual, let K be a commutative ring with 1, and let UT_n (K) = UT_n and T_n (K) = T_n be the full unitriangular and triangular matrix groups of degree n over K, respectively. Denote by

$$ {\rm{u}}{{\rm{t}}_{\rm{n}}}{\rm{ = u}}{{\rm{t}}_{\rm{n}}}\left( {\rm{K}} \right){\rm{ = }}\left( {{{\rm{K}}^{\rm{n}}}{\rm{,U}}{{\rm{T}}_{\rm{n}}}\left( {\rm{K}} \right)} \right){\rm{ and }}{{\rm{t}}_{\rm{n}}}{\rm{ = }}{{\rm{t}}_{\rm{n}}}\left( {\rm{K}} \right){\rm{ = }}\left( {{{\rm{K}}^{\rm{n}}}{\rm{,}}{{\rm{T}}_{\rm{n}}}\left( {\rm{K}} \right)} \right) $$

their canonical representations in the free K-module of rank n. These classical objects deserve to be studied from various positions; in particular, from the standpoint of identities and varieties. Naturally, the first problem one should solve here is the following: to describe the varieties var ut_n and var t _n or, equivalently, to find bases for the identities of the representations ut _n and t _n.