# Applications

• Samuel M. Vovsi
Part of the Progress in Mathematics book series (PM, volume 17)

## Abstract

This section is devoted to a very concrete problem. As usual, let K be a commutative ring with 1, and let UTn (K) = UTn and Tn (K) = Tn 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 utn and var tn or, equivalently, to find bases for the identities of the representations utn and tn.

## Keywords

Radical Class Nilpotent Group Wreath Product Faithful Representation Dedekind Domain
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