On monomial linearisation and supercharacters of pattern subgroups

Column closed pattern subgroups U of the finite upper unitriangular groups U_n(q) are defined as sets of matrices in U_n(q) having zeros in a prescribed set of columns besides the diagonal ones. We explain Jedlitschky's construction of monomial linearisation in his thesis and apply this to CU yielding a generalisation of Yan's coadjoint cluster representations. Then we give a complete classification of the resulting supercharacters,by describing the resulting orbits and determining the Hom-spaces between orbit modules.     

Automorphisms and normal structure of unipotent subgroups of finitary Chevalley groups

We propose a method of constructing vector fields with certain vortex properties by means of transformations that change the value of the field vector at every point, the form of the field lines, and their mutual position. We discuss and give concrete examples of the prospects of using the method in applications involving solution of partial differential equations, including nonlinear ones.     

Mixing of the upper triangular matrix walk

We study a natural random walk over the upper triangular matrices, with entries in the field ${\mathbb{Z}_2}$ , generated by steps which add row i + 1 to row i. We show that the mixing time of the lazy random walk is O(n ²) which is optimal up to constants. Our proof makes key use of the linear structure of the group and extends to walks on the upper triangular matrices over the fields ${\mathbb{Z}_q}$ for q prime.     

Involutions for upper triangular matrix algebras

In this paper we describe completely the involutions of the first kind of the algebra UT_n(F) of n×n upper triangular matrices. Every such involution can be extended uniquely to an involution on the full matrix algebra. We describe the equivalence classes of involutions on the upper triangular matrices. There are two distinct classes for UT_n(F) when n is even and a single class in the odd case.Furthermore we consider the algebra UT₂(F) of the 2×2 upper triangular matrices over an infinite field F of characteristic different from 2. For every involution *, we describe the *-polynomial identities for this algebra. We exhibit bases of the corresponding ideals of identities with involution, and compute the Hilbert (or Poincaré) series and the codimension sequences of the respective relatively free algebras.Then we consider the *-polynomial identities for the algebra UT₃(F) over a field of characteristic zero. We describe a finite generating set of the ideal of *-identities for this algebra. These generators are quite a few, and their degrees are relatively large. It seems to us that the problem of describing the *-identities for the algebra UT_n(F) of the n×n upper triangular matrices may be much more complicated than in the case of ordinary polynomial identities.     

Constructing arithmetic subgroups of unipotent groups

Let G be a unipotent algebraic subgroup of some defined over . We describe an algorithm for finding a finite set of generators of the subgroup . This is based on a new proof of the result (in more general form due to Borel and Harish-Chandra) that such a finite generating set exists.     

Involutions and characters of upper triangular matrix groups

We study the realizability over of representations of the group of upper-triangular matrices over . We prove that all the representations of are realizable over if , but that if , has representations not realizable over . This theorem is a variation on a result that can be obtained by combining work of J. Arregi and A. Vera-López and of the authors, but the proof of the theorem in this paper is much more natural.

     

Jordan isomorphisms of upper triangular matrix rings

Let R be a 2-torsionfree ring with identity 1 and let T_n(R), n ? 2, be the ring of all upper triangular n × n matrices over R. We describe additive Jordan isomorphisms of T_n(R) onto an arbitrary ring and generalize several results on this line.     

Jordan homomorphisms of upper triangular matrix rings

In this paper we investigate Jordan homomorphisms of upper triangular matrix rings and give a sufficient condition under which they are necessarily homomorphisms or anti-homomorphisms.     

Commuting traces of upper triangular matrix rings

Let \(T_n(R)\) be the upper triangular matrix ring over a unital ring R. Suppose that \(B:T_n(R)\times T_n(R) \rightarrow T_n(R)\) is a biadditive map such that \(B(X,X)X = XB(X,X)\) for all \(X \in T_n(R)\). We consider the problem of describing the form of the map \(X\mapsto B(X,X)\).     

Polynomial functions on upper triangular matrix algebras

There are two kinds of polynomial functions on matrix algebras over commutative rings: those induced by polynomials with coefficients in the algebra itself and those induced by polynomials with scalar coefficients. In the case of algebras of upper triangular matrices over a commutative ring, we characterize the former in terms of the latter (which are easier to handle because of substitution homomorphism). We conclude that the set of integer-valued polynomials with matrix coefficients on an algebra of upper triangular matrices is a ring, and that the set of null-polynomials with matrix coefficients on an algebra of upper triangular matrices is an ideal.     

Functional identities in upper triangular matrix rings revisited

The aim of this paper is to give an improvement of a result on functional identities in upper triangular matrix rings obtained by Eremita, which presents a short proof of Eremita’s result.