Branching rules and commuting probabilities for Triangular and Unitriangular matrices

This paper concerns the enumeration of simultaneous conjugacy classes of [Formula: see text]-tuples of commuting matrices in the upper triangular group [Formula: see text] and unitriangular group [Formula: see text] over the finite field [Formula: see text] of odd characteristic. This is done for [Formula: see text] and [Formula: see text], by computing the branching rules. Further, using the branching matrix thus computed, we explicitly get the commuting probabilities [Formula: see text] for [Formula: see text] in each case.

p-power conjugacy classes in U(n,q) and T(n,q)

Let [Formula: see text] be a [Formula: see text]-power where [Formula: see text] is a fixed prime. In this paper, we look at the [Formula: see text]-power maps on unitriangular group [Formula: see text] and triangular group [Formula: see text]. In the spirit of Borel dominance theorem for algebraic groups, we show that the image of this map contains large size conjugacy classes. For the triangular group we give a recursive formula to count the image size.

Distortion of embeddings of a torsion-free finitely generated nilpotent group into a unitriangular group

In this paper, we study distortion of various well-known embeddings of finitely generated torsion-free nilpotent groups [Formula: see text] into unitriangular groups [Formula: see text]. In particular, we show that there is no undistorted embeddings of [Formula: see text]-dimensional Heisenberg group into [Formula: see text]. We also provide a polynomial time algorithm for finding distortion of a given subgroup of [Formula: see text].

Triangular Automorphisms

This chapter focuses on triangular automorphisms, which can be analyzed by Lie techniques. Throughout the discussion K is a commutative ring containing ℚ as a subring. A formalism is introduced to analyze triangular automorphisms of such a polynomial algebra by means of their logarithms, the triangular derivations. After presenting some definitions and simple facts about filtered modules, filtered algebras, and graded algebras, the chapter considers triangular linear maps and the Lie algebra of an algebraic unitriangular group. It then describes derivations on the ring of column-finite matrices, along with iteration matrices and Riordan matrices. It also explains derivations on polynomial rings and concludes by applying triangular automorphisms to differential polynomials.

FIELD OF INVARIANTS OF BORELEAN GROUP OF ADJOINT REPRESENTATION OF GL(n,K)

The paper is devoted to invariant theory problems, in particular to the problem of finding generators of invariant fields in an explicit form. The set of generators is given for invariant field of unitriangular group concerning the ad-joint representation of GL(n, K) group. Moreover, the set of generators of Borel group for the field of invariants is constructed and their algebraic independence is proved. Lie group;adjoint representation;field of invariant;generators of the field of invariants;Borel group;

CLASS PRESERVING AUTOMORPHISMS OF UNITRIANGULAR GROUPS

Let UTn(K) be a unitriangular group over a field K. We prove that the group of all class preserving automorphisms of UTn(K) is equal to Inn ( UTn(K)) if and only if K is a prime field. Let [Formula: see text], where γ3( UTn(𝔽pm)) denotes the third term of the lower central series of UTn(𝔽pm). We calculate the group of all class preserving automorphisms and class preserving outer automorphisms of [Formula: see text].