Problems In Invariant Theory Research Articles

Non-Commutative Arithmetic Circuits with Division.

We initiate the study of the complexity of arithmetic circuits with division gates over non-commuting variables. Such circuits and formulas compute non-commutative functions, which, despite their name, can no longer be expressed as ratios of polynomials. We prove some lower and upper bounds, completeness and simulation results, as follows. If X is n x n matrix consisting of n2 distinct mutually non-commuting variables, we show that: (i). X-1 can be computed by a circuit of polynomial size, (ii). every formula computing some entry of X-1 must have size at least 2Ω(n). We also show that matrix inverse is complete in the following sense: (i). Assume that a non-commutative function f can be computed by a formula of size s. Then there exists an invertible 2s x 2s-matrix A whose entries are variables or field elements such that f is an entry of A-1. (ii). If f is a non-commutative polynomial computed by a formula without inverse gates then A can be taken as an upper triangular matrix with field elements on the diagonal. We show how divisions can be eliminated from non-commutative circuits and formulae which compute polynomials, and we address the non-commutative version of the rational function identity testing problem. As it happens, the complexity of both of these procedures depends on a single open problem in invariant theory.

An application of global Weyl modules of [formula omitted] to invariant theory

We identify the sl n + 1 isotypical components of the global Weyl modules W ( k ω 1 ) with certain natural subspaces of the polynomial ring in k variables. We then apply the representation theory of current algebras to classical problems in invariant theory.

Invariants, equivariants and characters in symmetric bifurcation theory

In the analysis of stability in bifurcation problems it is often assumed that the (appropriate reduced) equations are in normal form. In the presence of symmetry, the truncated normal form is an equivariant polynomial map. Therefore, the determination of invariants and equivariants of the group of symmetries of the problem is an important step. In general, these are hard problems of invariant theory and, in most cases, they are tractable only through symbolic computer programs. Nevertheless, it is desirable to obtain some of the information about invariants and equivariants without actually computing them, for example, the number of linearly independent homogeneous invariants or equivariants of a certain degree. Generating functions for these dimensions are generally known as ‘Molien functions’. We obtain formulae for the number of linearly independent homogeneous invariants or equivariants for Hopf bifurcation in terms of characters. We also show how to construct Molien functions for invariants and equivariants for Hopf bifurcation. Our results are then applied to the computation of the number of invariants and equivariants for Hopf bifurcation for several finite groups and the continuous group O(3).

On the cohomology and the Chow ring of the classifying space of PGL p

We investigate the integral cohomology ring and the Chow ring of the classifying space of the complex projective linear group PGL_p, when p is an odd prime. In particular, we determine its additive structure completely, and we reduce the problem of determing its multiplicative structure to a problem in invariant theory.