ON Product Representations of Powers, I.

The solvability of the equation a(1)a(2)... a(k) = x(2), a(1), a(2),..., a(k) epsilon A is Studied for fixed k and 'dense' sets A of positive integers. In particular, it is shown that if k is even and k greater than or equal to 4, and A is of positive upper density, then this equation can be solved. (C) 1995 Academic Press Limite

Kronecher powers and character polynomials

In this talk, I will present joint works with Cedric Chauve and Adriano Garsia. With C. Chauve, we studied Kronecker powers of the irreducible representation of Sn indexed with (n-1,1). We gave a combinatorial interpretation and a generating function for the coefficients of any irreducible representation in a k-th Kronecker power ( χ(n-1,1) )⊗k. With A. Garsia, we studied character polynomials qλ(x1,…,xn) which are polynomials in several variables with the fundamental property that their evaluation on the multiplicities (m1,m2, …,mn) of a partition µ of n gives the value of the irreducible character χ( n- | λ | , λ ) of the symmetric group Sn on the conjugacy class Cµ . Character polynomials are closely related to the problem of decomposition of Kronecker product of representations of Sn. They were defined by Specht in 1960. Since then they received little attention from the combinatorics community. I will show how character polyomials are related to Kronecker products, how to produce them, their algebraic structure and show some applications

k-String tensions and the 1/N expansion

We address the question of whether the large-N expansion in pure SU(N) gauge theories requires that k-string tensions must have a power series expansion in 1/N^2, as in the sine law, or whether 1/N contributions are also allowable, as in Casimir scaling. We find that k-string tensions may, in fact, have 1/N corrections, and consistency with the large-N expansion in the open-string sector depends crucially on an exact cancellation, which we will prove, among terms involving odd powers of 1/N in particular combinations of Wilson loops. It is shown how these cancellations are fulfilled, and consistency with the large-N expansion achieved, in a concrete example, namely, strong-coupling lattice gauge theory with the heat-kernel action. This is a model which has both a 1/N^2 expansion and Casimir scaling of the k-string tensions. Analysis of the closed string channel in this model confirms our conclusions, and provides further insights into the large-N dependence of energy eigenstates and eigenvalues.Comment: RevTeX4, 21 pages. Typos corrected, references added, some discussions expanded; conclusions unchanged. Version to appear on PR

k-String tensions and their large-N dependence

We consider whether the 1/N corrections to k-string tensions must begin at order 1/N^2, as in the Sine Law, or whether odd powers of 1/N, as in Casimir Scaling, are also acceptable. The issue is important because different models of confinement differ in their predictions for the representation-dependence of k-string tensions, and corrections involving odd powers of 1/N would seem to be ruled out by the large-N expansion. We show, however, that k-string tensions may, in fact, have leading 1/N corrections, and consistency with the large-N expansion, in the open string sector, is achieved by an exact pairwise cancellation among terms involving odd powers of 1/N in particular combinations of Wilson loops. It is shown how these cancellations come about in a concrete example, namely, strong coupling lattice gauge theory with the heat-kernel action, in which k-string tensions follow the Casimir scaling rule.Comment: Talk presented at the XXIX International Symposium on Lattice Field Theory - Lattice 2011, July 10-16, 2011, Squaw Valley, Lake Tahoe, Californi

Remarks on the symmetric powers of cusp forms on GL(2)

In this paper we prove the following conditional result: Let F be a number field, and pi a cusp form on GL(2)/F which is not solvable polyhedral. Assume that all the symmetric powers sym^m(pi) are modular, i.e., define automorphic forms on GL(m+1)/F. If sym^6(pi) is cuspidal, then all the symmetric powers are cuspidal, for all m. Moreover, sym^6(pi) is Eisenteinian iff sym^5(pi) is an abelian twist of the functorial product of pi with the symmetric square of a cusp form pi' on GL(2)/F.Comment: A sentence has been modified in the Introduction. It has nothing to do with the main result of the pape

On simplicity of reduced C*-algebras of groups

A countable group is C*-simple if its reduced C*-algebra is a simple algebra. Since Powers recognised in 1975 that non-abelian free groups are C*-simple, large classes of groups which appear naturally in geometry have been identified, including non-elementary Gromov hyperbolic groups and lattices in semisimple groups. In this exposition, C*-simplicity for countable groups is shown to be an extreme case of non-amenability. The basic examples are described and several open problems are formulated.Comment: 23 page