Related papers: Counter-example to conjectured SU(N) character asy…

Let \chi be an irreducible character of the finite group G. If g is an element of G and \chi(g) is not zero, then we conjecture that the order of g divides |G|/\chi(1). The conjecture is a generalization of the classical fact that…

Let $\Gamma$ be a finitely generated group and $G$ a real form of $\mathrm{SL}_n(\mathbb{C})$. We propose a definition for the $G$-character variety of $\Gamma$ as a subset of the $\mathrm{SL}_n(\mathbb{C})$-character variety of $\Gamma$.…

We generalize the methods in previous work to provide a program for proving Singer's Conjecture for Coxeter systems. Specifically, we consider even Coxeter systems with nerves that are flag triangulations of $\BS^{n-1}$, $n=2k$. We prove…

In this paper we prove a conjecture of Kudla and Rallis. Let $\chi$ be a unitary character, $s\in \mathbb{C}$ and $W$ a symplectic vector space over a non-archimedean field with symmetry group $G(W)$. Denote by $I(\chi,s)$ the degenerate…

In previous work of this author it was conjectured that the sum of power sums $p_\lambda,$ for partitions $\lambda$ ranging over an interval $[(1^n), \mu]$ in reverse lexicographic order, is Schur-positive. Here we investigate this…

Let $\mathbb{G}$ be a connected reductive group over $\mathcal{O}$, a complete discrete valuation ring with finite residue field $\mathbb{F}_q$. Let $R_{T_r,U_r}^{\theta}$ be a level $r$ Deligne--Lusztig representation of…

We establish the existence of an irreducible representation of $A_n$ whose dimension does not occur as the dimension of an irreducible representation of $S_n$, and vice versa. This proves a conjecture by Tong-Viet. The main ingredient in…

Recently, Keating and the second author of this paper devised a heuristic for predicting asymptotic formulas for moments of the Riemann zeta-function $\zeta(s)$. Their approach indicates how lower twisted moments of $\zeta(s)$ may be used…

Nous refutons, sous une certaine hypothese combinatoire, la "nonrevisiting path conjecture". Abstract: In this article, we give, under some hypothesis, a couterexample to the nonrevisiting path conjecture.

The covering number of a non-linear character $\chi$ of a finite group $G$ is the least positive integer $k$ such that every irreducible character of $G$ occurs in $\chi^k$. We determine the covering numbers of irreducible characters of the…

We compute the asymptotics of matrix elements in canonical bases of irreducible representations of the unitary group as the highest weight goes to infinity, in terms of the symplectic geometry of the associated coadjoint orbit. This uses…

We study (asymmetric) $U$-statistics based on a stationary sequence of $m$-dependent variables; moreover, we consider constrained $U$-statistics, where the defining multiple sum only includes terms satisfying some restrictions on the gaps…

We investigate the Reshetikhin--Turaev invariants associated to SU(2) for the 3-manifolds M obtained by doing any rational surgery along the figure 8 knot. In particular, we express these invariants in terms of certain complex double…

We consider a family of character sums as multiplicative analogues of Kloosterman sums. Using Gauss sums, Jacobi sums and Deligne's bound for hyper-Kloosterman sums, we establish asymptotic formulae for any real (positive) moments of the…

String representations of the Wilson loop are constructed in the SU(N)-version of compact QED in three and four dimensions. This is done exactly in the case of the fundamental Wilson loop and in the large-N limit in the case of the adjoint…

We find an intriguing relation between the chiral algebra and the mixed Hodge structure of the Coulomb branch of four dimensional $\mathcal{N} = 2$ superconformal field theories. We identify the space of irreducible characters of the…

We establish an explicit formula for the character of an irreducible finite-dimensional representation of $\mathfrak{gl}(m|n)$. The formula is a finite sum with integer coefficients in terms of a basis $\mathcal{E}_{\mu}$ (Euler characters)…

Let $\mathcal{SS}_k(n)$ be the family of {\it shuffle squares} in $[k]^{2n}$, words that can be partitioned into two disjoint identical subsequences. Let $\mathcal{RSS}_k(n)$ be the family of {\it reverse shuffle squares} in $[k]^{2n}$,…

We calculate the large quantum level asymptotic expansion of the RT-invariants associated to SU(2) of all oriented Seifert 3-manifolds X with orientable base or non-orientable base with even genus. Moreover, we identify the Chern-Simons…

The Wehrl entropy conjecture for coherent (highest weight) states in representations of the Heisenberg group, which was proved in 1978 and recently extended by us to the group $SU(2)$, is further extended here to symmetric representations…