Related papers: Combinatorial Proof of Selberg's Integral Formula
The classical Selberg integral contains a power of the Vandermonde determinant. When that power is a square, it is easy to prove Selberg's identity by interpreting it as a determinant of one-variable integrals. We give similar proofs of…
In this paper, we describe a general method for computing Selberg-like integrals based on a formula, due to Kaneko, for Selberg-Jack integrals. The general principle consists in expanding the integrand \emph{w.r.t.} the Jack basis, which is…
We extend Venkatesh's proof of the converse theorem for classical holomorphic modular forms to arbitrary level and character. The method of proof, via the Petersson trace formula, allows us to treat arbitrary degree 2 gamma factors of…
A set of recursive relations satisfied by Selberg-type integrals involving monomial symmetric polynomials are derived, generalizing previously known results. These formulas provide a well-defined algorithm for computing Selberg-Schur…
We prove a generalization of the $q$-Selberg integral evaluation formula. The integrand is that of $q$-Selberg integral multiplied by a factor of the same form with respect to part of the variables. The proof relies on the quadratic norm…
The Selberg integral is an important integral first evaluated by Selberg in 1944. Stanley found a combinatorial interpretation of the Selberg integral in terms of permutations. In this paper, new combinatorial objects "Young books" are…
We prove a generalization of the topological Tverberg theorem. One special instance of our general theorem is the following: Let $\Delta$ denote the 8-dimensional simplex viewed as an abstract simplicial complex, and suppose that its…
A new approach to the Selberg trace formula, and more precisely to its spectral side, is developed. The approach relies on a notion of "Plancherel decomposition" of "asymptotically finite functions", and may generalize to obtain a general…
We prove an $\mathbb F_p$-Selberg integral formula, in which the $\mathbb F_p$-Selberg integral is an element of the finite field $\mathbb F_p$ with odd prime number $p$ of elements. The formula is motivated by analogy between…
This note contains a new combinatorial proof of Cramer's rule based on the Gessel-Viennot-Lindstrom Lemma.
Recently, Grynkiewicz et al. [{\it Israel J. Math.} {\bf 193} (2013), 359--398], using tools from additive combinatorics and group theory, proved necessary and sufficient conditions under which the linear congruence $a_1x_1+\cdots…
We prove the Strengthened Hanna Neumann Conjecture, in its common graph theoretic formulation. Our original approach to this conjecture used cohomology of sheaves on graphs, although here we give a short combinatorial proof that we found in…
We present an elliptic version of Selberg's integral formula.
We prove the universality theorem for the iterated integrals of logarithms of $L$-functions in the Selberg class on some line parallel to the real axis.
Selberg introduced his beautiful integral formula in 1944, see [Sel]. Evans [E1] conjectured a finite field analog of Selberg integral formula in 1980. And Anderson [An] proved a major case of it in 1981 and his ideas was used to obtained…
In the paper we introduce the new approach how to use an orthonormality relation of coefficients of Dirichlet series defining given L-functions from the Selberg class to prove joint universality.
The topological Tverberg theorem claims that for any continuous map of the (q-1)(d+1)-simplex to R^d there are q disjoint faces such that their images have a non-empty intersection. This has been proved for affine maps, and if $q$ is a…
We present an elliptic version of Selberg's integral formula.
The goal of this paper is to lay the foundations for a combinatorial study, via orthogonal functions and intertwining operators, of category O for the rational Cherednik algebra of type G(r,p,n). As a first application, we give a…
In a recent paper Richards and Zheng compute the determinant of a matrix whose entries are given by beta-type integrals, thereby generalising an earlier result by Dixon and Varchenko. They then use their result to obtain a generalisation of…