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 introduced and shown to have a connection with the Selberg integral. This connection gives an enumeration formula for Young books. It is shown that special cases...

We present a generalisation of the famous Selberg integral. This confirms the g = An case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral for each simple Lie algebra g. Résumé. On présente une généralisation de la bien connue intégrale de Selberg. Cette généralisation vérifie le cas g = An de la conjecture de Mukhin et Varchenko concernant l’existence d’un...

A fundamental problem in number theory is to estimate the values of L-functions at the center of the critical strip. The Langlands program predicts that all L-functions arise from automorphic representations of GL(N) over a number field, and moreover that such L-functions can be decomposed as a product of primitive L-functions arising from irreducible cuspidal representations of GL(n) over Q. T...

Let K be a function field of odd characteristic, and let π (resp., η) be a cuspidal automorphic representation of GL2(AK ) (resp., GL1(AK )). Then we show that a weighted sum of the twists of L(s, π) by quadratic characters χD , ∑ D L(s, π ⊗ χD) a0(s, π, D) η(D) |D|, is a rational function and has a finite, nonabelian group of functional equations. A similar construction in the noncuspidal case...

In this paper we consider Selberg-type square matrices integrals with focus on Kummer-beta types I & II integrals. For generality of the results for real normed division algebras, the generalized matrix variate Kummer-beta types I & II are defined under the abstract algebra. Then Selberg-type integrals are calculated under orthogonal transformations.

in this paper we consider selberg-type square matrices integrals with focus on kummer-beta types i & ii integrals. for generality of the results for real normed division algebras, the generalized matrix variate kummer-beta types i & ii are defined under the abstract algebra. then selberg-type integrals are calculated under orthogonal transformations.

In a previous article [35] an algebraicity result for the central critical value for L-functions for GLn × GLn−1 over Q was proved assuming the validity of a nonvanishing hypothesis involving archimedean integrals. The purpose of this article is to generalize [35, Thm. 1.1] for all critical values for L-functions for GLn×GLn−1 over any number field F while using the period relations of [37] and...

For cofinite Kleinian groups, with finite-dimensional unitary representations, we derive the Selberg trace formula. As an application we define the corresponding Selberg zeta-function and compute its divisor, thus generalizing results of Elstrodt, Grunewald and Mennicke to non-trivial unitary representations. We show that the presence of cuspidal elliptic elements sometimes adds ramification po...

It has been remarked that a fair measure of the impact of Atle Selberg’s work is the number of mathematical terms that bear his name. One of these is the Selberg integral, an n-dimensional generalization of the Euler beta integral. We trace its sudden rise to prominence, initiated by a question to Selberg from Enrico Bombieri, more than thirty years after its initial publication. In quick succe...