A simple integral formula as an iterated residue is presented for the Baker-Akhiezer function related to $A_n$ type root system both in the rational and trigonometric cases. We present also a formula for the Baker-Akhiezer function as a Selberg-type integral and generalise it to the deformed $A_{n,1}$-case. These formulas can be interpreted as new cases of explicit evaluation of Selberg-type integrals.

2008-07-24math-phmath.MPmath.QA

The importance of the Selberg integral

It has been remarked that a fair measure of the impact of Atle Selberg's work is the number of mathematical terms which 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 publication. In quick succession the Selberg integral was used to prove an outstanding conjecture in random matrix theory, and cases of the Macdonald conjectures. It further initiated the study of q-analogues, which in turn enriched the Macd…

2007-10-22math-phmath.CAmath.CO

Elliptic Selberg integrals and conformal blocks

We present an elliptic version of Selberg's integral formula.

2002-10-03math.CAmath.QA

Elliptic Selberg integrals

We present an elliptic version of Selberg's integral formula.

2002-10-02math.CAmath.QA

Selberg Integrals

The paper is written for Kluwer's Encyclopaedia of Mathematics.

2004-10-11math.QA

A generalization of Selberg integral

We analyze the situation which is related to zonal spherical functions of type $A_n$ and obtain a generalization of Selberg integral.

1995-07-17math.QAq-alg

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Selberg integral and multiple zeta values

In this paper, we show that the coefficient of the Taylor expansion of Selberg integrals with respect to exponent variables are expressed as a linear combination of multiple zeta values. We use beta-nbc base so that the Selberg integral is holomorphic with respect to the exponent variables.

1999-08-11math.AGmath.QA

Variation on a theme of Selberg integral

In this paper we calculate some Generalized Selberg integrals. The answer is expressed in terms of $Γ$-functions. Integrals of this type serve as normalization constants or directly via undoing 2-D integrals for determination of structural constants of operator algebra.

1996-11-12math.QAq-alg

The sl_3 Selberg integral

Using an extension of the well-known evaluation symmetry, a new Cauchy-type identity for Macdonald polynomials is proved. After taking the classical limit this yields a new sl_3 generalisation of the famous Selberg integral. Closely related results obtained in this paper are an sl_3-analogue of the Askey-Habsieger-Kadell q-Selberg integral and an extension of the q-Selberg integral to a transformation between q-integrals of different dimensions.

2010-06-17math.CAmath.COmath.QA

A Selberg integral for the Lie algebra A_n

A new q-binomial theorem for Macdonald polynomials is employed to prove an A_n analogue of the celebrated Selberg integral. This confirms the g=A_n case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral for every simple Lie algebra g.

2007-08-08math.CAmath.COmath.QA

A Selberg Integral Type Formula for an sl_2 One-Dimensional Space of Conformal Blocks

For distinct complex numbers $z_1,...,z_{2N}$, we give a polynomial $P(y_1,...,y_{2N})$ in the variables $y_1,...,y_{2N}$, which is homogeneous of degree $N$, linear with respect to each variable, $sl_2$-invariant with respect to a natural $sl_2$-action, and is of order $N-1$ at $(y_1,...,y_{2N})=(z_1,...,z_{2N})$. We give also a Selberg integral type formula for the associated one-dimensional space of conformal blocks.

2009-05-23math.AGmath.QA

Extended Joseph polynomials, quantized conformal blocks, and a q-Selberg type integral

We consider the tensor power $V=(C^N)^{\otimes n}$ of the vector representation of $gl_N$ and its weight decomposition $V=\oplus_{λ=(λ_1,...,λ_N)}V[λ]$. For $λ= (λ_1 \geq ... \geq λ_N)$, the trivial bundle $V[λ]\times \C^n\to\C^n$ has a subbundle of q-conformal blocks at level l, where $l = λ_1-λ_N$ if $λ_1-λ_N> 0$ and l=1 if $λ_1-λ_N=0$. We construct a polynomial section $I_λ(z_1,...,z_n,h)$ of the subbundle. The section is the main object of the paper. We identify the section with the generating function $J_λ(z_1,...,z_n,h)$ of the extended Joseph polynomials of orbital varieties, defined in…

2012-06-28math-phmath.AGmath.MP

Selberg Integrals, Multiple Zeta Values and Feynman Diagrams

We prove that there is an isomorphism between the Hopf Algebra of Feynman diagrams and the Hopf algebra corresponding to the Homogenous Multiple Zeta Value ring H in C<<X,Y>> . In other words, Feynman diagrams evaluate to Multiple Zeta Values in all cases. This proves a recent conjecture of Connes-Kreimer, and others including Broadhurst and Kontsevich. The key step of our theorem is to present the Selberg integral as discussed in Terasoma [22] as a Functional from the Rooted Trees Operad to the Hopf algebra of Multiple Zeta Values. This is a new construction which provides illumination to the…

2002-06-05hep-thmath.NTmath.QA