In their 2011 paper on the AGT conjecture, Alba, Fateev, Litvinov and Tarnopolsky (AFLT) obtained a closed-form evaluation for a Selberg integral over the product of two Jack polynomials, thereby unifying the well-known Kadell and Hua--Kadell integrals. In this paper we use a variety of symmetric functions and symmetric function techniques to prove generalisations of the AFLT integral. These include (i) an $\mathrm{A}_n$ analogue of the AFLT integral, containing two Jack polynomials in the integrand; (ii) a generalisation of (i) for $γ=1$ (the Schur or GUE case), containing a product of $n+1$ Schur functions; (iii) an elliptic generalisation of the AFLT integral in which the role of the Jack polynomials is played by a pair of elliptic interpolation functions; (iv) an AFLT integral for Macdonald polynomials.
In the previous paper (J. Combin. Theory Ser. A, 120, 2013, 1263--1284) H. Tagawa and the two authors proposed an algebraic method to compute certain Pfaffians whose form resemble to Hankel determinants associated with moment sequences of the classical orthogonal polynomials. At the end of the paper they offered several conjectures. In this work we employ a completely different method to evaluate this type of Pfaffians. The idea is to apply certain de Bruijn type formula and convert the evaluation of the Pfaffians to the certain Selberg type integrals. This approach works not only for Pfaffian…
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 multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo $p$.
We study $β$-Jacobi diffusion processes on alcoves in $\mathbb R^N$, depending on 3 parameters. Using elementary symmetric functions, we present space-time-harmonic functions and martingales for these processes $(X_t)_{t\ge0}$ which are independent from one parameter. This leads to a formula for $\mathbb E(\prod_{i=1}^N (y-X_{t,i}))$ in terms of classical Jacobi polynomials. For $t\to\infty$ this yields a corresponding formula for Jacobi ensembles and thus Aomoto's Selberg integral.
We prove an $\mathbb F_p$-Selberg integral formula of type $A_n$, 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 multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo $p$. For the type $A_1$ the formula was proved in a previous paper by the authors.
We study Muttalib--Borodin ensembles --- particular eigenvalue PDFs on the half-line --- with classical weights, i.e. Laguerre, Jacobi or Jacobi prime. We show how the theory of the Selberg integral, involving also Jack and Schur polynomials, naturally leads to a multi-parameter generalisation of these particular Muttalib--Borodin ensembles, and also to the explicit form of underlying biorthogonal polynomials of a single variable. A suitable generalisation of the original definition of the Muttalib--Borodin ensemble allows for negative eigenvalues. In the cases of generalised Gaussian, symmetr…
There is a natural left and right invariant Haar measure associated with the matrix groups GL${}_N(\mathbb R)$ and SL${}_N(\mathbb R)$ due to Siegel. For the associated volume to be finite it is necessary to truncate the groups by imposing a bound on the norm, or in the case of SL${}_N(\mathbb R)$, by restricting to a fundamental domain. We compute the asymptotic volumes associated with the Haar measure for GL${}_N(\mathbb R)$ and SL${}_N(\mathbb R)$ matrices in the case of that the operator norm lies between $R_1$ and $1/R_2$ in the former, and this norm, or alternatively the 2-norm, is bound…
We present how explicit eigenfunctions of the principal Hamiltonian for the $BC_{m}$ relativistic Calogero-Moser-Sutherland model, due to van Diejen, can be constructed using gauge and integral transformations. In particular, we find that certain $BC$-type elliptic hypergeometric integrals, including elliptic Selberg integrals, of both Type I and Type II arise as eigenfunctions of the van Diejen model under some parameter restrictions. Among these are also joint eigenfunctions of so-called modular pairs of van Diejen operators. Furthermore, these transformations are related to reflections of t…
