Pieri integral formula and asymptotics of Jack unitary characters

Abstract

We introduce Jack (unitary) characters and prove two kinds of formulas that are suitable for their asymptotics, as the lengths of the signatures that parametrize them go to infinity. The first kind includes several integral representations for Jack characters of one variable. The second identity we prove is the Pieri integral formula for Jack characters which, in a sense, is dual to the well known Pieri rule for Jack polynomials. The Pieri integral formula can also be seen as a functional equation for irreducible spherical functions of virtual Gelfand pairs.As an application of our formulas, we study the asymptotics of Jack characters as the corresponding signatures grow to infinity in the sense of Vershik-Kerov. We prove the existence of a small δ > 0 such that the Jack characters of m variables have a uniform limit on the δ-neighborhood of the m-dimensional torus. Our result specializes to a theorem of Okounkov and Olshanski. For θ = 1, the Jack polynomials are the well known Schur polynomials.The main object of our study are the Jack unitary characters. We define the Jack unitary character of rank N , with m variables and parametrized by λ to bewhere 1 k , k ∈ N, stands for a string of k ones. Our terminology is based on the fact that, when θ = 1 and N = m, the Jack unitary character J λ (x 1 , . . . , x N ; N, θ) becomes the normalized character of the unitary group U (N ) corresponding to its irreducible representation parametrized by the highest weight λ. Therefore Jack unitary characters are natural one-parameter deformations of the normalized characters of the unitary groups. We will call Jack unitary characters simply Jack characters, for convenience. The goal of this paper is to establish several formulas for Jack characters 1 arXiv:1704.02430v2 [math.RT]

Asymptotic Formulas for Macdonald Polynomials and the Boundary of the (q,t)-Gelfand-Tsetlin Graph

“…A similar degenerate question in the setting of random matrix theory was studied in [42]. Some of the tools in this paper can be degenerated easily to these scenarios and they may provide an alternative approach to their proof as well; for example, the special case of our toolbox in the case t = q → 1 was used in [24] to study the corresponding boundary problem, and in [15] we also study refine the asymptotic result that is needed to solve the boundary problem of [37].…”

“…The paper [15] is this article's companion, in which the author studies Jack characters, a natural degeneration of Macdonald characters and obtains their asymptotics in the Vershik-Kerov limit regime. The approach to the study of asymptotics of Jack characters is different from the approach we use here to study the asymptotics of Macdonald characters; in particular, it relies heavily on the Pieri integral formula, see [15] for further details.…”

“…The paper [15] is this article's companion, in which the author studies Jack characters, a natural degeneration of Macdonald characters and obtains their asymptotics in the Vershik-Kerov limit regime. The approach to the study of asymptotics of Jack characters is different from the approach we use here to study the asymptotics of Macdonald characters; in particular, it relies heavily on the Pieri integral formula, see [15] for further details. The tools from this paper and [15] afford us a very strong control over the asymptotics of Macdonald characters, Jack characters and Bessel functions [36,43], if the number of variables remains fixed and the rank tends to infinity.…”

“…The approach to the study of asymptotics of Jack characters is different from the approach we use here to study the asymptotics of Macdonald characters; in particular, it relies heavily on the Pieri integral formula, see [15] for further details. The tools from this paper and [15] afford us a very strong control over the asymptotics of Macdonald characters, Jack characters and Bessel functions [36,43], if the number of variables remains fixed and the rank tends to infinity. As another application of the developed toolbox, we have studied a Jack-Gibbs model of lozenge tilings in the spirit of [7,25].…”

Universal behavior of the corners of Orbital Beta Processes

“…See Theorem 5.4 and Section 8 for more details on this result. The asymptotics of multivariate Bessel functions in this regime are studied with the help of (degenerations of) recent formulas proved by the author in [Cu18], [Cu17]; see also [GP] for the θ = 1 case.…”