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

Cuenca, Cesar

doi:10.3842/sigma.2018.001

Cited by 9 publications

(17 citation statements)

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“…where the Sign N -valued random variable X N is defined by X N (ω) = r(ω N ) for any ω ∈ Ω. This is nothing but a (q, t 2 )-central probability measure (see [5,Definition 6.4]). With these preparations, by Formula (2.9), we define the (q, t)-analogue F ν (q, t) to be Using the branching rule for Macdonald polynomials, we can confirm that Tr(F ν (q, t)) = P ν (t N −1 , t N −3 , .…”

Section: Appendix B (Q T)-central Probability Measuresmentioning

confidence: 99%

“…It may be important to notice that the consideration here suggests that any Stratila-Voiculescu AF-flow on the same Stratila-Voiculescu AF-algebra as that of U q (or equivalently that U (∞)) can be interpreted as a possible deformation of the Gelfand-Tsetlin graph in the algebraic level. On the other hand, according to the consideration here, we can regard a generating function of extremal (q, t 2 )-central probability measure (see Theorem 5.5 (or Section 6) in [5]) as (q, t)-analogue of Voiculescu function.…”

Section: Appendix B (Q T)-central Probability Measuresmentioning

confidence: 99%

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Quantized Vershik–Kerov theory and quantized central measures on branching graphs

Satô

2019

Journal of Functional Analysis

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We propose a natural quantized character theory for inductive systems of compact quantum groups based on KMS states on AF-algebras following Stratila-Voiculescu's work [27] (or [8]), and give its serious investigation when the system consists of quantum unitary groups Uq(N ) with q ∈ (0, 1). The key features of this work are: The "quantized trace" of a unitary representation of a compact quantum group can be understood as a quantized character associated with the unitary representation and its normalized one is captured as a KMS state with respect to a certain one-parameter automorphism group related to the so-called scaling group. In this paper we provide a Vershik-Kerov type approximation theorem for extremal quantized characters (called the ergodic method) and also compare our quantized character theory for the inductive system of Uq(N ) with Gorin's theory on q-Gelfand-Tsetlin graphs [11].

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Section: Appendix B (Q T)-central Probability Measuresmentioning

confidence: 99%

Section: Appendix B (Q T)-central Probability Measuresmentioning

confidence: 99%

Quantized Vershik–Kerov theory and quantized central measures on branching graphs

Satô

2019

Journal of Functional Analysis

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“…. , x n ; q, t), which feature two parameters q, t and specialize to Hall-Littlewood polynomials when q = 0; see [BO12,Cue18,Gor12,OO98,Ols21]. In special cases these combinatorial results take on additional significance in representation theory and harmonic analysis; the Schur case was already mentioned, and two other special cases of the result of [OO98] for the Jack polynomial case specialize to statements about the infinite symmetric spaces U (∞)/O(∞) and U (2∞)/Sp(∞).…”

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confidence: 99%

“…The boundary classification results of [Gor12] are generalized in [Cue18] to the Macdonald case with cotransition probabilities P µ/λ (t n ; q, t = q k ) P λ (1, . .…”

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confidence: 99%

Hall-Littlewood polynomials, boundaries, and $p$-adic random matrices

Peski¹

2021

Preprint

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We prove that the boundary of the Hall-Littlewood t-deformation of the Gelfand-Tsetlin graph is parametrized by infinite integer signatures, extending results of Gorin [Gor12] and Cuenca [Cue18] on boundaries of related deformed Gelfand-Tsetlin graphs. In the special case when 1/t is a prime p we use this to recover results of Bufetov-Qiu [BQ17] and Assiotis [Ass20] on infinite p-adic random matrices, placing them in the general context of branching graphs derived from symmetric functions.Our methods rely on explicit formulas for certain skew Hall-Littlewood polynomials. As a separate corollary to these, we obtain a simple expression for the joint distribution of the cokernels of products A1, A2A1, A3A2A1, . . . of independent Haar-distributed matrices Ai over the p-adic integers Zp. This expression generalizes the explicit formula for the classical Cohen-Lenstra measure on abelian p-groups. Contents 1. Introduction 2. Hall-Littlewood polynomials 3. Principally specialized skew Hall-Littlewood polynomials 4. The t-deformed Gelfand-Tsetlin graph and its boundary 5. Infinite p-adic random matrices and corners 6. Ergodic decomposition of p-adic Hua measures. 7. Products of finite p-adic random matrices Appendix A. Commuting dynamics and projection to the boundary References

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“…It was natural to consider the degeneration of the parameters q, t used for Macdonald characters in the "Jack regime" t = q θ , q → 1, and ask whether the formulas from [Cu] degenerate into formulas for Jack characters. It will be shown that certain formulas do admit a degeneration, but not all: the multiplicative formula for Macdonald characters, [Cu,Thm. 4.1], cannot be degenerated.…”

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confidence: 99%

Pieri integral formula and asymptotics of Jack unitary characters

Cuenca

2017

Sel. Math. New Ser.

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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]

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

Cited by 9 publications

References 49 publications

Quantized Vershik–Kerov theory and quantized central measures on branching graphs

Quantized Vershik–Kerov theory and quantized central measures on branching graphs

Hall-Littlewood polynomials, boundaries, and $p$-adic random matrices

Pieri integral formula and asymptotics of Jack unitary characters

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