Representations of classical Lie groups and quantized free convolution

Bufetov, Alexander I.; Gorin, Vadim

doi:10.1007/s00039-015-0323-x

Cited by 49 publications

(81 citation statements)

References 69 publications

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“…Therefore, λ ≤ k in the dominance order as desired, and we arrive at a sum of the form (2.18). The fact that α k (k) = 0 again follows from its explicit computation by [BuG,Lemma 5.5].…”

Section: 3mentioning

confidence: 95%

“…, 18 ALEXEY BUFETOV AND VADIM GORIN which follows from the explicit formula for this expression of [BuG,Lemma 5.5].…”

Section: 3mentioning

confidence: 95%

“…Let us recall that the branching rules for Schur polynomials match the combinatorics of the lozenge tilings of trapezoids (see e.g. [BuG,Section 3.2], [BuG2,Section 3.5]). In particular, if a signature µ encodes q lozenges along the vertical line x = q, then the conditional law Prob(µ | λ) can be found from the decomposition (4.45)…”

Section: Moreover the Random Variablesmentioning

confidence: 99%

“…, 1 N ). which was called Schur generating function (SGF) of P in [BuG,BuG2]. It is again straightforward to show that the correspondence is bijective: different measures correspond to different functions.…”

mentioning

confidence: 99%

“…For (2.9) this check is contained in the proof of [BuG,Lemma 6.1] -it reduces to the Lagrange inversion formula.…”

mentioning

confidence: 99%

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Fourier transform on high-dimensional unitary groups with applications to random tilings

Bufetov¹,

Gorin²

2019

Duke Math. J.

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A combination of direct and inverse Fourier transforms on the unitary group U (N ) identifies normalized characters with probability measures on N -tuples of integers. We develop the N → ∞ version of this correspondence by matching the asymptotics of partial derivatives at the identity of logarithm of characters with Law of Large Numbers and Central Limit Theorem for global behavior of corresponding random N -tuples.As one application we study fluctuations of the height function of random domino and lozenge tilings of a rich class of domains. In this direction we prove the Kenyon-Okounkov's conjecture (which predicts asymptotic Gaussianity and exact form of the covariance) for a family of non-simply connected polygons.Another application is a central limit theorem for the U (N ) quantum random walk with random initial data.

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Section: 3mentioning

confidence: 95%

“…, 18 ALEXEY BUFETOV AND VADIM GORIN which follows from the explicit formula for this expression of [BuG,Lemma 5.5].…”

Section: 3mentioning

confidence: 95%

Section: Moreover the Random Variablesmentioning

confidence: 99%

mentioning

confidence: 99%

“…For (2.9) this check is contained in the proof of [BuG,Lemma 6.1] -it reduces to the Lagrange inversion formula.…”

mentioning

confidence: 99%

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Fourier transform on high-dimensional unitary groups with applications to random tilings

Bufetov¹,

Gorin²

2019

Duke Math. J.

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Determinantal Structures in Space-Inhomogeneous Dynamics on Interlacing Arrays

Assiotis

2020

Ann. Henri Poincaré

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We introduce a space inhomogeneous generalization of the dynamics on interlacing arrays considered by Borodin and Ferrari in [7]. We show that for a certain class of initial conditions the point process associated to the dynamics has determinantal correlation functions and we calculate explicitly, in the form of a double contour integral, the correlation kernel for one of the most classical initial conditions, the densely packed. En route to proving this we obtain some results of independent interest on non-intersecting general pure-birth chains, that generalize the Charlier process, the discrete analogue of Dyson's Brownian motion. Finally, these dynamics provide a coupling between the inhomogeneous versions of the TAZRP and PushTASEP particle systems which appear as projections on the left and right edges of the array respectively⋆ ⋆ + 1 ⋆ + 2 ⋆ + 3 ⋆ + 4 ⋆ + 5 ⋆ + 6 Figure 1: A configuration of particles in GT 4 . If the clock of the particle labelled X 3,1 4 rings, which happens at rate λ(⋆ + 1), then the move is blocked since interlacing with X 2,1 4 would be violated. On the other hand, if the clock of the particle X 2,2 4 rings, which happens at rate λ(⋆ + 3), then it jumps to the right by one and instantaneously pushes both X 3,3 4 and X 4,4 4 to the right by one as well, for otherwise the interlacing would break.The right edge particle system is called inhomogeneous PushTASEP, see [7], [6] and also [16] for a q-deformation of the homogeneous case. The fact that this particle system has an underlying determinantal structure is new as well, but is also obtained in the independent work of Petrov [46] that uses different methods.

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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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Representations of classical Lie groups and quantized free convolution

Cited by 49 publications

References 69 publications

Fourier transform on high-dimensional unitary groups with applications to random tilings

Fourier transform on high-dimensional unitary groups with applications to random tilings

Determinantal Structures in Space-Inhomogeneous Dynamics on Interlacing Arrays

Pieri integral formula and asymptotics of Jack unitary characters

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