A New Proof of Harish-Chandra’s Integral Formula

McSwiggen, Colin

doi:10.1007/s00220-018-3259-9

Cited by 10 publications

(8 citation statements)

References 20 publications

(56 reference statements)

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“…This approximation is justified by the geometric observation that, since the equations and inequalities defining the BZ polytope depend linearly on pλ, µ, νq, the difference between the number of integral points in H sν sλ sµ and in H sν´k sλ sµ must be lower order than s dr for s large. Finally using relation (31), we obtain ÿ κPK r κ C sν sλ sµ κ « C sν sλ sµ " P ν λµ psq " a dr s dr `lower-order terms, whence the identification J pλ, µ; νq " a dr " Vol rel pH ν λµ q .…”

Section: J and The Euclidean Volume Of The Bz Polytopementioning

confidence: 97%

On Horn’s Problem and Its Volume Function

Coquereaux

McSwiggen

Zuber

2019

Commun. Math. Phys.

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We consider an extended version of Horn's problem: given two orbits O α and O β of a linear representation of a compact Lie group, let A P O α , B P O β be independent and invariantly distributed random elements of the two orbits. The problem is to describe the probability distribution of the orbit of the sum A`B. We study in particular the familiar case of coadjoint orbits, and also the orbits of self-adjoint real, complex and quaternionic matrices under the conjugation actions of SOpnq, SUpnq and USppnq respectively. The probability density can be expressed in terms of a function that we call the volume function. In this paper, (i) we relate this function to the symplectic or Riemannian geometry of the orbits, depending on the case; (ii) we discuss its non-analyticities and possible vanishing; (iii) in the coadjoint case, we study its relation to tensor product multiplicities (generalized Littlewood-Richardson coefficients) and show that it computes the volume of a family of convex polytopes introduced by Berenstein and Zelevinsky. These considerations are illustrated by a detailed study of the volume function for the coadjoint orbits of B 2 " sop5q.

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Section: J and The Euclidean Volume Of The Bz Polytopementioning

confidence: 97%

On Horn’s Problem and Its Volume Function

Coquereaux

McSwiggen

Zuber

2019

Commun. Math. Phys.

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“…The Harish-Chandra-Itzykson-Zuber (HCIZ in the following) integral [1] [2] [3] is defined in the RMT setting as:…”

Section: Harish-chandra-itzykson-zuber Integrals and Free Summentioning

confidence: 99%

Asymptotic behavior of the multiplicative counterpart of the Harish-Chandra integral and the $S$-transform

Mergny,

Potters

2020

Preprint

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In this note, we study the asymptotic of spherical integrals, which are analytical extension in index of the normalized Schur polynomials for β = 2 , and of Jack symmetric polynomials otherwise. Such integrals are the multiplicative counterparts of the Harish-Chandra-Itzykson-Zuber (HCIZ) integrals, whose asymptotic are given by the so-called R-transform when one of the matrix is of rank one. We argue by a saddle-point analysis that a similar result holds for all β > 0 in the multiplicative case, where the asymptotic is governed by the logarithm of the S-transform. As a consequence of this result one can calculate the asymptotic behavior of complete homogeneous symmetric polynomials.

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“…In this subsection we introduce a finite difference operator called the box spline Laplacian, which allows us to give a convenient new representation of the discrete convolution with b. This leads in turn to a reformulation of the convolution identity in Corollary 5.1 and to a representation ofÂ(Φ + ) as a finite difference operator on the space D(Φ + ) defined in(28), both of which will be useful in Section 5.4 below. For τ in the weight lattice, let ∆ τ and ∇ τ denote respectively the forwards and backwards finite difference operators in the direction of τ :…”

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Box splines, tensor product multiplicities and the volume function

McSwiggen¹

2021

Algebraic Combinatorics

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We study the relationship between the tensor product multiplicities of a compact semisimple Lie algebra g and a special function J associated to g, called the volume function. The volume function arises in connection with the randomized Horn's problem in random matrix theory and has a related significance in symplectic geometry. Building on box spline deconvolution formulae of Dahmen-Micchelli and De Concini-Procesi-Vergne, we develop new techniques for computing the multiplicities from J , answering a question posed by Coquereaux and Zuber. In particular, we derive an explicit algebraic formula for a large class of Littlewood-Richardson coefficients in terms of J . We also give analogous results for weight multiplicities, and we show a number of further identities relating the tensor product multiplicities, the volume function and the box spline. To illustrate these ideas, we give new proofs of some known theorems.

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A New Proof of Harish-Chandra’s Integral Formula

Abstract: We present a new proof of Harish-Chandra's formula [1] Π(h1)Π(h2)

Cited by 10 publications

References 20 publications

On Horn’s Problem and Its Volume Function

On Horn’s Problem and Its Volume Function

Asymptotic behavior of the multiplicative counterpart of the Harish-Chandra integral and the $S$-transform

Box splines, tensor product multiplicities and the volume function

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