Hyperdeterminantal computation for the Laughlin wavefunction

Boussicault, Adrien; Luque, Jean-Gabriel; Tollu, Christophe

doi:10.1088/1751-8113/42/14/145301

Cited by 6 publications

(13 citation statements)

References 27 publications

(45 reference statements)

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“…. , 1), we note that (22) perfectly matches the value for the average of the determinant for a Jacobi ensemble with γ = 1, as computed from the Selberg integral (5):…”

Section: Hyperdeterminant Formula For Statistics Of Permanents Fosupporting

confidence: 77%

Nonlinear random matrix statistics, symmetric functions and hyperdeterminants

Luque¹,

Vivo²

2010

J. Phys. A: Math. Theor.

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Nonlinear statistics (i.e. statistics of permanents) on the eigenvalues of invariant random matrix models are considered for the three Dyson's symmetry classes β = 1, 2, 4. General formulas in terms of hyperdeterminants are found for β = 2. For specific cases and all βs, more computationally efficient results are obtained, based on symmetric functions expansions. As an application, we consider the case of quantum transport in chaotic cavities extending results from [D.V. Savin, H.-J. Sommers and W. Wieczorek, Phys. Rev. B 77, 125332 (2008)].

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“…. , 1), we note that (22) perfectly matches the value for the average of the determinant for a Jacobi ensemble with γ = 1, as computed from the Selberg integral (5):…”

Section: Hyperdeterminant Formula For Statistics Of Permanents Fosupporting

confidence: 77%

Nonlinear random matrix statistics, symmetric functions and hyperdeterminants

Luque¹,

Vivo²

2010

J. Phys. A: Math. Theor.

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“…The study of ϕ m L as a symmetric function generated literature for the purpose of the expansion in the Schur basis and the monomial basis [10,33]. This first case is particularly interesting because it is also related to rectangular Jack polynomials and hyperdeterminants [2,6,26]. Notice that, even in this (simplest) case, Macdonald polynomial provides a more regular framework for the study of these functions.…”

Section: The Quasistaircase Partitionmentioning

confidence: 99%

“…, z τ l . Moreover, this identity is thus invariant under any permutation of these variables, and so h S (x(Y)s i ) = 0 in (6).…”

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confidence: 99%

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Connections between vector-valued and highest weight Jack and Macdonald polynomials

Colmenarejo¹,

Dunkl²,

Luque³

2019

Preprint

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We analyze conditions under which a projection from the vector-valued Jack or Macdonald polynomials to scalar polynomials has useful properties, specially commuting with the actions of the symmetric group or Hecke algebra, respectively, and with the Cherednik operators for which these polynomials are eigenfunctions. In the framework of representation theory of the symmetric group and the Hecke algebra, we study the relation between singular nonsymmetric Jack and Macdonald polynomials and highest weight symmetric Jack and Macdonald polynomials. Moreover, we study the quasistaircase partition as a continuation of our study on the conjectures of Bernevig and Haldane on clustering properties of symmetric Jack polynomials.Key words and phrases. Macdonald and Jack Polynomials, singular polynomials, highest weight polynomials, vector-valued polynomials, representation theory of symmetric group and Hecke algebra.This work is partially supported by the "European Regional Development Fund" (ERDF) via the regional (GRR) project MOUSTIC. 1 VECTOR-VALUED AND HIGHEST WEIGHT JACK AND MACDONALD POLYNOMIALS 2 5.3. The symmetric group case 20 5.3.1. Singularity of quasistaircase nonsymmetric Jack polynomials 20 5.3.2. From singular nonsymmetric Jack polynomials to highest weight symmetric Jack polynomials 23 5.4. The Hecke algebra case 23 5.4.1. Singularity of quasistaircase nonsymmetric Macdonald polynomials 24 5.4.2. From singular nonsymmetric Macdonald polynomials to highest weight symmetric Macdonald polynomials 25 5.4.3. A note on specializations 27 6. Factorizations at special points 27 6.1. The symmetric group case 28 6.2. The Hecke algebra case 29 6.2.1. The general polynomials of isotype τ 34 6.2.2. Application to projections and factorizations 35 7. Conclusion and perspective 38 References 39

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“…Date: August 8, 2018. However, a combinatorial interpretation for the coefficient of a given Schur function is still unknown. Recently, Boussicault, Luque and Tollu [2] provided a purely numerical algorithm for computing the coefficient of a given Schur function in the decomposition without computing the other coefficients. The algorithm uses hyperdeterminants and their Laplace expansion.…”

Section: Introductionmentioning

confidence: 99%

Powers of the Vandermonde determinant, Schur functions and recursive formulas

Ballantine¹

2012

J. Phys. A: Math. Theor.

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Since every even power of the Vandermonde determinant is a symmetric polynomial, we want to understand its decomposition in terms of the basis of Schur functions. We investigate several combinatorial properties of the coefficients in the decomposition. In particular, we give recursive formulas for the coefficient of the Schur function s µ in the decomposition of an even power of the Vandermonde determinant in n + 1 variables in terms of the coefficient of the Schur function s λ in the decomposition of the same even power of the Vandermonde determinant in n variables if the Young diagram of µ is obtained from the Young diagram of λ by adding a tetris type shape to the top or to the left. An extended abstract containing the statement of the results presented here appeared in the Proceedings of FPSAC11 [1]. Mathematics Standard Classification: 05E05, 15A15

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Hyperdeterminantal computation for the Laughlin wavefunction

Cited by 6 publications

References 27 publications

Nonlinear random matrix statistics, symmetric functions and hyperdeterminants

Nonlinear random matrix statistics, symmetric functions and hyperdeterminants

Connections between vector-valued and highest weight Jack and Macdonald polynomials

Powers of the Vandermonde determinant, Schur functions and recursive formulas

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