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The square of the Vandermonde determinant and its q-generalisation

The square of the Vandermonde determinant and its q-generalisation
The square of the Vandermonde determinant and its q-generalisation
The Vandermonde determinant plays a crucial role in the quantum Hall effect via Laughlin's wavefunction ansatz. Herein the properties of the square of the Vandermonde determinant as a symmetric function are explored in detail. Important properties satisfied by the coefficients arising in the expansion of the square of the Vandermonde determinant in terms of Schur functions are developed and generalized to q-dependent coefficients via the q-discriminant. Algorithms for the efficient calculation of the q-dependent coefficients as finite polynomials in q are developed. The properties, such as the factorization of the q-dependent coefficients, are exposed. Further light is shed upon the vanishing of certain expansion coefficients at q = 1. The q-generalization of the sum rule for the squares of the coefficients is derived. A number of compelling conjectures are stated.
0305-4470
735-767
King, R.C.
76ae9fb3-6b19-449d-8583-dbf1d7ed2706
Toumazet, F.
e49842d7-7844-4104-a7d4-be92a2659a6a
Wybourne, B.G.
527c5e72-79db-40a6-901c-284a3bd788dc
King, R.C.
76ae9fb3-6b19-449d-8583-dbf1d7ed2706
Toumazet, F.
e49842d7-7844-4104-a7d4-be92a2659a6a
Wybourne, B.G.
527c5e72-79db-40a6-901c-284a3bd788dc

King, R.C., Toumazet, F. and Wybourne, B.G. (2004) The square of the Vandermonde determinant and its q-generalisation. Journal of Physics A: Mathematical and General, 37, 735-767. (doi:10.1088/0305-4470/37/3/015).

Record type: Article

Abstract

The Vandermonde determinant plays a crucial role in the quantum Hall effect via Laughlin's wavefunction ansatz. Herein the properties of the square of the Vandermonde determinant as a symmetric function are explored in detail. Important properties satisfied by the coefficients arising in the expansion of the square of the Vandermonde determinant in terms of Schur functions are developed and generalized to q-dependent coefficients via the q-discriminant. Algorithms for the efficient calculation of the q-dependent coefficients as finite polynomials in q are developed. The properties, such as the factorization of the q-dependent coefficients, are exposed. Further light is shed upon the vanishing of certain expansion coefficients at q = 1. The q-generalization of the sum rule for the squares of the coefficients is derived. A number of compelling conjectures are stated.

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Published date: 2004
Organisations: Applied Mathematics

Identifiers

Local EPrints ID: 29534
URI: http://eprints.soton.ac.uk/id/eprint/29534
ISSN: 0305-4470
PURE UUID: 343e7843-2409-4b12-95e0-a23d58a43405

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Date deposited: 12 May 2006
Last modified: 08 Jan 2022 15:54

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Contributors

Author: R.C. King
Author: F. Toumazet
Author: B.G. Wybourne

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