Total positivity, Gramian matrices, and Schur polynomials

Pablo Díaz; Esmeralda Mainar; Beatriz Rubio; Pablo Díaz; Esmeralda Mainar; Beatriz Rubio

doi:10.3934/math.2025110

AIMS Mathematics

2025, Volume 10, Issue 2: 2375-2391. doi: 10.3934/math.2025110

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Total positivity, Gramian matrices, and Schur polynomials

Departamento de Matemática Aplicada/IUMA, Universidad de Zaragoza, España

Received: 07 October 2024 Revised: 11 January 2025 Accepted: 24 January 2025 Published: 10 February 2025
MSC : 05E05, 15A23, 65F05

This paper investigated the total positivity of Gramian matrices associated with general bases of functions. It demonstrated that the total positivity of collocation matrices for totally positive bases extends to their Gramian matrices. Additionally, a bidiagonal decomposition of these Gramian matrices, derived from integrals of symmetric functions, was presented. This decomposition enables the design of algorithms with high relative accuracy for solving linear algebra problems involving totally positive Gramian matrices. For polynomial bases, compact and explicit formulas for the bidiagonal decomposition were provided, involving integrals of Schur polynomials. These integrals, known as Selberg-like integrals, arise naturally in various contexts within Physics and Mathematics.
- Gramian matrices,
- bidiagonal decompositions,
- Schur functions,
- Selberg-like integrals,
- totally positive matrices
Citation: Pablo Díaz, Esmeralda Mainar, Beatriz Rubio. Total positivity, Gramian matrices, and Schur polynomials[J]. AIMS Mathematics, 2025, 10(2): 2375-2391. doi: 10.3934/math.2025110

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Abstract

This paper investigated the total positivity of Gramian matrices associated with general bases of functions. It demonstrated that the total positivity of collocation matrices for totally positive bases extends to their Gramian matrices. Additionally, a bidiagonal decomposition of these Gramian matrices, derived from integrals of symmetric functions, was presented. This decomposition enables the design of algorithms with high relative accuracy for solving linear algebra problems involving totally positive Gramian matrices. For polynomial bases, compact and explicit formulas for the bidiagonal decomposition were provided, involving integrals of Schur polynomials. These integrals, known as Selberg-like integrals, arise naturally in various contexts within Physics and Mathematics.

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