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arXiv · 2312.15680

Bialternant formula for Schur polynomials with repeating variables

Luis Angel González-Serrano·Egor A. Maximenko
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Abstract

We consider polynomials of the form $\operatorname{s}_\lambda(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]})$, where $\lambda$ is an integer partition, $\operatorname{s}_\lambda$ is the Schur polynomial associated to $\lambda$, and $y_j^{[\varkappa_j]}$ denotes $y_j$ repeated $\varkappa_j$ times. We represent $\operatorname{s}_\lambda(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]})$ as a quotient whose the denominator is the determinant of the confluent Vandermonde matrix, and the numerator is the determinant of some generalized confluent Vandermonde matrix. We give three algebraic proofs of this formula.

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Luis Angel González-Serrano, Egor A. Maximenko. 2023-12-25. Bialternant formula for Schur polynomials with repeating variables. https://doi.org/10.1080/03081087.2025.2464639

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