Bialternant formula for Schur polynomials with repeating variables

TL;DR

This work generalizes Schur polynomials to settings where variables repeat, defining $\operatorname{s}_\lambda(y^{[\varkappa]})$ as a quotient of determinants $\frac{\det G_\lambda(y,\varkappa)}{\det G_{\emptyset}(y,\varkappa)}$, with a generalized confluent Vandermonde matrix $G_\lambda(y,\varkappa)$. It establishes three algebraic proofs—via Jacobi–Trudi and matrix multiplications, via partial reduction/elimination of determinants, and via the classical bialternant formula—each deriving the same determinantal quotient. The paper also provides practical computation methods for constructing $G_\lambda(y,\varkappa)$, analyzes algorithmic complexity, and links the special case of equal multiplicities to plethysm, enriching connections between symmetric polynomials and interpolation problems with repeating nodes. These results extend determinant identities for Schur polynomials to cases with repeated variables and open avenues for interpolation, Toeplitz-type structures, and plethystic interpretations in symmetric function theory.

Abstract

We consider polynomials of the form $\operatorname{s}_λ(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]})$, where $λ$ is an integer partition, $\operatorname{s}_λ$ is the Schur polynomial associated to $λ$, and $y_j^{[\varkappa_j]}$ denotes $y_j$ repeated $\varkappa_j$ times. We represent $\operatorname{s}_λ(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]})$ 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.

Theorems & Definitions (43)

• Theorem 1.1
• Corollary 1.2: bialternant formula for complete homogeneous polynomials with repeated variables
• Definition 2.1
• Example 2.2
• Definition 2.3
• Example 2.4
• Proposition 3.1: Lita da Silva, 2018
• proof
• Proposition 3.2
• proof