Some new identities for Schur polynomials

Abstract

The Schur polynomials play a central role in combinatorics, representation theory, and symmetric functions. Classical identities such as the Cauchy and Littlewood formulas establish fundamental relationships between these polynomials. In this article, we present novel generalizations of these identities for bounded and unbounded cases. Specifically, we prove that a family of Schur polynomial expansions, parameterized by arbitrary polynomial sequences, satisfies a determinant-based factorization property. This result extends the classical Cauchy identities and provides a unifying framework for bounded analogues. Additionally, we derive new bounded identities for Schur polynomials, which refine Macdonald's earlier results and incorporate constraints on partitions. These findings include determinant representations that encapsulate both classical and bounded settings, enabling further generalizations. Applications of these results to combinatorial structures and the theory of symmetric functions are also discussed.