2024/05/02 by Milo Bechtloff Weising, Weising, Milo Bechtloff
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2405.01049
openalex publication_date 2024/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce and study a generalization s(μ|λ) of the Schur functions called the almost symmetric Schur functions. These functions simultaneously generalize the finite variable key polynomials and the infinite variable Schur functions. They form a homogeneous basis for the space of almost symmetric functions and are defined using a family of recurrences involving the isobaric divided difference operators and limits of Weyl symmetrization operators. The s(μ|λ) are the q=t=0 specialization of the stable limit non-symmetric Macdonald functions \widetildeE(μ|λ) defined by the author in previous work. We find a combinatorial formula for these functions simultaneously generalizing well known formulas for the Schur functions and the key polynomials. Further, we prove positivity results for the coefficients of the almost symmetric Schur functions expanded into the monomial basis and into the monomial-Schur basis of the space of almost symmetric functions. The latter positivity result follows after realizing the almost symmetric Schur functions s(μ|λ) as limits of characters of representations of parabolic subgroups in type GL.