A parking function interpretation for (-1)^k∇ m_2^k1^l

2023/12/28 by Menghao Qu, Guoce Xin, Qu, Menghao +1
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2312.16824

openalex publication_date 2023/12/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Haglund, Morse, and Zabrocki introduced a family of creation operators of Hall-Littlewood polynomials, \C_a\ for any a∈ ℤ, in their compositional refinement of the shuffle (ex-)conjecture. For any α\vDash n, the combinatorial formula for ∇ C_α is a weighted sum of parking functions. These summations can be converted to a weighted sum of certain LLT polynomials. Thus ∇ C_α is Schur positive since Grojnowski and Haiman proved that all LLT polynomials are Schur positive. In this paper, we obtain a recursion that implies the C-positivity of (-1)^k m_2^k1^l, and hence prove the Schur positivity of (-1)^k∇ m_2^k1^l. As a corollary, a parking function interpretation for (-1)^k∇ m_2^k1^l is obtained by using the compositional shuffle theorem of Carlsson and Mellit.

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