2024/08/23 by A. S. Khanna, Khanna, Aditya, Nicholas A. Loehr +1 · 2 citations
Computer Science · #05A17 #05E05 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2408.13404
openalex publication_date 2024/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Asvin G and Andrew O'Desky recently introduced the graded algebra PΛ of polysymmetric functions as a generalization of the algebra Λ of symmetric functions. This article develops combinatorial formulas for some multiplication rules and transition matrix entries for PΛ that are analogous to well-known classical formulas for Λ. In more detail, we consider pure tensor bases \s⊗τ\, \p⊗τ\, and \m⊗τ\ for PΛ that arise as tensor products of the classical Schur basis, power-sum basis, and monomial basis for Λ. We find expansions in these bases of the non-pure bases \Pδ\, \Hδ\, \E+δ\, and \Eδ\ studied by Asvin G and O'Desky. The answers involve tableau-like structures generalizing semistandard tableaux, rim-hook tableaux, and the brick tabloids of Eğecioğlu and Remmel. These objects arise by iteration of new Pieri-type rules that give expansions of products such as s⊗σHδ, p⊗σEδ, etc.