A new Plethystic Symmetric Function Operator and The rational Compositional Shuffle Conjecture at t=1/q

2015/01/04 by Adriano M. Garsia, A. M. Garsia, E. Leven +9
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.1501.00631

arxiv created 2015/01/04 · openalex publication_date 2015/01/04 · arxiv updated 2015/01/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Our main result here is that the specialization at t=1/q of the Q_km,kn operators studied in [4] may be given a very simple plethystic form. This discovery yields elementary and direct derivations of several identities relating these operators at t=1/q to the Rational Compositional Shuffle conjecture of [3]. In particular we show that if m,n and k are positive integers and (m,n) is a coprime pair then q^{(km-1)(kn-1)+k-1\over 2} Q_km,kn(-1)^kn|_t=1/q = \textstyle[k]_q\over [km]_q e_km[ X[km]_q] where as customarily, for any integer s ≥ 0 and indeterminate u we set [s]_u=1+u+⋯ +u^s-1. We also show that the symmetric polynomial on the right hand side is always Schur positive. Moreover, using the Rational Compositional Shuffle conjecture, we derive a precise formula expressing this polynomial in terms of Parking functions in the km× kn lattice rectangle.

Citations

• Compositional (km,kn)-Shuffle Conjectures
• Science Fiction and Macdonald's Polynomials
• A residue theorem for Malcev–Neumann series
• A simpler formula for the number of diagonal inversions of an (m,n)-Parking Function and a returning Fermionic formula
• Affine Springer fibers of type A and combinatorics of diagonal coinvariants

Related

• Schur Function Expansions and the Rational Shuffle Theorem
• A combinatorial model for ∇ m_μ
• A refinement of the Shuffle Conjecture with cars of two sizes and t=1/q
• A proof of the shuffle conjecture
• Compositional (km,kn)-Shuffle Conjectures