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

A. M. Garsia; E. Leven; N. Wallach; G. Xin

paper

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

arXiv:1501.00631

Abstract

Our main result here is that the specialization at $t = 1/ q$ of the $Q_{k m, k n}$ 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 $q^{\frac{( k m - 1 ) ( k n - 1 ) + k - 1}{2}} Q_{k m, k n} (- 1)^{k n}_{t = 1/ q} = \frac{[ k ] _{q}}{[ k m ] _{q}} e_{k m} [X [k m]_{q}]$ where as customarily, for any integer $s \geq 0$ and indeterminate $u$ we set $[s]_{u} = 1 + u + \dots + u^{s - 1}$ . 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 $k m \times k n$ lattice rectangle.

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

Abstract

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