A new plethystic symmetric function operator and the rational compositional shuffle conjecture at t=1/q

作者：Garsia, Adriano^*; Leven, Emily Sergel; Wallach, Nolan; Xin, Guoce

来源：Journal of Combinatorial Theory - Series A, 2017, 145: 57-100.

DOI：10.1016/j.jcta.2016.07.001

• Parking function
• Shuffle conjecture
• Plethysm

Our main result here is that the specialization at t = 1/q of the Q(km,kn) operators studied in Bergeron et al. [2] 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 Bergeron et al. [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-1/2) Q(km,kn)(-1)(kn)vertical bar(t=1/q) = [k](q)/[km](q) e(km) [X[km](q)]
where as customarily, for any integer s >= 0 and indeterminate u we set [s](u) = 1 + u + center dot center dot center dot + 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 x kn lattice rectangle.

• 出版日期2017-1
• 单位首都师范大学