A symplectic refinement of shifted Hecke insertion

Abstract

Buch, Kresch, Shimozono, Tamvakis, and Yong defined Hecke insertion to formulate a combinatorial rule for the expansion of the stable Grothendieck polynomials G π indexed by permutations in the basis of stable Grothendieck polynomials G λ indexed by partitions. Patrias and Pylyavskyy introduced a shifted analogue of Hecke insertion whose natural domain is the set of maximal chains in a weak order on orbit closures of the orthogonal group acting on the complete flag variety. We construct a generalization of shifted Hecke insertion for maximal chains in an analogous weak order on orbit closures of the symplectic group. As an application, we identify a combinatorial rule for the expansion of "orthogonal" and "symplectic" shifted analogues of G π in Ikeda and Naruse's basis of K-theoretic Schur P -functions.

Shifted insertion algorithms for primed words

“…The latter, in turn, is a special case of the shifted Hecke insertion algorithm from [14,33]. As explained in Remark 2.3, the correspondence a → (P O EG (a), Q O EG (a)) is the "orthogonal" counterpart to a "symplectic" shifted insertion algorithm studied in [16,28,29].…”

“…Here, a semistandard shifted tableau is allowed to have primed entries in diagonal positions; for the precise definition, see Section 2. Restricting a → (P O EG (a), Q O EG (a)) to unprimed words gives the map called involution Coxeter-Knuth insertion in [12,28] and orthogonal Edelman-Greene insertion in [29]. The latter, in turn, is a special case of the shifted Hecke insertion algorithm from [14,33].…”

“…Moreover, it always holds that x φ = x Q Sp (φ) . This bijection is called symplectic Hecke insertion in [28]. If a = a 1 a 2 • • • a k ∈ H Sp (z) then we set P Sp (a) = P Sp (φ) and Q Sp (a) = Q Sp (φ) for φ = (a 1 , a 2 , .…”

“…This is also why we will often include the adjective "orthogonal" with constructions involving R + inv (z). There is a parallel "symplectic" story for a different analogue of reduced words corresponding to the orbits of Sp 2n (C) acting on Fl 2n (see, e.g., [13,28,31,40]).…”

Principal specializations of Schubert polynomials in classical types

“…Let InvHecke(z) denote the set of involution Hecke words for an element z in I n or I FPF n . This set was denoted as either H O (z) for z ∈ I n or H Sp (z) for z ∈ I FPF n in [12]. Also define (z) = min{ (a) : a ∈ InvHecke(z)}.…”