Shifted Hecke insertion and the K-theory of OG(n,2n+ 1)

Abstract

Patrias and Pylyavskyy introduced shifted Hecke insertion as an application of their theory of dual filtered graphs. We use shifted Hecke insertion to construct symmetric function representatives for the K-theory of the orthogonal Grassmannian. These representatives are closely related to the shifted Grothendieck polynomials of Ikeda and Naruse. We then recover the K-theory structure coefficients of Clifford-Thomas-Yong/Buch-Samuel by introducing a shifted K-theoretic Poirier-Reutenauer algebra. Our proofs depend on the theory of shifted K-theoretic jeu de taquin and the weak K-Knuth relations.1 representatives for the Grassmannian. We will see in Proposition 3.5 that the K λ and the GP λ of Ikeda and Naruse are closely related. As a corollary, we derive a new proof of the symmetry of GP λ . Surprisingly, while GP λ satisfies the Q-cancellation property, we show that K λ does not, hence cannot be expressed as a sum of Schur P -functions.Using weak K-Knuth equivalence, we define a shifted K-theoretic Poirier-Reutenauer algebra. This generalizes the shifted Poirier-Reutenauer algebra of Jang and Li [10], a shifted analogue of the Poirier-Reutenauer Hopf algebra [14]. Our approach follows work of Patrias and Pylyavskyy on a K-theoretic Poirier-Reutenauer bialgebra [13]. The shifted K-theoretic Poirier-Reutenauer algebra does not have the coalgebra structure analogous to that of shifted Poirier-Reutenauer and K-theoretic Poirier Reutenauer. Using the shifted Ktheoretic Poirier-Reutenauer algebra, we define a Littlewood-Richardson rule for the product K λ · K µ . This rule coincides up to sign with the rule of Clifford, Thomas, and Yong [5] and Buch and Samuel [4]. As a consequence, we have our main result.The remainder of the paper is structured as follows. In the next section, we define shifted Hecke insertion and show how it relates to shifted K-theoretic jeu de taquin. This allows us to demonstrate the relationship between shifted Hecke insertion and the weak K-Knuth equivalence relations. The third section is devoted to defining the K λ , expressing them in terms of shifted Hecke insertion and proving Theorem 1.1. In the fourth and final section, we develop the shifted K-theoretic Poirier-Reutenauer algebra and prove Theorem 1.2. Shifted Hecke Insertion and Weak K-Knuth EquivalenceWe show that the shifted Hecke insertion given in [12] respects the weak K-Knuth equivalence given in [4]. Before presenting our argument, we review previous work on increasing shifted tableaux, shifted Hecke insertion, and shifted K-jeu de taquin.2.1. Increasing shifted tableaux. To each strict partition λ = (λ 1 > λ 2 > . . . > λ k ) we associate the shifted shape, which is an array of boxes where the ith row has λ i boxes and is indented i − 1 units. A shifted tableau is a filling of the shifted shape with positive integers. A shifted tableau is increasing if the labels are strictly increasing from left to right along rows and top to bottom down columns. The reading word of an increasing shifted tableau T , denoted row(T ), is the w...

Dual filtered graphs

“…The result of such insertion agrees with that of K-theoretic jeu de taquin rectification described in [8]. See [12].…”

Shifted insertion algorithms for primed words

“…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]. 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].…”

“…. ) and let KP λ := T x T where the sum is over all (semistandard) weak set-valued shifted tableaux of shape λ with no primed entries on the diagonal, as described in [14,Def. 3.1] (and with x T as defined in the same place).…”

Linear Compactness and Combinatorial Bialgebras

“…Nevertheless, some interesting "Hopf algebras" that can be identified with K (∼) P when ∼ is inhomogeneous have appeared in the literature [16,24,36,37]. A secondary, expos-itory goal of this paper is to describe explicitly the monoidal category containing such objects, which in general is not the usual category of bialgebras over a field.…”