Zachary Hamaker

Involution words: counting problems and connections to Schubert calculus for symmetric orbit closures

Abstract Involution words are variations of reduced words for involutions in Coxeter groups, first studied under the name of “admissible sequences” by Richardson and Springer. They are maximal chains in Richardson and Springers weak order on involutions. This article is the first in a series of papers on involution words, and focuses on their enumerative properties. We define involution analogues of several objects associated to permutations, including Rothe diagrams, the essential set, Schubert polynomials, and Stanley symmetric functions. These definitions have geometric interpretations for certain intervals in the weak order on involutions. In particular, our definition of “involution Schubert polynomials” can be viewed as a Billey–Jockusch–Stanley type formula for cohomology class representatives of O n - and Sp 2 n -orbit closures in the flag variety, defined inductively in recent work of Wyser and Yong. As a special case of a more general theorem, we show that the involution Stanley symmetric function for the longest element of a finite symmetric group is a product of staircase-shaped Schur functions. This implies that the number of involution words for the longest element of a finite symmetric group is equal to the dimension of a certain irreducible representation of a Weyl group of type B.

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

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.

Schur P-positivity and involution Stanley symmetric functions

The involution Stanley symmetric functions

Transition formulas for involution Schubert polynomials

\hat{F}_y

Involution words II: braid relations and atomic structures

are the stable limits of the analogues of Schubert polynomials for the orbits of the orthogonal group in the flag variety. These symmetric functions are also generating functions for involution words, and are indexed by the involutions in the symmetric group. By construction each

Doppelg\"angers: Bijections of Plane Partitions

\hat{F}_y

Fixed-point-free involutions and Schur P-positivity

is a sum of Stanley symmetric functions and therefore Schur positive. We prove the stronger fact that these power series are Schur

Pattern avoidance and quasisymmetric functions.

P

Subwords and Plane Partitions

-positive. We give an algorithm to efficiently compute the decomposition of

Weak order and descents for monotone triangles.

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