Brendan Pawlowski

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.

Permutation patterns, Stanley symmetric functions, and generalized Specht modules

Abstract Generalizing the notion of a vexillary permutation, we introduce a filtration of S ∞ by the number of terms in the Stanley symmetric function, with the kth filtration level called the k-vexillary permutations. We show that for each k, the k-vexillary permutations are characterized by avoiding a finite set of patterns. A key step is the construction of a Specht series, in the sense of James and Peel, for the Specht module associated with the diagram of a permutation. As a corollary, we prove a conjecture of Liu on diagram varieties for certain classes of permutation diagrams. We apply similar techniques to characterize multiplicity-free Stanley symmetric functions, as well as permutations whose diagram is equivalent to a forest in the sense of Liu.

Schur P-positivity and involution Stanley symmetric functions

The involution Stanley symmetric functions

Transition formulas for involution Schubert polynomials

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Efficient Finite Permutation Groups and Homomesy Computation in Common Lisp

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

Involution words II: braid relations and atomic structures

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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

Catalan matroid decompositions of certain positroids

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

Cohomology classes of rank varieties and a conjecture of Liu

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