Permutation patterns, Stanley symmetric functions and generalized Specht modules

TL;DR

This paper introduces k-vexillary permutations, characterized by pattern avoidance, and constructs Specht series to analyze Stanley symmetric functions, proving a conjecture and classifying multiplicity-free cases.

Contribution

It generalizes vexillary permutations via a new filtration, characterizes them through pattern avoidance, and constructs Specht series for permutation diagrams.

Findings

k-vexillary permutations are characterized by avoiding finite pattern sets

Constructed Specht series for permutation diagrams

Proved Liu's conjecture on diagram varieties

Abstract

Generalizing the notion of a vexillary permutation, we introduce a filtration of S_infinity by the number of Schur function 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 to 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.