Permutation patterns, Stanley symmetric functions and generalized Specht modules

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.

29 pages; corrected typos in section 3, fixed arguments and improved results in section 6

References in corpus (4)

• Degeneracy Loci, Pfaffians, and Vexillary Signed Permutations in Types B, C, and D
• Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams
• Algorithms for discovering and proving theorems about permutation patterns
• Shape and pattern containment of separable permutations