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Permutation patterns, Stanley symmetric functions, and generalized Specht modules. (English) Zbl 1300.05313

Summary: Generalizing the notion of a vexillary permutation, we introduce a filtration of \(S_\infty\) by the number of terms in the Stanley symmetric function, with the \(k\)th 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 G. James and M. Peel [J. Algebra 56, 343–364 (1979; Zbl 0398.20016)], 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.

MSC:

05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
20C30 Representations of finite symmetric groups

Citations:

Zbl 0398.20016
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References:

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