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Stanley symmetric functions for signed involutions

Authors: Marberg, Eric Paul MATH; Pawlowski, Brendan;

Stanley symmetric functions for signed involutions

Abstract

An involution in a Coxeter group has an associated set of involution words, a variation on reduced words. These words are saturated chains in a partial order first considered by Richardson and Springer in their study of symmetric varieties. In the symmetric group, involution words can be enumerated in terms of tableaux using appropriate analogues of the symmetric functions introduced by Stanley to accomplish the same task for reduced words. We adapt this approach to the group of signed permutations. We show that involution words for the longest element in the Coxeter group $C_n$ are in bijection with reduced words for the longest element in $A_n = S_{n+1}$, which are known to be in bijection with standard tableaux of shape $(n, n-1, \ldots, 2, 1)$.

26 pages, 3 figures; v2: fixed several typos; v3: some minor corrections, a few added remarks, final version

Keywords

Signed permutations, Exact enumeration problems, generating functions, Schur S-functions, Linear algebraic groups over arbitrary fields, Involution words, Reflection and Coxeter groups (group-theoretic aspects), Coxeter systems, FOS: Mathematics, Mathematics - Combinatorics, Stanley symmetric functions, shifted tableaux, Representation Theory (math.RT), Permutations, words, matrices, Symmetric functions and generalizations, Symmetric groups, transition equations, Reduced words, signed permutations, reduced words, 0-Hecke monoids, involution words, Shifted tableaux, Schur \(S\)-functions, Combinatorics (math.CO), Mathematics - Representation Theory

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
2
Average
Average
Average
Green
bronze