
arXiv: 1806.11208
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
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
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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