Elements of minimal length and Bruhat order on fixed point cosets of Coxeter groups

2023/11/12 by Chapelier-Laget, Nathan, Gobet, Thomas
#Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2311.06827

Abstract

We study the restriction of the strong Bruhat order on an arbitrary Coxeter group W to cosets x W_L^θ, where x is an element of W and W_L^θ the subgroup of fixed points of an automorphism θ of order at most two of a standard parabolic subgroup W_L of W. When θ\neqid, there is in general more than one element of minimal length in a given coset, and we explain how to relate elements of minimal length. We also show that elements of minimal length in cosets are exactly those elements which are minimal for the restriction of the Bruhat order.

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