On reduced expressions for core double cosets
arxiv(2024)
摘要
The notion of a reduced expression for a double coset in a Coxeter group was
introduced by Williamson, and recent work of Elias and Ko has made this theory
more accessible and combinatorial. One result of Elias-Ko is that any coset
admits a reduced expression which factors through a reduced expression for a
related coset called its core. In this paper we define a class of cosets called
atomic cosets, and prove that every core coset admits a reduced expression as a
composition of atomic cosets. This leads to an algorithmic construction of a
reduced expression for any coset. In types A and B we prove that the
combinatorics of compositions of atomic cosets matches the combinatorics of
ordinary expressions in a smaller group. In other types the combinatorics is
new, as explored in a sequel by Ko.
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