Abstracts
for the Seminar

Spring 2016

**Speaker: **Sara Billey, UW

**Title: **Enumeration of Parabolic Double Cosets for Coxeter Groups

**Time:** 2:30 PM, Monday, March 14, 2016

**Place:** Malott 206

**Abstract:** Parabolic subgroups $W_I$ of Coxeter systems $(W,S)$ and their ordinary and
double cosets $W / W_I$ and $W_I \backslash W / W_J$ appear in many contexts in
combinatorics and Lie theory, including the geometry and topology of
generalized flag varieties and the symmetry groups of regular polytopes. The
set of ordinary cosets $w W_I$, for $I \subseteq S$, forms the Coxeter complex of $W$,
and is well-studied. In this talk, we look at a less studied object: the set of
all parabolic double cosets $W_I w W_J$ for $I, J \subseteq S$.
Each double coset can be presented by many different triples
$(I,w,J)$. We describe what we call the lex-minimal presentation
and prove that there exists a unique such choice for each double coset.
Lex-minimal presentations can be enumerated via a
finite automaton depending on the Coxeter graph for $(W,S)$.
In particular, we present a formula for the number of
parabolic double cosets with a fixed minimal element when $W$
is the symmetric group $S_n$. In that case, parabolic subgroups are also
known as Young subgroups. Our formula is almost always linear time computable in $n$, and the formula can be generalized to any Coxeter group.
This is talk is based on joint work with Matjaz Konvalinka, T. Kyle Petersen, William Slofstra and Bridget Tenner.

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