Sara Billey ; Matjaz Konvalinka ; T. Kyle Petersen ; William Slofstra ; Bridget Tenner

Parabolic double cosets in Coxeter groups
dmtcs:6347 
Discrete Mathematics & Theoretical Computer Science,
April 22, 2020,
DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)

https://doi.org/10.46298/dmtcs.6347
Parabolic double cosets in Coxeter groups
Authors: Sara Billey ^{1}; Matjaz Konvalinka ^{2}; T. Kyle Petersen ^{3}; William Slofstra ^{4}; Bridget Tenner ^{3}
NULL##0000000207396744##NULL##NULL##NULL
Sara Billey;Matjaz Konvalinka;T. Kyle Petersen;William Slofstra;Bridget Tenner
1 Department of Mathematics [Seattle]
2 Departement of Mathematics [Slovenia]
3 Department of Mathematical Sciences [Chicago]
4 Institute for Quantum Computing [Waterloo]
Parabolic subgroups WI of Coxeter systems (W,S) and their ordinary and double cosets W/WI and WI\W/WJ 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 wWI , for I ⊆ S, forms the Coxeter complex of W , and is wellstudied. In this extended abstract, we look at a less studied object: the set of all double cosets WIwWJ for I,J ⊆ S. Each double coset can be presented by many different triples (I, w, J). We describe what we call the lexminimal presentation and prove that there exists a unique such choice for each double coset. Lexminimal 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 Sn. 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.