Geodesic
Parabolic double cosets in Coxeter groups
Discrete mathematics & theoretical computer science, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) (2020).

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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 well-studied. 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 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 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. 
@article{DMTCS_2020_special_379_a29,
     author = {Billey, Sara and Konvalinka, Matjaz and Petersen, T. Kyle and Slofstra, William and Tenner, Bridget},
     title = {Parabolic double cosets in {Coxeter} groups},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)},
     year = {2020},
     doi = {10.46298/dmtcs.6347},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.6347/}
}
TY  - JOUR
AU  - Billey, Sara
AU  - Konvalinka, Matjaz
AU  - Petersen, T. Kyle
AU  - Slofstra, William
AU  - Tenner, Bridget
TI  - Parabolic double cosets in Coxeter groups
JO  - Discrete mathematics & theoretical computer science
PY  - 2020
VL  - DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)
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DO  - 10.46298/dmtcs.6347
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%0 Journal Article
%A Billey, Sara
%A Konvalinka, Matjaz
%A Petersen, T. Kyle
%A Slofstra, William
%A Tenner, Bridget
%T Parabolic double cosets in Coxeter groups
%J Discrete mathematics & theoretical computer science
%D 2020
%V DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.6347/
%R 10.46298/dmtcs.6347
%G en
%F DMTCS_2020_special_379_a29
Billey, Sara; Konvalinka, Matjaz; Petersen, T. Kyle; Slofstra, William; Tenner, Bridget. Parabolic double cosets in Coxeter groups. Discrete mathematics & theoretical computer science, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) (2020). doi : 10.46298/dmtcs.6347. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.6347/

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