coxeter systems
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2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Sara Billey ◽  
Matjaz Konvalinka ◽  
T. Kyle Petersen ◽  
William Slofstra ◽  
Bridget Tenner

International audience 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.


2019 ◽  
Vol 527 ◽  
pp. 437-446
Author(s):  
James Parkinson ◽  
Yeeka Yau
Keyword(s):  

2018 ◽  
Vol 22 (6) ◽  
pp. 1479-1512
Author(s):  
Tianyuan Xu
Keyword(s):  

2017 ◽  
Vol 2018 (7) ◽  
pp. 2070-2098 ◽  
Author(s):  
Misha V Feigin ◽  
Alexander P Veselov

Abstract It is shown that the description of certain class of representations of the holonomy Lie algebra $\mathfrak g_{\Delta}$ associated with hyperplane arrangement $\Delta$ is essentially equivalent to the classification of $\vee$-systems associated with $\Delta.$ The flat sections of the corresponding $\vee$-connection can be interpreted as vector fields, which are both logarithmic and gradient. We conjecture that the hyperplane arrangement of any $\vee$-system is free in Saito's sense and show this for all known $\vee$-systems and for a special class of $\vee$-systems called harmonic, which includes all Coxeter systems. In the irreducible Coxeter case the potentials of the corresponding gradient vector fields turn out to be Saito flat coordinates, or their one-parameter deformations. We give formulas for these deformations as well as for the potentials of the classical families of harmonic $\vee$-systems.


2017 ◽  
Vol 11 (2) ◽  
pp. 469-498 ◽  
Author(s):  
Hao Chen ◽  
Jean-Philippe Labbé
Keyword(s):  

2016 ◽  
Vol 144 ◽  
pp. 326-360 ◽  
Author(s):  
Bernhard Mühlherr ◽  
Koji Nuida
Keyword(s):  

2014 ◽  
Vol 174 (1) ◽  
pp. 43-73 ◽  
Author(s):  
Hao Chen ◽  
Jean-Philippe Labbé

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