scholarly journals Minimal length elements in some double cosets of Coxeter groups

2007 ◽  
Vol 215 (2) ◽  
pp. 469-503 ◽  
Author(s):  
Xuhua He
Keyword(s):  
Coxeter Groups ◽  
Minimal Length ◽  
Double Cosets

scholarly journals Minimal length elements of finite Coxeter groups

2012 ◽  
Vol 161 (15) ◽  
pp. 2945-2967 ◽  
Author(s):  
Xuhua He ◽  
Sian Nie
Keyword(s):  
Coxeter Groups ◽  
Minimal Length

scholarly journals Zero excess and minimal length in finite Coxeter groups

2012 ◽  
Vol 15 (4) ◽  
Author(s):  
Sarah B. Hart ◽  
Peter J. Rowley
Keyword(s):  
Coxeter Groups ◽  
Minimal Length

Abstract.LetWhen

scholarly journals Minimal Length Elements in Twisted Conjugacy Classes of Finite Coxeter Groups

2000 ◽  
Vol 229 (2) ◽  
pp. 570-600 ◽  
Author(s):  
Meinolf Geck ◽  
Sungsoon Kim ◽  
Götz Pfeiffer
Keyword(s):  
Coxeter Groups ◽  
Minimal Length ◽  
Conjugacy Classes ◽  
Twisted Conjugacy

scholarly journals A Two-Sided Analogue of the Coxeter Complex

10.37236/8015 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
T. Kyle Petersen
Keyword(s):  
Generating Function ◽  
Joint Distribution ◽  
Coxeter Groups ◽  
Double Coset ◽  
Minimal Length ◽  
Coxeter System ◽  
Coxeter Complex ◽  
Eulerian Polynomial ◽  
Left And Right ◽  
Simplicial Poset

For any Coxeter system $(W,S)$ of rank $n$, we study an abstract boolean complex (simplicial poset) of dimension $2n-1$ that contains the Coxeter complex as a relative subcomplex. For finite $W$, this complex is first described in work of Hultman. Faces are indexed by triples $(I,w,J)$, where $I$ and $J$ are subsets of the set $S$ of simple generators, and $w$ is a minimal length representative for the parabolic double coset $W_I w W_J$. There is exactly one maximal face for each element of the group $W$. The complex is shellable and thin, which implies the complex is a sphere for the finite Coxeter groups. In this case, a natural refinement of the $h$-polynomial is given by the "two-sided" $W$-Eulerian polynomial, i.e., the generating function for the joint distribution of left and right descents in $W$.

scholarly journals A two-sided analogue of the Coxeter complex

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
T. Kyle Petersen
Keyword(s):  
Generating Function ◽  
Joint Distribution ◽  
Coxeter Groups ◽  
Minimal Length ◽  
Coxeter System ◽  
Coxeter Complex ◽  
Eulerian Polynomial ◽  
International Audience ◽  
Left And Right ◽  
Simplicial Poset

International audience For any Coxeter system (W, S) of rank n, we introduce an abstract boolean complex (simplicial poset) of dimension 2n − 1 which contains the Coxeter complex as a relative subcomplex. Faces are indexed by triples (J,w,K), where J and K are subsets of the set S of simple generators, and w is a minimal length representative for the double parabolic coset WJ wWK . There is exactly one maximal face for each element of the group W . The complex is shellable and thin, which implies the complex is a sphere for the finite Coxeter groups. In this case, a natural refinement of the h-polynomial is given by the “two-sided” W -Eulerian polynomial, i.e., the generating function for the joint distribution of left and right descents in W .

scholarly journals Parabolic double cosets in Coxeter groups

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Sara Billey ◽  
Matjaz Konvalinka ◽  
T. Kyle Petersen ◽  
William Slofstra ◽  
Bridget Tenner
Keyword(s):  
Coxeter Groups ◽  
Linear Time ◽  
Minimal Element ◽  
Double Coset ◽  
Flag Varieties ◽  
Parabolic Subgroups ◽  
Double Cosets ◽  
Coxeter Graph ◽  
Coxeter Systems ◽  
Minimal Presentations

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.

scholarly journals Parabolic Double Cosets in Coxeter Groups

10.37236/6741 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Sara C. Billey ◽  
Matjaž Konvalinka ◽  
T. Kyle Petersen ◽  
William Slofstra ◽  
Bridget E. Tenner
Keyword(s):  
Coxeter Groups ◽  
Linear Time ◽  
Minimal Element ◽  
Double Coset ◽  
Weyl Groups ◽  
Flag Varieties ◽  
Parabolic Subgroups ◽  
Double Cosets ◽  
Affine Weyl Groups ◽  
Coxeter Graph

Parabolic subgroups $W_I$ of Coxeter systems $(W,S)$, as well as their ordinary and double quotients $W / W_I$ 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 article we look at a less studied object: the set of all double cosets $W_I w W_J$ for $I, J \subseteq S$. Double cosets are not uniquely presented by triples $(I,w,J)$. We describe what we call the lex-minimal presentation, and prove that there exists a unique such object for each double coset. Lex-minimal presentations are then used to enumerate double cosets via a finite automaton depending on the Coxeter graph for $(W,S)$. As an example, 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 this case, parabolic subgroups are also known as Young subgroups). Our formula is almost always linear time computable in $n$, and we show how it can be generalized to any Coxeter group with little additional work. We spell out formulas for all finite and affine Weyl groups in the case that $w$ is the identity element.

scholarly journals Counting Parabolic Double Cosets in Symmetric Groups

10.37236/9988 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Thomas Browning
Keyword(s):  
Asymptotic Formula ◽  
Polynomial Time ◽  
Coxeter Groups ◽  
Symmetric Groups ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Double Cosets ◽  
New Formula

Billey, Konvalinka, Petersen, Solfstra, and Tenner recently presented a method for counting parabolic double cosets in Coxeter groups, and used it to compute $p_n$, the number of parabolic double cosets in $S_n$, for $n\leq13$. In this paper, we derive a new formula for $p_n$ and an efficient polynomial time algorithm for evaluating this formula. We use these results to compute $p_n$ for $n\leq5000$ and to prove an asymptotic formula for $p_n$ that was conjectured by Billey et al.

Minimum space length

2019 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Minimal Length ◽  
Space Length

We follow some logical principles to conclude there is a minimal length.

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