How to get the weak order out of a digraph ?

We construct a poset from a simple acyclic digraph together with a valuation on its vertices, and we compute the values of its Möbius function. We show that the weak order on Coxeter groups $A$<sub>$n-1$</sub>, $B$<sub>$n$</sub>, $Ã$<sub>$n$</sub>, and the flag we...

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Main Author: Francois Viard
Format: Article
Language:English
Published: Discrete Mathematics & Theoretical Computer Science 2015-01-01
Series:Discrete Mathematics & Theoretical Computer Science
Subjects:
Online Access:https://dmtcs.episciences.org/2538/pdf
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author Francois Viard
author_facet Francois Viard
author_sort Francois Viard
collection DOAJ
description We construct a poset from a simple acyclic digraph together with a valuation on its vertices, and we compute the values of its Möbius function. We show that the weak order on Coxeter groups $A$<sub>$n-1$</sub>, $B$<sub>$n$</sub>, $Ã$<sub>$n$</sub>, and the flag weak order on the wreath product &#8484;<sub>$r$</sub> &#8768; $S$<sub>$n$</sub> introduced by Adin, Brenti and Roichman (2012), are special instances of our construction. We conclude by briefly explaining how to use our work to define quasi-symmetric functions, with a special emphasis on the $A$<sub>$n-1$</sub> case, in which case we obtain the classical Stanley symmetric function.
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spelling doaj.art-cc29fa61af7a43e999a5bb3cb91cf77b2024-03-07T15:01:26ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502015-01-01DMTCS Proceedings, 27th...Proceedings10.46298/dmtcs.25382538How to get the weak order out of a digraph ?Francois Viard0Combinatoire, théorie des nombresWe construct a poset from a simple acyclic digraph together with a valuation on its vertices, and we compute the values of its Möbius function. We show that the weak order on Coxeter groups $A$<sub>$n-1$</sub>, $B$<sub>$n$</sub>, $Ã$<sub>$n$</sub>, and the flag weak order on the wreath product &#8484;<sub>$r$</sub> &#8768; $S$<sub>$n$</sub> introduced by Adin, Brenti and Roichman (2012), are special instances of our construction. We conclude by briefly explaining how to use our work to define quasi-symmetric functions, with a special emphasis on the $A$<sub>$n-1$</sub> case, in which case we obtain the classical Stanley symmetric function.https://dmtcs.episciences.org/2538/pdftableauxdigraphsposetscoxeter groupsweak orderreduced decompositions[info.info-dm] computer science [cs]/discrete mathematics [cs.dm]
spellingShingle Francois Viard
How to get the weak order out of a digraph ?
Discrete Mathematics & Theoretical Computer Science
tableaux
digraphs
posets
coxeter groups
weak order
reduced decompositions
[info.info-dm] computer science [cs]/discrete mathematics [cs.dm]
title How to get the weak order out of a digraph ?
title_full How to get the weak order out of a digraph ?
title_fullStr How to get the weak order out of a digraph ?
title_full_unstemmed How to get the weak order out of a digraph ?
title_short How to get the weak order out of a digraph ?
title_sort how to get the weak order out of a digraph
topic tableaux
digraphs
posets
coxeter groups
weak order
reduced decompositions
[info.info-dm] computer science [cs]/discrete mathematics [cs.dm]
url https://dmtcs.episciences.org/2538/pdf
work_keys_str_mv AT francoisviard howtogettheweakorderoutofadigraph