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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Format: | Article |
Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2015-01-01
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Series: | Discrete Mathematics & Theoretical Computer Science |
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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 ℤ<sub>$r$</sub> ≀ $S$<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. |
first_indexed | 2024-04-25T02:01:04Z |
format | Article |
id | doaj.art-cc29fa61af7a43e999a5bb3cb91cf77b |
institution | Directory Open Access Journal |
issn | 1365-8050 |
language | English |
last_indexed | 2024-04-25T02:01:04Z |
publishDate | 2015-01-01 |
publisher | Discrete Mathematics & Theoretical Computer Science |
record_format | Article |
series | Discrete Mathematics & Theoretical Computer Science |
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 ℤ<sub>$r$</sub> ≀ $S$<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 |