Koszulity of dual braid monoid algebras via cluster complexes

The dual braid monoid was introduced by Bessis in his work on complex reflection arrangements. The goal of this work is to show that Koszul duality provides a nice interplay between the dual braid monoid and the cluster complex introduced by Fomin and Zelevinsky. Firstly, we prove koszulity of the d...

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Published in:Annales mathématiques Blaise Pascal Vol. 30; no. 2; pp. 141 - 188
Main Authors: Nadeau, Philippe, Josuat-Vergès, Matthieu
Format: Journal Article
Language:English
Published: Université Blaise-Pascal - Clermont-Ferrand 30.04.2024
Subjects:
Combinatorics
Mathematics
ISSN:1259-1734, 2118-7436
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Abstract The dual braid monoid was introduced by Bessis in his work on complex reflection arrangements. The goal of this work is to show that Koszul duality provides a nice interplay between the dual braid monoid and the cluster complex introduced by Fomin and Zelevinsky. Firstly, we prove koszulity of the dual braid monoid algebra, by building explicitly the minimal free resolution of the ground field. This is done explicitly using some chains complexes defined in terms of the positive part of the cluster complex. Secondly, we derive various properties of the quadratic dual algebra. We show that it is naturally graded by the noncrossing partition lattice. We get an explicit basis, naturally indexed by positive faces of the cluster complex. Moreover, we find the structure constants via a geometric rule in terms of the cluster fan. Eventually, we realize this dual algebra as a quotient of a Nichols algebra. This latter fact makes a connection with results of Zhang, who used the same algebra to compute the homology of Milnor fibers of reflection arrangements. Le monoïde dual des tresses a été introduit par Bessis dans le contexte des arrangements d’hyperplanscomplexes. Le but de ce travail est de montrer que la dualité de Koszul fournit une interaction remarquableavec le complexe d’amas introduit par Fomin et Zelevinsky. Premièrement, nous démontrons la koszulitéde l’algèbre du monoïde dual des tresses, en donnant explicitement la résolution libre minimale ducorps de base. Cette construction utilise des complexes de chaînes définis grâce à la partie positive ducomplexe d’amas. Deuxièmement, nous examinons diverses propriétés de l’algèbre quadratique duale.Nous démontrons qu’elle est naturellement graduée par le treillis des partitions non-croisées. Nousobtenons une base explicite, indicée par les faces positives du complexe d’amas. Les constantes destructure peuvent être décrites explicitement en termes de l’éventail des amas. Enfin, nous réalisons cettealgèbre duale comme un quotient d’une algèbre de Nichols. Ce dernier point se relie aux travaux deZhang, qui a utilisé cette algèbre pour un calcul d’homologie des fibres de Milnor d’un arrangement deCoxeter.
AbstractList The dual braid monoid was introduced by Bessis in his work on complex reflection arrangements. The goal of this work is to show that Koszul duality provides a nice interplay between the dual braid monoid and the cluster complex introduced by Fomin and Zelevinsky. Firstly, we prove koszulity of the dual braid monoid algebra, by building explicitly the minimal free resolution of the ground field. This is done explicitly using some chains complexes defined in terms of the positive part of the cluster complex. Secondly, we derive various properties of the quadratic dual algebra. We show that it is naturally graded by the noncrossing partition lattice. We get an explicit basis, naturally indexed by positive faces of the cluster complex. Moreover, we find the structure constants via a geometric rule in terms of the cluster fan. Eventually, we realize this dual algebra as a quotient of a Nichols algebra. This latter fact makes a connection with results of Zhang, who used the same algebra to compute the homology of Milnor fibers of reflection arrangements. Le monoïde dual des tresses a été introduit par Bessis dans le contexte des arrangements d’hyperplanscomplexes. Le but de ce travail est de montrer que la dualité de Koszul fournit une interaction remarquableavec le complexe d’amas introduit par Fomin et Zelevinsky. Premièrement, nous démontrons la koszulitéde l’algèbre du monoïde dual des tresses, en donnant explicitement la résolution libre minimale ducorps de base. Cette construction utilise des complexes de chaînes définis grâce à la partie positive ducomplexe d’amas. Deuxièmement, nous examinons diverses propriétés de l’algèbre quadratique duale.Nous démontrons qu’elle est naturellement graduée par le treillis des partitions non-croisées. Nousobtenons une base explicite, indicée par les faces positives du complexe d’amas. Les constantes destructure peuvent être décrites explicitement en termes de l’éventail des amas. Enfin, nous réalisons cettealgèbre duale comme un quotient d’une algèbre de Nichols. Ce dernier point se relie aux travaux deZhang, qui a utilisé cette algèbre pour un calcul d’homologie des fibres de Milnor d’un arrangement deCoxeter.
Author Nadeau, Philippe
Josuat-Vergès, Matthieu
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  surname: Josuat-Vergès
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Snippet The dual braid monoid was introduced by Bessis in his work on complex reflection arrangements. The goal of this work is to show that Koszul duality provides a...
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Title Koszulity of dual braid monoid algebras via cluster complexes
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