Invariant theory for the face algebra of the braid arrangement
The faces of the braid arrangement form a monoid. The associated monoid algebra -- the face algebra -- is well-studied, especially in relation to card shuffling and other Markov chains. In this paper, we explore the action of the symmetric group on the face algebra from the perspective of invariant...
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Zusammenfassung: | The faces of the braid arrangement form a monoid. The associated monoid
algebra -- the face algebra -- is well-studied, especially in relation to card
shuffling and other Markov chains. In this paper, we explore the action of the
symmetric group on the face algebra from the perspective of invariant theory.
Bidigare proved the invariant subalgebra of the face algebra is
(anti)isomorphic to Solomon's descent algebra. We answer the more general
question: what is the structure of the face algebra as a simultaneous
representation of the symmetric group and Solomon's descent algebra? Special
cases of our main theorem recover the Cartan invariants of Solomon's descent
algebra discovered by Garsia-Reutenauer and work of Uyemura-Reyes on certain
shuffling representations. Our proof techniques involve the homology of
intervals in the lattice of set partitions. |
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DOI: | 10.48550/arxiv.2404.00536 |