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Invariant theory for the face algebra of the braid arrangement
Commins, Patricia1
1 University of Minnesota School of Mathematics 206 Church St SE Minneapolis MN 55455 (USA)
Algebraic Combinatorics, Volume 8 (2025) no. 2, pp. 421-468.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.412
Classification: 05E10, 05E18, 16W22, 16GXX
Keywords: Solomon’s descent algebra, higher Lie characters, plethysm, finite dimensional algebras, poset topology, reflection arrangements, Tits semigroup, left regular band

Commins, Patricia 1

1 University of Minnesota School of Mathematics 206 Church St SE Minneapolis MN 55455 (USA)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Commins, Patricia. Invariant theory for the face algebra of the braid arrangement. Algebraic Combinatorics, Volume 8 (2025) no. 2, pp. 421-468. doi : 10.5802/alco.412. https://alco.centre-mersenne.org/articles/10.5802/alco.412/

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