Séminaire Lotharingien de Combinatoire, 91B.16 (2024), 12 pp.
Darij Grinberg and Nadia Lafrenière
The Somewhere-To-Below Shuffles in The Symmetric Group and Hecke Algebras
Abstract.
We introduce and study the somewhere-to-below shuffles,
which are elements of the group algebra of the symmetric group Sn defined
as sums of cycles.
We show that these elements are
simultaneously triangularizable (in an easily-defined basis of
k[Sn]), and compute their joint
eigenvalues with multiplicities.
We furthermore discuss some identities between them, a
card shuffling interpretation and its probabilistic properties, and a possible
generalization to the Hecke algebra.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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