The Eigenvalues of Hyperoctahedral Descent Operators and Applications to Card-Shuffling

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Abstract

We extend an algebra of Mantaci and Reutenauer, acting on the free associative algebra, to a vector space of operators acting on all graded connected Hopf algebras. These operators are convolution products of certain involutions, which we view as hyperoctahedral variants of Patras's descent operators. We obtain the eigenvalues and multiplicities of all our new operators, as well as a basis of eigenvectors for a subclass akin to Adams operations. We outline how to apply this eigendata to study Markov chains, and examine in detail the case of card-shuffles with flips or rotations.
Original languageEnglish
Article numberP1.32
Pages (from-to)1-50
Number of pages50
JournalThe Electronic Journal of Combinatorics
Volume29
Issue number1
DOIs
Publication statusPublished - 25 Feb 2022

Scopus Subject Areas

  • Theoretical Computer Science
  • Applied Mathematics
  • Discrete Mathematics and Combinatorics
  • Geometry and Topology
  • Computational Theory and Mathematics

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