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

C. Y. Amy Pang

doi:10.37236/10678

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

C. Y. Amy Pang^*

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

1 Citation (Scopus)

67 Downloads (Pure)

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 language	English
Article number	P1.32
Number of pages	50
Journal	Electronic Journal of Combinatorics
Volume	29
Issue number	1
DOIs	https://doi.org/10.37236/10678
Publication status	Published - 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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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.",

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N2 - 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.

AB - 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.

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

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