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We generalize random-to-random shuffling from a Markov chain on the symmetric group to one on the Type A Iwahori Hecke algebra, and show that its eigenvalues are polynomials in q with non-negative integer coefficients. Setting q=1 recovers results of Dieker and Saliola, whose computation of the spectrum of random-to-random in the symmetric group resolved a nearly 20 year old conjecture by Uyemura-Reyes. Our methods simplify their proofs by drawing novel connections to the Jucys-Murphy elements of the Hecke algebra, Young seminormal forms, and the Okounkov-Vershik approach to representation theory., Comment: Reorganized, streamlined arguments, improved exposition, and addressed/clarified subtleties from v1

We develop the idempotent theory for algebras over a class of semigroups called left regular bands of groups (LRBGs), which simultaneously generalize group algebras of finite groups and left regular band (LRB) algebras. Our techniques weave together the representation theory of finite groups and LRBs, opening the door for a systematic study of LRBGs in an analogous way to LRBs. We apply our results to construct complete systems of primitive orthogonal idempotents in the Mantaci--Reutenauer algebra ${\sf{MR}}_n[G]$ associated to any finite group $G$. When $G$ is abelian, we give closed form expressions for these idempotents, and when $G$ is the cyclic group of order two, we prove that these recover idempotents introduced by Vazirani., Comment: To appear in Journal of Algebra

We examine from an invariant theory viewpoint the monoid algebras for two monoids having large symmetry groups. The first monoid is the free left-regular band on $n$ letters, defined on the set of all injective words, that is, the words with at most one occurrence of each letter. This monoid carries the action of the symmetric group. The second monoid is one of its $q$-analogues, considered by K. Brown, carrying an action of the finite general linear group. In both cases, we show that the invariant subalgebras are semisimple commutative algebras, and characterize them using Stirling and $q$-Stirling numbers. We then use results from the theory of random walks and random-to-top shuffling to decompose the entire monoid algebra into irreducibles, simultaneously as a module over the invariant ring and as a group representation. Our irreducible decompositions are described in terms of derangement symmetric functions introduced by D\'esarm\'enien and Wachs., Comment: final version, to appear in the Pacific Journal of Mathematics

We give a Type $B$ analog of Whitehouse's lifts of the Eulerian representations from $S_n$ to $S_{n+1}$ by introducing a family of $B_{n}$-representations that lift to $B_{n+1}$. As in Type $A$, we interpret these representations combinatorially via a family of orthogonal idempotents in the Mantaci-Reutenauer algebra, and topologically as the graded pieces of the cohomology of a certain $\mathbb{Z}_{2}$-orbit configuration space of $\mathbb{R}^{3}$. We show that the lifted $B_{n+1}$-representations also have a configuration space interpretation, and further parallel the Type $A$ story by giving analogs of many of its notable properties, such as connections to equivariant cohomology and the Varchenko-Gelfand ring., Comment: final version, to appear in Mathematische Zeitschrift

The Eulerian idempotents, first introduced for the symmetric group and later extended to all reflection groups, generate a family of representations called the Eulerian representations that decompose the regular representation. In Type $A$, the Eulerian representations have many elegant but mysterious connections to rings naturally associated with the braid arrangement. In this paper, we unify these results and show that they hold for any reflection group of coincidental type -- that is, $S_{n}$, $B_{n}$, $H_{3}$ or the dihedral group $I_{2}(m)$ -- by giving six characterizations of the Eulerian representations, including as components of the associated graded of the Varchenko-Gelfand ring $\mathcal{V}$. As a consequence, we show that Solomon's descent algebra contains a commutative subalgebra generated by sums of elements with the same number of descents if and only if $W$ is coincidental. More generally, when $W$ is any finite real reflection group, we give a case-free construction of a family of Eulerian representations described by a flat-decomposition of the ring $\mathcal{V}$., Comment: Final version, to appear in the Journal of the London Mathematical Society

Fulton and Kraskiewicz gave a presentation of Specht modules as a quotient of the space of column tabloids by dual Garnir relations. We simplify this presentation by showing that it can be generated by a single relation for each pair of columns of a tableau with ordered columns, thereby significantly reducing the number of generators given in the original construction. Our presentation applies to all Specht modules, and is of a similar nature to a recent result by Friedmann-Hanlon-Stanley-Wachs that applies to staircase partitions. We show that our presentation implies the Friedmann-Hanlon-Stanley-Wachs presentation., Comment: Final version appearing in the Journal of Pure and Applied Algebra