Shuffles on Coxeter groups

TL;DR

This paper extends classical card shuffling methods to Coxeter groups of types B and D, introducing new shuffle models for signed decks and developing a shuffle algebra framework linked to Coxeter group geometry.

Contribution

It introduces new shuffles for Coxeter groups of types B and D and formalizes a shuffle algebra connecting algebraic and geometric structures.

Findings

Defined shuffles for signed decks corresponding to types B and D

Developed a shuffle algebra capturing algebraic and geometric relations

Discussed generalizations to buildings and q-analogues

Abstract

The random-to-top and the riffle shuffle are two well-studied methods for shuffling a deck of cards. These correspond to the symmetric group $S_n$, i.e., the Coxeter group of type $A_{n-1}$. In this paper, we give analogous shuffles for the Coxeter groups of type $B_n$ and $D_n$. These can be interpreted as shuffles on a ``signed'' deck of cards. With these examples as motivation, we abstract the notion of a shuffle algebra which captures the connection between the algebraic structure of the shuffles and the geometry of the Coxeter groups. We also briefly discuss the generalisation to buildings which leads to q-analogues.