Shuffles on Coxeter groups

2001/08/13 by Swapneel Mahajan, Mahajan, Swapneel · 1 citation
Computer Science · Mathematics · #05A99 #60J10 #Advanced Combinatorial Mathematics #Algorithms and Data Compression #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #math.CO #math.PR #msc:05A99 #msc:60J10

paper · pdf · doi:10.48550/arxiv.math/0108094

arxiv created 2001/08/13 · openalex publication_date 2001/08/13 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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