The one-sided cycle shuffles in the symmetric group algebra
Year: 2024
Type: article
Abstract: We study an infinite family of shuffling operators on the symmetric group S n , which includes the well-studied top-to-random shuffle. The general shuffling scheme consists of removing one card at a time from the deck (according to some probability distribution) and re-inserting it at a position chosen uniformly at random among the positions below. Rewritten in terms of the group algebra ℝ S n , our shuffle corresponds to right multiplication by a linear combination of the elements t ℓ : = cyc ℓ + cyc ℓ , ℓ + 1 + cyc ℓ , ℓ + 1 , ℓ + 2 + ⋯ + cyc ℓ , ℓ + 1 , ... , n ∈ ℝ S n for all ℓ ∈ 1 , 2 , ... , n (where cyc i 1 , i 2 , ... , i p denotes the permutation in S n that cycles through i 1 , i 2 , ... , i p ). We compute the eigenvalues of these shuffling operators and of all their linear combinations. In particular, we show that the eigenvalues of right multiplication by a linear combination λ 1 t 1 + λ 2 t 2 + ⋯ + λ n t n (with λ 1 , λ 2 , ... , λ n ∈ ℝ ) are the numbers
Source: Algebraic Combinatorics
DOI: 10.5802/alco.346
Authors: Darij Grinberg, Nadia Lafrenière
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4396228930