Skew hook Schur functions and the cyclic sieving phenomenon

2022/11/25 by Nishu Kumari, Kumari, Nishu
Computer Science · Mathematics · #05E05 #05E10 #05E18 #Combinatorics (math.CO) #FOS: Mathematics #History and Theory of Mathematics #Mathematical Dynamics and Fractals #Polynomial and algebraic computation #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2211.14093

openalex publication_date 2022/11/25 · openalex created_date 2022/11/30 · openalex updated_date 2026/07/28

Abstract

Fix an integer t ≥ 2 and a primitive t^th root of unity ω. We consider the specialized skew hook Schur polynomial hs_λ/μ(X,ωX,…,ω^t-1X/Y,ωY,…,ω^t-1Y), where ω^k X=(ω^k x₁, …, ω^k x_n), ω^k Y=(ω^k y₁, …, ω^k y_m) for 0 ≤ k ≤ t-1. We characterize the skew shapes λ/μ for which the polynomial vanishes and prove that the nonzero polynomial factorizes into smaller skew hook Schur polynomials. Then we give a combinatorial interpretation of hs_λ/μ(1,ω^d,…,ω^d(tn-1)/1,ω^d,…,ω^d(tm-1)), for all divisors d of t, in terms of ribbon supertableaux. Lastly, we use the combinatorial interpretation to prove the cyclic sieving phenomenon on the set of semistandard supertableaux of shape λ/μ for odd t. Using a similar proof strategy, we give a complete generalization of a result of Lee--Oh (arXiv: 2112.12394, 2021) for the cyclic sieving phenomenon on the set of skew SSYT conjectured by Alexandersson--Pfannerer--Rubey--Uhlin (Forum Math. Sigma, 2021).

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