2020/10/26 by Tair Akhmejanov, Balázs Elek, Akhmejanov, Tair +1
Mathematics · #05E10 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematical and Theoretical Analysis #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2010.13930
openalex publication_date 2020/10/26 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
Let δ=(δ1,…,δn) be a string of letters h and v. We define a Young tableau to be δ-semistandard if the entries are weakly increasing along rows and columns, and the entries i form a horizontal strip if δi=h and a vertical strip if δi=v. We define δ-promotion on such tableaux via a modified jeu-de-taquin. The first main result is that δ-promotion has period n on rectangular δ-semistandard tableaux, generalizing the results of Haiman and Rhoades for standard and semistandard tableaux. The second main result states that the set of rectangular δ-semistandard tableaux for fixed δ and content γ exhibits the cyclic sieving phenomenon with the generalized Kostka polynomial. To do so we follow Fontaine-Kamnitzer and associate to (δ,γ) an SLm-invariant space Inv(Vλ1⊗⋯⊗ Vλn) where each Vλi is an alternating or symmetric representation. We show that the Satake basis of the corresponding invariant space is indexed by the set of tableaux corresponding to (δ,γ) and is permuted by rotation of tensor factors. We then diagonalize the rotation action using the fusion product. This cyclic sieving generalizes the result of Rhoades, and of Fontaine-Kamnitzer (in type A), and is closely related to that of Westbury.