2013/09/25 by Matthew Fayers, Fayers, Matthew · 1 citation
Chemistry · Mathematics · #05E10 #20C08 #20C30 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Molecular spectroscopy and chirality #Representation Theory (math.RT) #math.CO #math.RT #msc:05E10 #msc:20C08 #msc:20C30
paper · pdf · doi:10.48550/arxiv.1309.6467
openalex publication_date 2013/09/25 · arxiv created 2015/01/29 · arxiv updated 2015/01/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Suppose λ and μ are integer partitions with λ⊇μ. Kenyon and Wilson have introduced the notion of a cover-inclusive Dyck tiling of the skew Young diagram λ∖μ, which has applications in the study of double-dimer models. We examine these tilings in more detail, giving various equivalent conditions and then proving a recurrence which we use to show that the entries of the transition matrix between two bases for a certain permutation module for the symmetric group are given by counting cover-inclusive Dyck tilings. We go on to consider the inverse of this matrix, showing that its entries are determined by what we call cover-expansive Dyck tilings. The fact that these two matrices are mutual inverses allows us to recover the main result of Kenyon and Wilson. We then discuss the connections with recent results of Kim et al, who give, a simple expression for the sum, over all μ, of the number of cover-inclusive Dyck tilings of λ∖μ. Our results provide a new proof of this result. Finally, we show how to use our results to obtain simpler expressions for the homogeneous Garnir relations for the universal Specht modules introduced by Kleshchev, Mathas and Ram for the cyclotomic quiver Hecke algebras.