Dyck tilings and the homogeneous Garnir relations for graded Specht modules

@article{Fayers2013DyckTA,
  title={Dyck tilings and the homogeneous Garnir relations for graded Specht modules},
  author={Matthew Fayers},
  journal={Journal of Algebraic Combinatorics},
  year={2013},
  volume={45},
  pages={1041-1082}
}
  • M. Fayers
  • Published 25 September 2013
  • Mathematics
  • Journal of Algebraic Combinatorics
Suppose $$\lambda $$λ and $$\mu $$μ are integer partitions with $$\lambda \supseteq \mu $$λ⊇μ. Kenyon and Wilson have introduced the notion of a cover-inclusive Dyck tiling of the skew Young diagram $$\lambda \backslash \mu $$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… 
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The number of tilings of skew Young diagrams by ribbon tiles shaped like Dyck paths, in which the tiles are "vertically decreasing", is studied to compute pairing probabilities in the double-dimer model.
Dyck tilings, linear extensions, descents, and inversions
Dyck tilings were introduced by Kenyon and Wilson in their study of double-dimer pairings. They are certain kinds of tilings of skew Young diagrams with ribbon tiles shaped like Dyck paths. We give
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The graded Specht module Sλ for a cyclotomic Hecke algebra comes with a distinguished generating vector zλ∈Sλ, which can be thought of as a ‘highest weight vector of weight λ’. This paper describes
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