Keys and Demazure crystals for Kac–Moody algebras

Abstract

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Quasi-solvable lattice models for and Demazure atoms and characters

“…This is in parallel to the models in [16,20] with Gelfand-Tsetlin patterns and reverse semistandard tableaux because we can only use one particular ordering of the spectral parameters. We conjecture that our key map agrees with the representation-theoretic description coming from the Kashiwara crystal structure [22,40] described in [38], the key map on Kashiwara-Nakashima (KN) tableaux [41] given in [71], and with the crystal structure defined recently on King tableaux by Lee [56]. A salient ingredient is a weight-preserving bijection from reverse King (respectively, Sundaram) tableaux to regular King (respectively, Sundaram) tableaux.…”

“…Additionally, applying the tableau-switching algorithm from is a crystal isomorphism with the crystal structure on given by Lee [56]. Moreover, the composition of applying the tableau-switching algorithm to interchange for all i , replacing entry-wise, and applying the Sheats bijection is a crystal isomorphism on the corresponding crystal atoms that sends to the key map defined in [38, 71].…”

Symplectic Keys and Demazure Atoms in Type C

“…In those questions, Azenhas identified some type C n Kashiwara-Nakashima tableaux as key tableaux, which match our identification, but it lacks a construction of the right key map, thus lacking a proof of the combinatorial description of type C Demazure characters. Note also that, during the preparation of this paper, Jacon and Lecouvey informed us about their paper [12], where they find the same key in type C, but their approach is different from ours.…”

Symplectic right keys -- Type C Willis' direct way

“…For a complete overview in type A, see [3] and the references therein. In type C n , the symplectic key tableaux are characterized in [2,14,15,6]. Using the Sheats symplectic jeu de taquin, a right key map is given, in [14,15,6], to send a Kashiwara-Nakashima (KN) tableau T to its right key tableau K + (T), that detects the Demazure atom crystal which contains T. They are also computed in the type C n alcove path model [11], as well as for reverse King tableau using the colored five vertex model [4].…”

“…Jacon and Lecouvey [6] have suggested that Willis' method should be adaptable to type C n . Motivated by Willis' direct inspection [17], we create an alternative algorithm, based on the direct inspection of a KN tableau, for the symplectic right key map, that does not use the symplectic jeu de taquin.…”