A trilogy of (super)crystals: the queer, the set-valued and the hook-valued
This dissertation compiles three main results concerning crystals or supercrystals. Firstly, we present a characterization for queer supercrystals introduced by Grantcharov et al. We also construct a type A crystal whose character is the stable Grothendieck polynomials for fully-commutative permutations. Finally, we describe an uncrowding map on hook-valued tableaux which intertwines the crystal operators on hook-valued tableaux to those of set-valued tableaux.