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Crystal Combinatorics and Grothendieck Polynomials


Crystals are models for representations of symmetrizable Kac-Moody Lie algebras. They have close connections to algebra and geometry via symmetric functions. In this dissertation, combinatorics related to two kinds of symmetric functions arising from Schubert calculus is discussed. The first one is the stable Grothendieck polynomial. We introduce a type A crystal structure for the combinatorial objects underlying the stable Grothendieck polynomials which we call *-crystal. This crystal is a K-theoretic generalization of the Morse-Schilling crystal on decreasing factorizations in the symmetric group. We prove that under the residue map the *-crystal intertwines with the crystal on set-valued tableaux introduced by Monical, Pechenik and Scrimshaw. We also define a new insertion from decreasing factorizations to pairs of semistandard Young tableaux and prove several properties, such as its relation to the Hecke insertion and the uncrowding algorithm. The new insertion also intertwines with the crystal operators.

The second one is the stable canonical Grothendieck polynomial. Whereas set-valued tableaux are the combinatorial objects associated to stable Grothendieck polynomials, hook-valued tableaux are associated to stable canonical Grothendieck polynomials. We define a novel uncrowding algorithm for hook-valued tableaux. The algorithm "uncrowds" the entries in the arm of the hooks and yields a set-valued tableau and a column-flagged increasing tableau. We prove that our uncrowding algorithm intertwines with crystal operators. An alternative uncrowding algorithm that "uncrowds" the entries in the leg instead of the arm of the hooks is also given. As an application of uncrowding, we obtain various expansions of the canonical Grothendieck polynomials.

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