Laurent polynomial Landau-Ginzburg models for cominuscule homogeneous spaces and mirror symmetry for the exceptional family

This thesis considers mirror symmetry for the small quantum cohomology of cominuscule homogeneous spaces. We present two main results: Firstly, in Theorem 2.2.7, we present a type-independent Laurent polynomial expression for Rietsch's Lie-theoretic mirror model [Rie08] restricted to an algebraic torus. Secondly, in Theorems 3.1.1 and 3.1.2, we present canonical mirror models for the exceptional family in terms of projective coordinates called (generalized) Plücker coordinates, and show that these are isomorphic to Rietsch's models. The Laurent polynomial expression resembles the potential for projective complete intersections given in [Giv96]: the sum of all the toric coordinates plus a "quantum term" consisting of a homogeneous polynomial in the toric coordinates divided by the product of all toric coordinates. This polynomial is initially enumerated by subexpressions of a given Weyl group element in another. In Corollary 2.5.12 we show that this enumeration can be replaced by diagrammatic combinatorics considering subsets of the quivers defined in [Per07, CMP08]. As we illustrate in the example of Grassmannians, these subsets can be considered as generalizations of Young diagrams. The canonical mirror models for the exceptional family are similar to the models found for other cominuscule families in [MR20, PR13, PRW16]. One notable difference is that we find cubic and quartic homogeneous polynomials in the Plücker coordinates, whereas these polynomials were found to be at most quadratic in the cases of Lagrangian Grassmannians and quadrics, and linear for Grassmannians. Analogously to the aforementioned papers, we use a presentation of the coordinate ring of a unipotent cell by [GLS11]. However, we give a type-independent criterion for the generators of this presentation to coincide (up to a constant) with Plücker coordinates in Proposition 3.2.5, and we show the isomorphism with Rietsch's mirror model using the algebraic torus where our Laurent polynomial expression holds.