On The Free and G-Saturated Weight Monoids of Smooth Affine Spherical Varieties For G=SL(n)

Let $X$ be an affine algebraic variety over $\mathbb{C}$ equipped with an action of a connected reductive group $G$. The weight monoid $\Gamma(X)$ of $X$ is the set of isomorphism classes of irreducible representations of $G$ that occur in the coordinate ring $\mathbb{C}[X]$ of $X$. Losev has shown that if $X$ is a smooth affine spherical variety, that is, if $X$ is smooth and $\mathbb{C}[X]$ is multiplicity-free as a representation of $G$, then $\Gamma(X)$ determines $X$ up to equivariant automorphism. Pezzini and Van Steirteghem have recently obtained a combinatorial characterization of the weight monoids of smooth affine spherical varieties, using the combinatorial theory of spherical varieties and a smoothness criterion due to R. Camus. The first part of this thesis gives an implementation in Sage of a special case of this combinatorial characterization: given a free and ``$G$-saturated monoid $\Gamma$ of dominant weights for $G=\mathrm{SL}(n)$, the algorithm decides whether there exists a smooth affine spherical $G$-variety $X$ such that $\Gamma(X) = \Gamma$. In the second part of the thesis, we apply Pezzini and Van Steirteghem\u27s characterization to determine which subsets of the set of fundamental weights of $\mathrm{SL}(n)$ generate a monoid that is the weight monoid of a smooth affine spherical $G$-variety