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Geometric Interpretation of the Integrable Crystals via Generalized Slices

Student: Vasily Krylov

Supervisor: Mikhail V. Finkelberg

Faculty: Faculty of Mathematics

Educational Programme: Mathematics (Bachelor)

Final Grade: 10

Year of Graduation: 2017

Let $G$ be a connected reductive algebraic group over $\BC$. Let $\Lap_{G}$ be the lattice of dominant coweights of $G$. It is known (see \cite{Jo}) that if $G$ is of adjoint type then there is a unique family of crystals of highest weight $\bB(\la)$ such that for every $\la_{1}, \la_{2} \in \Lap_{G}$ there exists the map of crystals $\bp_{\la_{1},\la_{2}}: \bB(\la_{1}) \otimes \bB(\la_{2}) \ra \bB(\la_{1}+\la_{2})\cup\{0\}.$ In the first part of the paper we construct crystals $\bB(\la)$ using the geometry of generalized transversal slices. After that we construct the maps $\bp_{\la_{1},\la_{2}}$ in terms of multiplication of generalized transversal slices. Let $L \subset G$ be a Levi subgroup inside $G$. In the third part of the paper we describe the restriction functor $\on{Res}^{\check{G}}_{\check{L}}$ using hyperbolic restriction functors between different subvarieties in generalized transversal slices.

Full text (added June 10, 2017)

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