Kiumars Kaveh

Crystal bases and Newton–Okounkov bodies

Let G be a connected reductive algebraic group. We prove that the string parametrization of a crystal basis for a finite dimensional irreducible representation of G coincides with a natural valuation on the field of rational functions on the flag variety G/B, constructed out of a sequence of (translated) Schubert varieties, or equivalently a coordi- nate system on a Bott-Samelson variety. This shows that the string polytopes associated to irreducible representations, can be realized as Newton-Okounkov bodies for the flag variety. This fully generalizes an earlier result of A. Okounkov for the Gelfand-Cetlin polytopes of the symplectic group. As another corollary we deduce a multiplicativity property of the canonical basis due to P. Caldero. We generalize the results to spherical varieties. From these the existence of SAGBI bases for the homogeneous coordinate rings of flag and spher- ical varieties, as well as toric degenerations for them follow.

Algebraic Equations and Convex Bodies

The well-known Bernstein–Kushnirenko theorem from the theory of Newton polyhedra relates algebraic geometry and the theory of mixed volumes. Recently, the authors have found a far-reaching generalization of this theorem to generic systems of algebraic equations on any algebraic variety. In the present note we review these results and their applications to algebraic geometry and convex geometry.

Convex bodies and multiplicities of ideals

We associate convex regions in ℝn to m-primary graded sequences of subspaces, in particular m-primary graded sequences of ideals, in a large class of local algebras (including analytically irreducible local domains). These convex regions encode information about Samuel multiplicities. This is in the spirit of the theory of Gröbner bases and Newton polyhedra on the one hand, and the theory of Newton-Okounkov bodies for linear systems on the other hand. We use this to give a new proof as well as a generalization of a Brunn-Minkowski inequality for multiplicities due to Teissier and Rees-Sharp.

Springer's Weyl Group Representation via Localization

Let

Okounkov Bodies and Applications

G

Mini-Workshop: New Developments in Newton-Okounkov Bodies

denote a reductive algebraic group over

Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory

\mathbb{C}

Integrable systems, toric degenerations and Okounkov bodies

and

Convex bodies and algebraic equations on affine varieties

x

CONVEX BODIES ASSOCIATED TO ACTIONS OF REDUCTIVE GROUPS

a nilpotent element of its Lie algebra