Degenerations of algebraic varieties, with applications to combinatorics and representation theory

Cornell University, Ithaca NY

Investigators

Abstract

Knutson's proposal makes use of degeneration of algebraic varieties to study algebraic combinatorics, particularly combinatorial representation theory. Littelmann's path model in representation theory already has such an interpretation: Chirivi gave a degeneration of each flag manifold G/P (plus ample line bundle) to a reduced, seminormal union of toric varieties. Unfortunately, this used properties of Lusztig's canonical basis, so was specific to representation theory applications. Knutson hopes to replace these properties with Samuel-Rees-Nagata degenerations, to achieve similar reducedness results for degenerations of much more general varieties. One specific application is in computing branching rules from a group G to a symmetric subgroup K, using a K-equivariant degeneration of G/P, and Knutson, jointly with Jiang-Hua Liu, states a conjectural rule based on this. Many interesting numbers and polynomials are associated to irreducible algebraic sets, where "irreducible" essentially means "not glued from smaller pieces in a nontrivial way". A circle x^2 + y^2 - 1 = 0 is such an example, as its equation does not factor, and one can associate the degree 2 to this equation. But if we degenerate it to x^2 + y^2 - 0 = 0, then the equation does factor (over the complex numbers), giving the union of two lines x = +/- iy. Those two equations are degree 1, and from the degenerative geometry we obtain the combinatorial result 2 = 1 + 1. Knutson uses this technique to study much more interesting irreducible sets, e.g. the space of all maximal nested chains of subspaces in a vector space; degenerating them to highly reducible unions of simple pieces replaces their geometric complexity with combinatorial complexity, and gives much more interesting formulae for their degrees (and generalizations thereof). In some of the work proposed, he specifically plans to pursue this technique to understand how quantum-mechanical systems with noncompact symmetry (e.g. under special relativity transformations, which allow boosts by speeds up to but not including light-speed) decompose when one only considers their compact symmetries (e.g. under rotation, by angles that are trapped on a circle and cannot run off to infinity). This is the same sort of problem (though the hoped-for results would be wholly complementary) as studied recently by the ATLAS team in the study of the representations of E_8.