Algebra and Combinatorics Seminar

The Voronoi cells of the A_d lattice are interesting geometric objects. For examples, the Voronoi cells of A₂ are regular hexagons, and those of A₃ are rhombic dodecahedrons. The Voronoi cells can be expressed as zonotopes with (d+1) generators. Discrete subgroups can be induced on them by sublattices. These subgroups are abelian and provide a fertile field for doing discrete Fourier analysis akin to the fast Fourier transform done on the unit circle. We will also discuss their dual groups and study the interactions between them. The group structure plays a very important role in the proof of the Bernstein inequality which has application in packing problems.

To decide whether an irreducible complex representation of a finite group that has a real-valued character can actually be realized by real-valued matrices, G. Frobenius and I. Schur introduced a certain indicator that involved a sum over the squares of the group elements. These indicators have been generalized in various stages, from squares to higher powers and from there to objects that depend on two indices, one of which is this higher power. These two-index objects are equivariant with respect to the canonical action of the modular group and are therefore called equivariant Frobenius-Schur indicators. As we explain in the talk, they can be used to show that the kernel of the action of the modular group on the center of the Drinfel'd double of a semisimple Hopf algebra is a congruence subgroup whose level is the exponent of the Hopf algebra. The talk is based on joint work with Yongchang Zhu.

Let A be an algebra over an order R in a number field K. Suppose that the additive R-module of A is free of finite rank. Suppose that we know the following:

• For each maximal ideal p of R, we know r_p, the smallest number of generators of A / pA as an R/p-algebra.
• We know r₀, the smallest number of generators of the K-algebra A ⊗_R K.

We will see how the smallest number of generators of the R-algebra A depends on r₀ and all the numbers r_p. The talk will be based on my joint work with R. Kravchenko and M. Mazur (arXiv:1001.2873).