There is a well known correspondence between lattices and codes via the classical “construction A.” With this, the weight enumerator for codes corresponds to the theta series for lattices, where one counts the number of codewords by composition, and the other counts the number of vectors in a lattice of a certain length. In this talk, we will explore how some of the attendant machinery of theta series are born out in this correspondence. In particular, we will consider the Kneser-Hecke operator, a code theoretic analogue of the classical Hecke operator.
Kneser-Hecke operators for codes over finite chain rings
Mar. 01, 2016 1pm (MATH 220)
Julie Linman (CU Boulder) Minimal functions on the random permutation, Part 2
Mar. 01, 2016 2pm (MATH 350)
Lie Theory
Jay Taylor (University of Padova)
X
The classical groups , , , have long been of interest in group theory; often because of their close relationship to finite simple groups. These groups are all examples of finite reductive groups. Such a group is obtained as the fixed point subgroup of a connected reductive algebraic group under a Frobenius endomorphism . Assuming the characteristic of the field defining is not too small (greater than 5 suffices) then Kawanaka has defined a family of complex representations of the finite group called Generalised Gelfand—Graev Representations (GGGRs). Using these representations Kawanaka conjectured that one could associate to every irreducible character of a well-defined -stable unipotent conjugacy class of which he called the wave front set. Under certain assumptions on the underlying characteristic Lusztig proved Kawanaka’s conjecture, asserting the existence of the wave front set. In this talk we will give an introduction to these ideas an present our result that Kawanaka’s conjecture holds whenever the GGGRs are defined.
Generalised Gelfand—Graev Representations and Wave Front Sets Sponsored by the Meyer Fund
A famous question of Dusa McDuff, often referred to as the "McDuff Conjecture", is whether there exists a non-Hamiltonian symplectic circle action with isolated fixed points on a compact symplectic manifold. Susan Tolman recently answered this question in the affirmative, constructing a -dimensional such space with exactly fixed points. A crucial ingredient to this construction involves Hamiltonian circle actions on complex manifolds and orbifolds in which the interaction between the complex structure and the symplectic form is fairly weak. Specifically, versions of Sjamaar's holomorphic slice theorem, the birational equivalence theorem of Guillemin and Sternberg, as well as reduction, cutting, and blow-up (all of which work in the Kaehler world) are required in this weaker setting.
All of these theorems and constructions are extended to this weaker setting in joint work by Tolman and myself. In particular, by weak, we mean that if is the vector field induced by the circle action, the symplectic form, and the complex structure, then on the complement of the fixed point set. This condition is sufficient for all of the theorems and constructions above except for the blow-up (which also requires tameness at the point to be blown-up).
In this talk, I will focus on the holomorphic slice theorem about a fixed point, reduction, and (time-permitting) the birational equivalence theorem proven in the joint paper.
Kneser-Hecke operators for codes over finite chain rings