Ottawa-Carleton joint algebra seminar (Winter 2014)

Bounds for the Euclidean minima of number fields and function fields.

Bounds for the Euclidean minima of number fields and function fields.

Abstract: The Euclidean division is a basic tool when dealing with the ordinary integers. It does not extend to rings of integers of algebraic number fields in general. It is natural to ask how to measure the "deviation" from the Euclidean property, and this leads to the notion of Euclidean minimum. The case of totally real number fields is of special interest, in particular because of a conjectured upper bound (conjecture attributed to Minkowski). The talk will present some recent results concerning abelian fields of prime power conductor. We will also define Euclidean minima for function fields and give some bounds for this invariant. We furthermore show that the results are analogous to those obtained in the number field case.
 

Unique factorization of tensor products for Kac-Moody algebras.

Unique factorization of tensor products for Kac-Moody algebras.

Abstract: Suppose V is a finite dimensional representation of a complex finite dimensional simple Lie algebra that can be written as a tensor product of irreducible representations. A theorem of C.S. Rajan states that the irreducible factors that occur in the tensor product factorization of V are uniquely determined, up to reordering, by the isomorphism class of V. I will present an elementary proof of Rajan's theorem, which generalizes with no extra effort to infinite dimensional (Kac-Moody) Lie algebras. This is joint work with S.Viswanath.
 

On the minimal degree of faithful representations of Chevalley groups.

On the minimal degree of faithful representations of Chevalley groups.

Abstract: Motivated by a question in additive combinations, we will study the minimal degree of all faithful representations of Chevalley groups, which extends the previous work of Bourgain and Gamburd. This is a joint work with Camelia Karimianpour, Keivan Mallahi-Karai and Hadi Salmasian.
 

Classification of toric varieties under the action of nonsplit tori.

Classification of toric varieties under the action of nonsplit tori.

Abstract: The classification of toric varieties under the action of a split torus is known since the 70s, due essentially to the work of Demazure. It is stated in term of combinatorial objects called fans. In this talk, we will provide a generalization of that classification to the case of nonsplit tori.

Time and location: 2:30 pm in KED B015
 

Towers of algebras categorify the Heisenberg double.

Towers of algebras categorify the Heisenberg double.

Abstract: A tower of algebras is a graded algebra such that each graded piece is itself an algebra (with a different, 'internal' multiplication). Classic examples include the towers of group algebras of symmetric groups, Hecke algebras of type A, and nilcoxeter algebras. It is known that the Grothendieck groups of towers of algebras satisfying some natural conditions are Hopf algebras, with the product and coproduct coming from induction and restriction functors. We will discuss how certain induction and restriction functors on the category of modules over a tower of algebras categorify the so-called Heisenberg double of the Hopf algebra associated to that tower. In addition, we prove a Stone—von Neumann type theorem in this general setting. As special cases of our categorification theorem, we recover results of Geissinger, Zelevinsky, and others (for the case of symmetric groups, where the Heisenberg double is the infinite rank Heisenberg algebra) and Khovanov (for the case of nilcoxeter algebras, where the Heisenberg double is the Weyl algebra). For the tower of 0-Hecke algebras, we obtain an algebra that we call the quasi-Heisenberg algebra. As an application of our Stone—von Neumann type theorem in this case, we obtain a new, representation theoretic, proof of the fact that the algebra of quasisymmetric functions is free as a module over the algebra of symmetric functions. This is joint work with Oded Yacobi.
 

PBW filtration and Kirillov-Reshetikhin crystals.

PBW filtration and Kirillov-Reshetikhin crystals.

Abstract: For a simple Lie algebra we study the PBW filtration on the highest weight representation $V(\lambda)$ and present the Feigin-Fourier-Littelmann polytope parameterizing a basis of the associated graded module (for types $A$ and $C$). We define an affine crystal sturcture on the $A$-polytope (via the promotion operator) and show that this leads to a realization of the Kirillov-Reshetikhin crystal. It turns out that the affine operators are remarkably simple as well as the combinatorial $R$-matrix and the energy function.
 

Finite multiplicity theorem for spherical pairs.

Finite multiplicity theorem for spherical pairs.

Abstract: Let $(G, H)$ be a spherical pair over a local field $k$ and let $\pi$ be an admissible representation of $G$. Kobayashi and Oshima, and independently Kroetz and Schlichtkrull have recently shown that, if $k=\mathbb{R}$, then the space of $H$-invariant functionals on $\pi$ is finite-dimensional. Both approaches use Casselman's theorem which says that $\pi$ can be presented as a quotient of a principal series representation. We will consider another approach that does not use this theorem. An important step in our proof is to show that the singular support of any $H$-spherical character is a Lagrangian in the cotangent bundle of $G$. In the future, we hope to generalize our proof to the $p$-adic case, where Casselman's theorem does not hold, and the finite multiplicity theorem is known only for certain kinds of spherical pairs (due to Delorme and to Sakellaridis-Venkatesh). The talk is based on an ongoing work with A. Aizenbud and D. Gourevitch. This talk will be held in KED B015 at 4:00pm.
 

Simple singularities and cluster algebras.

Simple singularities and cluster algebras.

Abstract: The aim of this talk is to present some of the recently found connections between isolated singularities and cluster algebras of finite type. This should not come as a surprise since both are classified by Dynkin diagrams, but King, Jensen, Su and others found beautiful ways to construct the cluster category from the ring describing the singularity in some cases.