Combinatorics/Algebra Seminar
Fall 2014 Archive:

December 1, 8

Marcus Bishop: Constructing quiver presentations of the descent algebras of Coxeter groups (cont.).

November 17

Yeansu Kim: Two main conjectures in the Langlands program.

Abstract: In this talk, I will introduce two main conjectures in the Langlands program: the local Langlands correspondence and the Langlands functoriality conjecture. Briefly, the local Langlands correspondence asserts that there exists a "natural" bijection between two different sets of objects: the arithmetic (Galois or Weil-Deligne) side and the analytic (representation-theoretic) side. On each side, we can define the L-functions that corresponds to these objects. The L-functions on the analytic side have been constructed by F. Shahidi as he developed the Langlands-Shahidi method. The L-functions on the arithmetic side are the so-called Artin L-functions. One of the important requirements of "naturality" imposed on the correspondence between the two sides is that the Artin L-function of an object on the arithmetic side should equal the L-function attached to the corresponding object on the analytic side. For example, in the case of GL, the local Langlands correspondence is formulated by the equality of the Rankin-Selberg L-functions (analytic side) and Artin L-functions (arithmetic side) due to Harris-Taylor and Henniart. The Langlands functoriality conjecture describes deep relationships among automorphic representations of different groups. If time permits, I will explain recent results about those two main conjectures in the case of the GSpin groups. (The talk will be accessible to graduate students).

November 3, 10

Marcus Bishop: Constructing quiver presentations of the descent algebras of Coxeter groups.

October 20, 27

Jonathan Axtell: Schur superalgebras and spin polynomial functors.

Abstract: We discuss categories Pol^I_d, Pol^II_d of strict polynomial functors defined on vector superspaces over any field of characteristic not equal 2. These categories are related to polynomial representations of the supergroups GL(m | n), Q(n) respectively. In particular, there is an equivalence between Pol^I_d, Pol^II_d and the category of finite dimensional supermodules over the Schur superalgebras S(m | n, d), Q(n, d) respectively provided m, n ≥ d. We also discuss some aspects of Sergeev duality from the viewpoint of the category Pol^II_d.

October 6, 13

Jonathan Smith: Left quantum quasigroups.

Abstract: Quantum quasigroups and loops are nonassociative versions of Hopf algebras, with a much simpler axiomatization. They include previous attempts at extending Hopf algebra structure to nonassociative situations, in particular the Moufang-Hopf algebras of Benkart et al., and the H-bialgebras of Pérez-Izquierdo. In these talks, we examine their one-sided versions: left quantum quasigroups and loops. An open problem is to establish the relationship between associative left quantum loops and the left Hopf algebras of Taft et al., which only satisfy one of the two conditions for an antipode.

September 22, 29

Anna Romanowska (Warsaw University of Technology): Constructing bisemilattices from lattices and semilattices.

Abstract: A bisemilattice is an algebra with two (possibly different) structures of a semilattice (idempotent commutative semigroup) defined on the same set. The class of bisemilattices contains lattices (defined by the absorption laws:  x + xy = x(x+y) = x), and stammered semilattices (defined by the law:  x+y = xy), where both semilattice operations coincide. In this talk we will be interested in bisemilattices close to lattices, called Birkhoff systems (defined by the "almost absorption" law:  x+xy = x(x+y)). I will discuss some methods of constructing Birkhoff systems from lattices and semilattices.

September 15

John Harding (New Mexico State University): Remarks on topological Boolean algebras.

Abstract: We provide a (nearly) self-contained, order-theoretic proof of a classic result about topological Boolean algebras. This result is usually proved using high-powered tools that obscure the reason why it is true, while the direct proof points to its underlying combinatorial nature. In this work, a combinatorial conjecture arises. Suggestions on this conjecture would be most welcome.

September 8

Jolie Roat: On 8p-dimensional Hopf algebras with the Chevalley property.

Abstract: The classification of 24-dimensional Hopf algebras remains predominantly open. In particular, little is known about those Hopf algebras with the Chevalley property. In this talk, we will classify the non-semisimple Hopf algebras of dimension 24 with the Chevalley property. In particular, we find such a Hopf algebra must either be pointed, or have a coradical of dimension 12. Further, the 12-dimensional coradical can only be isomorphic to the dual of the dihedral group algebra, the dual of the dicyclic group algebra, or the self-dual, non-commutative semisimple Hopf algebra  A₊  of dimension 12 determined by  G(A₊) = Z₂ x Z₂. These results can be extended to Hopf algebras of dimension 8p with the Chevalley property.

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