
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 WeilDeligne) side and the analytic (representationtheoretic) side. On each side, we can define the Lfunctions that corresponds to these objects. The Lfunctions on the analytic side have been constructed by F. Shahidi as he developed the LanglandsShahidi method. The Lfunctions on the arithmetic side are the socalled Artin Lfunctions. One of the important requirements of "naturality" imposed on the correspondence between the two sides is that the Artin Lfunction of an object on the arithmetic side should equal the Lfunction 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 RankinSelberg Lfunctions (analytic side) and Artin Lfunctions (arithmetic side) due to HarrisTaylor 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 MoufangHopf algebras of Benkart et al., and the Hbialgebras of PérezIzquierdo. In these talks, we examine their onesided 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) selfcontained, ordertheoretic proof of a classic result about topological Boolean algebras. This result is usually proved using highpowered 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 8pdimensional Hopf algebras with the Chevalley property.
Abstract:
The classification of 24dimensional 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 nonsemisimple 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 12dimensional coradical can only be isomorphic to the dual of the dihedral group algebra, the dual of the dicyclic group algebra, or the selfdual, noncommutative semisimple Hopf algebra A_{+} of dimension 12 determined by G(A_{+}) = Z_{2} x Z_{2}. These results can be extended to Hopf algebras of dimension 8p with the Chevalley property.
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