David Hill
Office Hours:
Tuesday/Thursday 12-1:30 or by appointment.
I am a postdoctoral fellow in the Research Training Group (RTG) in the Interactions of Representation theory, Geometry and Combinatorics at the Department of Mathematics, University of California, Berkeley. I received my PhD at the University of Oregon under the supervision of Alexander Kleshchev.
Office: 785 Evans Hall
Email: dhill1@math.berkeley.edu
Phone: 415-642-6923
Research
My research interests are in representation theory of Lie algebras, finite groups and related objects such as quantum groups and Hecke algebras. I am also interested in the applications of representation theory to categorification.
Here’s my CV and Research Statement.
Here are some slides: Overview of Categorification of Quantum Groups; Representation Theory of Quiver Hecke Algebras via Lyndon Bases.
Teaching
Publications/Preprints
Elementary Divisors of the Shapovalov Form on the Basic Representation of Kac-Moody Algebras, J. Algebra 319 (2008) 5208-5246. pdf.
The integral form of the basic representation of an (untwisted) affine Kac-Moody algebra of type A encodes information about the representation theory of symmetric groups and Iwahori-Hecke algebras in positive quantum characteristic, l. In particular, weight spaces of the basic representation correspond to blocks of the algebras and the Shapovalov form corresponds to the Cartan pairing between projective modules. We calculate the invariant factors of the Gram matrix of the Shapovalov form when each prime factor p of l occurs with multiplicity at most p.
A note on Weyl modules gl∞ and a∞, Communications in Algebra, Volume 36 Issue 12 (2008), 4375-4385. pdf.
To each dominant integral weight for the Kac-Moody algebra gl∞ one may associate a finite dimensional cyclotomic quotient of the affine Hecke algebra of type A with quantum characteristic 0. The associated irreducible highest weight module for gl∞ encodes the representation theory of the cyclotomic Hecke algebra. In particular, weight spaces of this representation correspond to blocks of the algebra and the Shapovalov form corresponds to the Cartan pairing between projective modules. In this paper, we explain how to extend Jantzen’s result on the determinant of the Shapovalov form on irreducible modules for gln to gl∞.
Cartan Invariants of Symmetric Groups and Iwahori-Hecke Algebras (w/ C. Bessenrodt), J. London Math. Soc., to appear. pdf.
Kulshammer, Olsson and Robinson (KOR) showed that many of the invariants of the usual block theory for symmetric groups in characteristic p are independent of p being a prime. Using character theoretic methods, they developed a theory of l-blocks of symmetric groups and conjectured that a certain set of numbers determined the invariant factors of the corresponding l-Cartan matrix. By a work of Donkin, these numbers agree with those for the Iwahori-Hecke algebra with parameter q an lth root of unity. In this paper, we build evidence for the conjecture in my first paper by showing that the invariant factors predicted there give the correct determinant, and that they agree with the numbers conjectured by KOR. In particular, my conjecture is a refinement of the KOR conjecture to blocks, and the conjecture is true provided each prime factor p of l occurs with multiplicity at most p.
Degenerate Affine Hecke-Clifford Algebras and Type Q Lie Superalgebras (w/ J. Kujawa and J. Sussan), preprint. pdf.
We construct the finite dimensional simple integral modules for the (degenerate) affine Hecke-Clifford algebra (AHCA). Our construction includes an analogue of Zelevinsky's segment representations, a complete combinatorial description of the simple calibrated modules, and a classification of the simple integral modules. Additionally, we construct an analogue of the Arakawa-Suzuki functor for the Lie superalgebra of type Q.
The Khovanov-Lauda 2-category and Categorifications of a Level Two quantum sl(n) Representation (w/ J. Sussan), pdf.
Representations of Quiver Hecke Algebras via Lyndon Bases (w/ G. Melvin and D. Mondragon), in preparation.