Description: Description: Description: Description: IMG_2004

 

David Hill

Office Hours: Monday-Friday 1-3pm.

 

 

 

 

 

 

I am currently a Whyburn instructor at the University of Virginia. Previously, I was a postdoc in the Research Training Group (RTG) in the Interactions of Representation theory, Geometry and Combinatorics at the University of California, Berkeley.  I received my PhD at the University of Oregon under the supervision of Alexander Kleshchev.

 

 

 

Office: 224 Kerchof Hall                                                                            

 

Email: dehill@virginia.edu

 

Phone: 434-924-4924

 

 

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 particularly interested in problems in representation theory related to categorification.

 

CV

Research Statement

Teaching Statement

 

 

 

Teaching

 

Previous Courses.

 

Abstract Algebra (Spring 2014)

 

Linear Algebra (Spring 2014)

 

                        Representation Theory (Fall 2013)

 

                         

Publications

 

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 for 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 glencodes 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., Volume 81 Part 1 (2010), 113-128. 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), Math. Z., 268 (2011), no. 3-4, pp. 1091-1158. 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), special issue `Categorification in Representation Theory’, Int. J. Math. Math. Sci. 2010 Art. Id 892387. Pdf.

We construct 2-functors from a 2-category categorifying quantum sl(n) to 2-categories categorifying the irreducible representation of highest weight 2wk.

 

Representations of Quiver Hecke Algebras via Lyndon Bases (w/ G. Melvin and D. Mondragon), J. Pure Appl. Algebra 216 (2012), pp. 1052-1079. Pdf.

A new class of algebras has been introduced by Khovanov and Lauda and independently by Rouquier. These algebras categorify one-half of the Quantum group associated to arbitrary Cartan data. In this paper, we use the combinatorics of Lyndon words to construct the irreducible representations of those algebras associated to Cartan data of finite type. This completes the classification of simple modules for the quiver Hecke algebra initiated by Kleshchev and Ram.

 

Categorification of Quantum Kac-Moody Superalgebras (w/ W. Wang), preprint. Pdf.

We introduce a non-degenerate bilinear form and use it to provide a new characterization of quantum Kac-Moody superalgebras with no isotropic odd simple roots. We show that the spin quiver Hecke algebras introduced by Kang-Kashiwara-Tsuchioka simultaneously provide a categorification of half the quantum Kac-Moody algebras and superalgebras, using the recent work of Ellis-Khovanov-Lauda. This categorification generalizes (part of) Brundan and Kleshchev’s categorification of the Kac-Moody algebra of type A2l(2) via the representation theory of Hecke-Clifford algebras. A new idea here is that the additional signs that appear in the defining relations of the Kac-Moody superalgbras are categorified as spin (i.e., the parity-shift functor).

 

Quantum Supergroups I: Foundations (w/ S. Clark and W. Wang), Transformation groups, to appear. Pdf.

In this paper, we follow the algebraic constructions of Lusztig  to study representations of quantum supergroups with nonisotropic odd roots. The representation theory of the quantum supergroups defined in this paper is strictly larger than that of quantum Kac-Moody superalgebras appearing in the literature, but there is an obvious full subcategory which recovers the older representation theory.

 

Quantum Supergroups II: Canonical Bases (w/ S. Clark and W. Wang), submitted. Pdf.

This work extend’s the grand loop argument of Kashiwara to the quantum supergroups in part 1. In this way we obtain (necessarily signed) canonical bases of quantum Kac-Moody superalgebras.

 

Quantum Shuffles and Quantum supergroups of Basic Type (w/ S. Clark and W. Wang), arXiv:1310.7523. Pdf.

We study half of the quantum group associated to a simple Lie superalgebra of basic type from the point of view of quantum shuffle algebras. Among other results, we construct a family of PBW bases for the quantum supergroup, one for each total ordering on the set of nodes for the associated Dynkin diagram, and show directly that each such basis is orthogonal with respect to the standard bilinear form. We go on to prove that in types gl(1|n), osp(1|2n), and osp(2|2n), the quantum supergroup admits a (signed) canonical basis.

 

Braid Group Actions on Quantum Kac-Moody Superalgebras (w/ S. Clark) in preparation. Pdf.

We construct a (spin) braid group action on the covering quantum Kac-Moody superalgebras and integrable modules defined in an earlier collaboration of the authors with W. Wang.