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.

__Teaching__

Abstract Algebra
(Spring 2014)

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 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 gl_{n} 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 2w_{k}.

*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 A_{2}_{l}^{(}^{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.