My research concerns complex reductive groups and their representations. I am interested in geometric and algebraic approaches to this study, in particular, the geometric Satake correspondence. Here are my papers, grouped by project.
Yangians and slices in the affine Grassmannian
Ben Webster, Alex Weekes, Oded Yacobi and I studied slices to spherical Schubert varieties in the affine Grassmannian. These slices carry natural Poisson structures and we gave conjectural quantizations of them using quotients of shifted Yangians. We conjectured that these slices are the symplectic duals to Nakajima quiver varieties. This conjecture was partially confirmed in an appendix that we wrote to a paper of Braverman-Finkelberg-Nakajima. In a followup paper, we gave a conjectural description of the set of highest weights for truncated shifted Yangians. Then, in two papers, we studied the natural moduli description of these slices and proved that they are reduced in type A. Finally, in a joint paper with Finkelberg, Pham, Rybnikov, and Weekes, we defined comultiplication for shifted Yangians.
Yangians and quantizations of slices in the affine Grassmannian, with B. Webster, A. Weekes and O. Yacobi, Alg. Num. Theory 8 (2014) 857-893.
Highest weights for truncated shifted Yangians and product monomial crystals, with P. Tingley, B. Webster, A. Weekes, and O. Yacobi.
Appendix to 3d N=4 quiver gauge theories and slices in the affine Grassmannian, with A. Braverman, M. Finkelberg, R. Kodera, H. Nakajima, B. Webster, and A. Weekes.
On a reducedness conjecture for spherical Schubert varieties and slices in the affine Grassmannian, with D. Muthiah and A. Weekes, to appear in Transformation Groups.
Reducedness of affine Grassmannian slices in type A, with D. Muthiah, A. Weekes, and O. Yacobi, to appear in Proceedings AMS.
Comultiplication for shifted Yangians and quantum open Toda lattice, with M. Finkelberg, K. Pham, L. Rybnikov, and A. Weekes, to appear in Advances Math.
Webs, buildings, components of Satake fibres, and skew Howe duality
Webs are planar graphs which give invariant vectors inside tensor products of representations. They are used to combinatorial realize the representation category of a semisimple (this realization is called a spider). Bruce Fontaine, Greg Kuperberg, and I developed a link between the theory of webs and components of Satake fibres, both of which can be used to describe invariant vectors in tensor products. Separately, Sabin Cautis, Scott Morrison and I studied webs using skew Howe duality. In particular, we gave an explicit presentation for the SL(n) spider. Later, Bruce and I studied rotation of components of Satake fibres and gave a geometric representation theory perspective on the cyclic sieving phenomenon. I also wrote a paper explaining how these components of Satake fibres could be used to give a geometric definition of the category of crystals, thus establishing a combinatorial geometric Satake equivalence. Finally, Cautis and I applied these ideas to give a version of quantum geometric Satake (for SL(n)) based on convolution algebras in equivariant K-theory.
Buildings, spiders, and geometric Satake, with B. Fontaine and G. Kuperberg, Compositio Math. 149 no. 11 (2013), 1871-1912.
Webs and quantum skew Howe duality, with S. Cautis and S. Morrison, Math. Ann. 360 no.1 (2014), 351-390.
Cyclic sieving, rotation, and geometric representation theory with B. Fontaine, Selecta Math.20 no. 2 (2014) 609-625.
A combinatorial geometric Satake equivalence, Advances Math. 300 (2016) 5-16.
Quantum K-theoretic geometric Satake, with S. Cautis, to appear in Compositio Math.
Components of quiver varieties and (affine) MV polytopes
Pierre Baumann and I applied the theory of MV polytopes to the study of components of Lusztig's nilpotent varieties. Along the way, we introduced reflection functors for modules over the non-deformed preprojective algebra of a quiver. The type A situation was described more explicitly in a paper with Chandrika Sadanand. After this, Pierre Baumann, Peter Tingley, and I generalized to the case of affine Lie algebras. This allowed us to define MV polytopes for simply-laced affine Lie algebras. In particular, we proved that they are determined by rank 2 conditions. Later, we (joint with Tom Dunlap) gave an explicit description of the rank 2 affine MV polytopes. Also, Pierre, Stephane Gaussent and I used this theory to prove a relationship between Lusztig data and reverse string data for any symmetric Kac-Moody algebra.
Preprojective algebras and MV polytopes, with P. Baumann, Represent. Theory 16 (2012), 152-188.
Modules with 1-dimensional socle and compenents of Lusztig quiver varieties in type A, with C. Sadanand, Combinatorial Aspects of Commutative Algebra and Algebraic Geometry, Abel Symposia, 2011.
Affine Mirkovic-Vilonen polytopes, with P. Baumann and P. Tingley, Publ. IHES 120 no. 1 (2014), 113-205.
Rank 2 affine MV polytopes, with P. Baumann, T. Dunlap and P.Tingley, Represent. theory 17 (2013) 442-468.
Reflexions dans un cristal, with P. Baumann and S. Gaussent, C. R. Math. Acad. Sci. Paris 350 (2012), no. 23-24, 999--002.
Geometric categorical g actions and equivalences of derived categories of coherent sheaves
Sabin Cautis, Anthony Licata, and I have written a series of papers applying the theory of categorical g actions to the construction of equivalences between derived categories of coherent sheaves. We developed a notion of "geometric categorical sl(2) action" and proved that such an action gives a strong categorical sl(2) action (which is almost the same as a 2-representation in the sense of Rouquier). We then extended a result of Chuang-Rouquier to prove that a strong categorical sl(2) action gives an equivalence of opposite weight categories. We applied these results to two related situations: cotangent bundles to Grassmannians and convolutions of smooth affine Schubert varieties for PSL(m). Later Cautis and I wrote a paper about geometric categorical g actions for any Kac-Moody Lie algebra g. We applied this to cotangent bundles of n-step flag varieties. Then, Cautis, Licata, and I defined categorical g action using Nakajima quiver varieties.
Categorical geometric skew Howe duality, with S. Cautis and A. Licata, Inventiones Mathematicae, 180, no. 1 (2010), 111-159.
Coherent sheaves and categorical sl(2) actions, with S. Cautis and A. Licata, Duke Math. J. 154, no. 1 (2010), 135-179.
Derived equivalences for cotangent bundles to Grasssmannians via categorical sl(2) actions with S. Cautis and A. Licata, to appear in J. Reine Angew. Math.
Braid groups and geometric categorical g actions, Compositio Math., 148 no. 2 (2012), 464-506.
Coherent Sheaves on Quiver Varieties and Categorification, with S. Cautis and A. Licata, Math. Ann. 357 no. 3 (2013), 805-854.
Knot homology via derived categories of coherent sheaves
Mikhail Khovanov has begun categorifying quantum knot invariants, such as the Jones polynomial. Sabin Cautis and I have proposed a program to accomplish this categorification using geometric representation theory. Specifically we study derived categories of coherent sheaves on varieties arising in the geometric Langlands programs. In our first paper, we study the case of sl(2) and recover the original Khovanov homology. In this case, our construction is related to that of Seidel-Smith by hyperKahler rotation. Our second paper deals with the case of sl(m) and the standard and dual representations. We conjecture that this recovers Khovanov-Rozansky homology. I also wrote a shorter paper advertising how one can develop as symplectic version of our theory generalizing the work of Seidel-Smith and Manolescu. Later, Cautis and I gave a description of an extension to this theory where we use coloured links; geometrically this involves dealing with the Beilinson-Drinfeld Grassmannian.
Knot homology via derived categories of coherent sheaves I, sl(2) case with S. Cautis, Duke Math. J. 142 no. 3 (2008), 511-588.
Knot homology via derived categories of coherent sheaves II, sl(m) case with S. Cautis, Inventiones Mathematicae, 174, no. 1 (2008), 165-232.
The Beilinson-Drinfeld Grassmannian and symplectic knot homology. Grassmannians, Moduli spaces and vector bundles, Clay Mathematics Proceedings, 2011
Mirkovic-Vilonen cycles and polytopes
By the geometric Satake Isomorphism, the Mirkovic-Vilonen cycles (certain subvarieties of the affine Grassmannian) give a basis for representations of complex reductive groups. In his thesis, Jared Anderson introduced MV polytopes and explained how they could be used to get some combinatorial objects out of MV cycles. In my thesis, I gave an explicit description of MV cycles and polytopes. This description used some combinatorics which had already been developed by Berenstein-Zelevinsky to describe Lusztig's canonical basis. In particular, this gives a combinatorial link between MV cycles and the canonical basis. I also studied the crystal structure on MV polytopes proving that the crystal operators coming from MV cycles (as defined by Braverman-Gaitsgory) and the crystal operators coming from the canonical basis coincide. Finally, I used this theory to construct a natural bijection between components of fibres of the convolution morphism for the affine Grassmannian of GL_n and combinatorial objects known as hives -- both objects count GL_n tensor product multiplicities.
Mirkovic-Vilonen cycles and polytopes, Annals of Mathematics, 171, (2010) no. 1, 245-294.
The crystal structure on the set of Mirkovic-Vilonen polytopes, Adv. Math. 215 (2007), 66-93.
Hives and the fibres of the convolution morphism, Selecta Math. N.S. 13 no. 3 (2007), 483-496.
Crystals, coboundary categories, and the topology of the moduli space of real curves
Andre Henriques and I began by studying the relationship between the octahedron recurrence and the category of gl(n) crystals. That lead us to defining a commuter for the category of crystals for an arbitrary reductive Lie algebra. We found that this category is a coboundary category and that the cactus group acts on tensor products in coboundary categories. The classifying space for the cactus group is the moduli space of marked genus 0 real curves and so we were lead to studying the topology of this space with Pavel Etingof and Eric Rains. More recently, Peter Tingley and I have picked up the study of the commutor. First, we showed that the commutor admits an alternate definition using Kashiwara's involution. Later, we proved that the crystal commutor arising as q=0 limit of Drinfeld's unitarized R-matrix.
Crystals and coboundary categories with A. Henriques, Duke Math. J. 132 (2006), 191-216.
The octaheron recurrence and gl(n) crystals with A. Henriques, Adv. Math. 206 (2006), 211-249.
The cohomology ring of the moduli space of stable curves of genus 0 with marked points with P. Etingof, A. Henriques, and E. Rains, Annals of Mathematics, 171, no. 2, (2010) 731-777.
A definition of the crystal commutor using Kashiwara's involution with P. Tingley, J. of Alg. Comb., 29 no. 2 (2009), 261-268.
The crystal commutor and Drinfeld's unitarized R-matrix with P. Tingley, J. of Alg. Comb., 29, no. 3 (2009), 315-335