Research

I look for similarities in different combinatorial/geometric/categorical constructions of representations.

Past projects

Refereed publications

Baumann, P. and Kamnitzer, J. and Knutson, A., The Mirkovic–Vilonen basis and Duistermaat–Heckman measures with an appendix by Dranowski, A., Kamnitzer, J. and Morton-Ferguson, C., Acta (2019) arXiv:1905.08460

Dranowski, A., Generalized orbital varieties for Mirkovic–Vybornov slices as affinizations of Mirkovic–Vilonen cycles, Transformation Groups (2019) arXiv:1905:18174

Burchard, A., Chambers, G. R. and Dranowski, A., Ergodic properties of folding maps on spheres, Discrete and Continuous Dynamical Systems-Series A (2015) arXiv:1509.02454

Ongoing projects

Reverse plane partitions and quivers

Reverse plane partitions are order reversing maps of minuscule posets of type ADE. We show that they form a crystal, and that this crystal is equivalent to that which is formed by a corresponding subvariety of Lusztig's nilpotent variety of modules over the preprojective algebra.

Reverse plane partitions have a number of interesting combinatorial properties, including toggling and RSK. Our bijection allows us to investigate these geometrically. We start by analyzing the effect of toggling on Lusztig's semicanonical basis.

Comparing the MV basis and the KLR basis

This project aims to make precise the relation between Lusztig’s canonical basis and the MV basis by using Hilbert series to study supports of certain modules over the parity KLRW algebra −𝒫 (a data enriched Webster variation on the Khovanov-Lauda-Rouquier algebra, whose simple modules are in crystal isomorphism with Lusztig’s canonical basis).

In KTTWY, the authors showed that the category of modules over −𝒫 and the category 𝒪 of a truncated and shifted Yangian are equivalent. Two consequences of interest to us are:

  1. a simple −𝒫 module L defines a simple module Θ(L) in 𝒪, and
  2. the highest weights for 𝒪 (and associated unique highest weight simple modules) are in bijection with a product monomial crystal ℳ (a certain subcrystal of Nakajima’s monomials introduced and studied by the same authors in their earlier work on highest weights and truncated shifted Yangians).

We are ultimately interested in the support of L because the truncated shifted Yangians arise as deformation quantizations of certain subvarieties of Gr. The associated graded of Θ(L) therefore defines a coherent sheaf on Gr and this sheaf is supported on a union of MV cycles.

Our first goal is to leverage the description of weight spaces of Θ(L) in terms of weight spaces of L provided by KTTWY to better understand this union: which MV cycles appear and with what multiplicity? Does the Hilbert series of Θ(L) recover the equivariant multiplicity of any of the MV cycles appearing in its support?

Polynomial solutions of qKZ via geometric Satake

Solutions to the divided difference version of the classical KZ equation, the so-called qKZ can be constructed using:

  1. quantum conformal blocks over configuration spaces as in Felder & Veselov (2007),
  2. stable envelopes for quiver varieties as in Maulik & Okounkov (2012), and
  3. MV cycles in the affine Grassmannian as in Zinn-Justin (2015).

Broadly, we would like to know to what extent does the oriented cohomology theory we work in govern the representations we construct. We are also interested in the combinatorial structure that can be gleaned by considering the effect of an R-matrix exchange on the dual semicanonical basis.

We start by deriving elliptic multidegrees and checking that they satisfy the elliptic version of the qKZ as predicted by Zinn-Justin (2015).

Convolutions of MV cycles and cluster algebras

Ideals of MV cycles

Derived geometric Satake