Research - Anne Schilling

Research Interests - Some of my papers

Research Interests

My research interests range from algebraic combinatorics, representation theory to mathematical physics. I have been interested in combinatorial properties of quantum algebras. Quantum algebras have their roots in two dimensional solvable lattice models in statistical mechanics. Kashiwara showed that at zero temperature the quantum algebras exhibit beautiful combinatorial properties. Mathematically these are formulated in the crystal base theory. Particularly nice crystals are Kirillov-Reshetikhin crystals; they are affine finite-dimensional crystals that can be used in the Kyoto path model to construct highest weight affine crystals. They are also in close relation to Demazure crystals as shown in work with Fourier and Shimozono. Together with Okado and Fourier, I recently showed that all Kirillov-Reshetikhin crystals of nonexceptional types exist and constructed them combinatorially exploiting certain automorphisms of the affine Dynkin diagrams. With Kirillov and Shimozono I have established a relation between crystal base theory and other combinatorial objects, called rigged configurations, which also arise from physics. A generalization of this result provides new fermionic formulas of the physical partition sums which, as follows by work of the Stony Brook group, encapture the statistics of the particle in the physics model (actually these statistics are similar to those found in the fractional quantum Hall effect; see here). In terms of q-series these yield generalizations of the famous Rogers-Ramanujan identities.

I am also part of the Focused Research Group on Affine Schubert Calculus: Combinatorial, geometric, physical, and computational aspects funded by the NSF. More information can be found at the FRG website.

I am especially fond of understanding relations between different areas in mathematics and mathematical physics. Often this helps to unravel hard problems in one area by exploiting the techniques of another area. I have also contributed to the open-source computer package Sage through Sage-Combinat by implementing crystal bases and Kirillov-Reshetikhin crystals (I previously contributed to MuPAD-Combinat). If you want to learn more on how to use crystals in Sage, there is a thematic tutorial on Lie methods and related combinatorics written in collaboration with Dan Bump from Stanford.

(Parabolic) quantum Bruhat graphs for A2 and A3 with J={1}

Some of my papers

Google Scholar
Comments and questions are welcome:
Back to main homepage