Final Report Summary - LIC (Loop models, integrability and combinatorics)
This project has shed some new light on the algebraic aspects (combinatorics, representation theory, algebraic geometry) of quantum integrable models / exactly solvable two-dimensional statistical mechanics models. The main directions of progress that have been accomplished during the 5 years of LIC are:
- Quantum Knizhnik-Zamolodchikov (qKZ) equation and combinatorics.
The quantum Knizhnik-Zamolodchikov equation is an important difference equation arising in various areas of mathematics physics and algebra. We have explored various special solutions of the qKZ equation and their combinatorial properties.
In particular, we have found a new and remarkably simple new expression for higher-spin polynomial solutions of the quantum Knizhnik-Zamolodchikov equation. This is interesting because it has many connections to recent results in the study of combinatorial properties of quantum integrable systems (loop models, supersymmetric spin chains) and because of the appearance of Macdonald polynomials in these expressions.
- Quantum integrable systems and geometry.
This is one of the most novel aspects of this project, and major progress has been made in this area. The general idea that quantum integrable systems can be built out of generalized cohomology theories has spread fast in the mathematical community, following in particular the work of Nekrasov and Shatashvili, and that of Maulik and Okounkov. In the framework of LIC, we have emphasized a particular approach which tries to stay as close to "naive" geometric ideas and find their translation into the language of quantum integrable systems. The qKZ equation again plays a key role, as shown in my work in collaboration with Varchenko et al. Loop models quite naturally appear, and we have described (still slightly conjecturally) a general connection to equivariant K-theory. Initially our work was concerned with the Brauer loop model (and the corresponding geometry, the Brauer loop scheme), though this is a somewhat exotic case; we did investigate it further, as planned, including variations to other boundary conditions.
- Alternating Sign Matrices and Descending Plane Partitions.
This was one of the goals of the project, and it was achieved with a proof of the (30 years old) ASM-DPP conjecture relating the enumeration of Alternating Sign Matrices and Descending Plane Partitions, as well as a completely new extension of it to include an additional statistic.
- Quantized affine algebras and discrete holomorphicity.
In the search for exactly computable quantities using the qKZ approach, a new connection has been found to discrete holomorphicity. In short, we have found that quantum integrable models, via their algebraic structure (quantized affine algebras), naturally provide discretely holomorphic observables (such as the ones defined and used by Smirnov et al). This also explains empirical observations of Cardy et al about "linearization" of the Yang-Baxter equation: via the formalism of nonlocal currents of Bernard and Leclair, we show that this is nothing but Jimbo's method for constructing solutions of the Yang--Baxter equation out of quantum groups.
- Finally, we have succesfully applied the methods of quantum integrable systems to the study of symmetric polynomials with my post-doc M. Wheeler, extending work of my own on the subject. In particular, we have derived new identities satisfied by Hall-Littlewood polynomials and their generalizations to other types, displaying a beautiful connection to the six-vertex model with domain wall boundary conditions and variations, itself connected to interesting combinatorics. Furthermore, we have obtained a new, simpler, Littlewood-Richardson rule for double Grothendieck polynomials (equivalently, we produce a particularly nice combinatorial formula for the structure constants of the equivariant K-theory ring of the Grassmannian).
- Quantum Knizhnik-Zamolodchikov (qKZ) equation and combinatorics.
The quantum Knizhnik-Zamolodchikov equation is an important difference equation arising in various areas of mathematics physics and algebra. We have explored various special solutions of the qKZ equation and their combinatorial properties.
In particular, we have found a new and remarkably simple new expression for higher-spin polynomial solutions of the quantum Knizhnik-Zamolodchikov equation. This is interesting because it has many connections to recent results in the study of combinatorial properties of quantum integrable systems (loop models, supersymmetric spin chains) and because of the appearance of Macdonald polynomials in these expressions.
- Quantum integrable systems and geometry.
This is one of the most novel aspects of this project, and major progress has been made in this area. The general idea that quantum integrable systems can be built out of generalized cohomology theories has spread fast in the mathematical community, following in particular the work of Nekrasov and Shatashvili, and that of Maulik and Okounkov. In the framework of LIC, we have emphasized a particular approach which tries to stay as close to "naive" geometric ideas and find their translation into the language of quantum integrable systems. The qKZ equation again plays a key role, as shown in my work in collaboration with Varchenko et al. Loop models quite naturally appear, and we have described (still slightly conjecturally) a general connection to equivariant K-theory. Initially our work was concerned with the Brauer loop model (and the corresponding geometry, the Brauer loop scheme), though this is a somewhat exotic case; we did investigate it further, as planned, including variations to other boundary conditions.
- Alternating Sign Matrices and Descending Plane Partitions.
This was one of the goals of the project, and it was achieved with a proof of the (30 years old) ASM-DPP conjecture relating the enumeration of Alternating Sign Matrices and Descending Plane Partitions, as well as a completely new extension of it to include an additional statistic.
- Quantized affine algebras and discrete holomorphicity.
In the search for exactly computable quantities using the qKZ approach, a new connection has been found to discrete holomorphicity. In short, we have found that quantum integrable models, via their algebraic structure (quantized affine algebras), naturally provide discretely holomorphic observables (such as the ones defined and used by Smirnov et al). This also explains empirical observations of Cardy et al about "linearization" of the Yang-Baxter equation: via the formalism of nonlocal currents of Bernard and Leclair, we show that this is nothing but Jimbo's method for constructing solutions of the Yang--Baxter equation out of quantum groups.
- Finally, we have succesfully applied the methods of quantum integrable systems to the study of symmetric polynomials with my post-doc M. Wheeler, extending work of my own on the subject. In particular, we have derived new identities satisfied by Hall-Littlewood polynomials and their generalizations to other types, displaying a beautiful connection to the six-vertex model with domain wall boundary conditions and variations, itself connected to interesting combinatorics. Furthermore, we have obtained a new, simpler, Littlewood-Richardson rule for double Grothendieck polynomials (equivalently, we produce a particularly nice combinatorial formula for the structure constants of the equivariant K-theory ring of the Grassmannian).