A pentagon of identities, graded tensor products and the Kirillov-Reshetikhin conjecture

This paper provides a brief review of the relations between the Feigin-Loktev conjecture on the dimension of graded tensor products of \g[t]-modules, the Kirillov-Reshetikhin conjecture, the combinatorial ``M=N" conjecture, their proofs for all simple Lie algebras, and a pentagon of identities which results from the proof.Comment: 21 page

Fusion products of Kirillov-Reshetikhin modules and fermionic multiplicity formulas

We give a complete description of the graded multiplicity space which appears in the Feigin-Loktev fusion product [FL99] of graded Kirillov-Reshetikhin modules for all simple Lie algebras. This construction is used to obtain an upper bound formula for the fusion coefficients in these cases. The formula generalizes the case of g=A_r [AKS06], where the multiplicities are generalized Kostka polynomials [SW99,KS02]. In the case of other Lie algebras, the formula is the the fermionic side of the X=M conjecture [HKO+99]. In the cases where the Kirillov-Reshetikhin conjecture, regarding the decomposition formula for tensor products of KR-modules, has been been proven in its original, restricted form, our result provides a proof of the conjectures of Feigin and Loktev regarding the fusion product multiplicites.Comment: 22 pages; v2: minor changes; v3: exposition clarifie

T-systems with boundaries from network solutions

In this paper, we use the network solution of the $A_r$ $T$-system to derive that of the unrestricted $A_\infty$ $T$-system, equivalent to the octahedron relation. We then present a method for implementing various boundary conditions on this system, which consists of picking initial data with suitable symmetries. The corresponding restricted $T$-systems are solved exactly in terms of networks. This gives a simple explanation for phenomena such as the Zamolodchikov periodicity property for $T$-systems (corresponding to the case $A_\ell\times A_r$) and a combinatorial interpretation for the positive Laurent property of the variables of the associated cluster algebra. We also explain the relation between the $T$-system wrapped on a torus and the higher pentagram maps of Gekhtman et al.Comment: 63 pages, 67 figure

Quantum Q systems: From cluster algebras to quantum current algebras

In this paper, we recall our renormalized quantum Q-system associated with representations of the Lie algebra $A_r$, and show that it can be viewed as a quotient of the quantum current algebra $U_q({\mathfrak n}[u,u^{-1}])\subset U_q(\widehat{\mathfrak sl}_2)$ in the Drinfeld presentation. Moreover, we find the interpretation of the conserved quantities in terms of Cartan currents at level 0, and the rest of the current algebra, in a non-standard polarization in terms of generators in the quantum cluster algebra.Comment: 38 pages, 2 figure

Difference equations for graded characters from quantum cluster algebra

We introduce a new set of $q$-difference operators acting as raising operators on a family of symmetric polynomials which are characters of graded tensor products of current algebra ${\mathfrak g}[u]$ KR-modules \cite{FL} for ${\mathfrak g}=A_r$. These operators are generalizations of the Kirillov-Noumi \cite{kinoum} Macdonald raising operators, in the dual $q$-Whittaker limit $t\to\infty$. They form a representation of the quantum $Q$-system of type $A$ \cite{qKR}. This system is a subalgebra of a quantum cluster algebra, and is also a discrete integrable system whose conserved quantities, analogous to the Casimirs of $U_q({\mathfrak sl}_{r+1})$, act as difference operators on the above family of symmetric polynomials. The characters in the special case of products of fundamental modules are class I $q$-Whittaker functions, or characters of level-1 Demazure modules or Weyl modules. The action of the conserved quantities on these characters gives the difference quantum Toda equations \cite{Etingof}. We obtain a generalization of the latter for arbitrary tensor products of KR-modules.Comment: 35 page