The Kirillov-Reshetikhin conjecture and solutions of T-systems

Proof of the Combinatorial Kirillov-Reshetikhin Conjecture

“…Therefore, if the conjecture 2.8 is true, then combined with Theorem 2.7 and the result of [13], it implies the completeness conjecture 2.4 of Kirillov and Reshetikhin.…”

“…The proof in each of these cases involves a bijection between combinatorial objects known as "rigged configurations" and crystal paths. [17] also introduced another, closely related conjecture, recently proven for all simple Lie algebras g [20,13] and some generalizations [12] (more precisely, the conjecture is concerned with finite-dimensional modules of U q ( g) or Y (g)).…”

“…Theorem 2.5 ( [17,20,13]). The characters Q α,i of the Kirillov-Reshetikhin modules of U q ( g) for any simple Lie algebra g satisfy the so-called Q-system (Equation (2.5) below).…”

“…The characters Q α,i of the Kirillov-Reshetikhin modules of U q ( g) for any simple Lie algebra g satisfy the so-called Q-system (Equation (2.5) below). In addition, they satisfy [13] the asymptotic conditions of [10] (condition C of Theorem 7.1 [10]), so that their decomposition into irreducible U q (g)-modules is given by Equation (2.9).…”

Type $${{\varvec{D}}}_{{\varvec{n}}}^\mathbf{(1)}$$ rigged configuration bijection

“…In this case, X turns out to be a branching number of KR modules with respect to the quantized enveloping algebra corresponding to the underlying finitedimensional simple Lie algebra. The relations between characters were shown to be Q-systems [Nak03,Her06]. They are known to imply the fermionic formula M at q = 1 in a weak sense, meaning that it may contain the binomial coefficient p+m m with p < 0 [HKO + 99,KNT02].…”