Remarks on fermionic formula

Abstract

Abstract. Fermionic formulae originate in the Bethe ansatz in solvable lattice models. They are specific expressions of some q-polynomials as sums of products of q-binomial coefficients. We consider the fermionic formulae associated with general non-twisted quantum affine algebra Uq(X (1) n ) and discuss several aspects related to representation theories and combinatorics. They include crystal base theory, one dimensional sums, spinon character formulae, Q-system and combinatorial completeness of the string hypothesis for arbitrary Xn.

Proof of the Combinatorial Kirillov-Reshetikhin Conjecture

“…Conjecture 2.13. In the case where V i are all of Kirillov-Reshetikhin-Chari type, the graded multiplicities M λ,{Vi} (q) are equal to the generalized Kostka polynomials or the fermionic sums M λ,n (q) of [17,10]. This conjecture implies 2.11 for these cases, because the polynomials are indepenent of the localization parameters.…”

“…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).…”

“…It was proved in [10] that if the characters {Q α,i } satisfy the Q-system plus a certain asymptotic condition, then the characters of their tensor products have an explicit "fermionic" expression Theorem 2.7 (Theorem 8.1 [10]). Define Q α,i to be the U q (g)-character of the KR-module corresponding to highest weight iω α .…”

“…That is, the N -sum has more terms, some of which are negative. In fact, [10] conjectured that the two sums are equal. We will describe something which we call the HKOTY-conjecture, which is slightly stronger than this.…”

“…We then define generating functions of fermionic sums which have a factorized form in terms of the Q-functions. This allows us to prove an equality of restricted generating functions, and the constant term of this identity is the conjectured M = N identity of [10].…”