The Bethe Equation at q = 0, the Möbius Inversion Formula, and Weight Multiplicities. II. The Xn Case

We study a family of power series characterized by a system of recursion relations (Q-system) with a certain convergence property. We show that the coefficients of the series are expressed by the numbers which formally count the off-diagonal solutions of the Uq(X(1)n) Bethe equation at q = 0. The series are conjectured to be the Xn-characters of a certain family of irreducible finite-dimensional Uq(X(1)n)-modules which we call the KR (Kirillov–Reshetikhin) modules. Under the above conjecture, these coefficients give a formula of the weight multiplicities of the tensor products of the KR modules, which is also interpreted as the formal completeness of the XXZ-type Bethe vectors.