ON FRENKEL-MUKHIN ALGORITHM FOR q-CHARACTER OF QUANTUM AFFINE ALGEBRAS

The q-character is a strong tool to study finite-dimensional representations of quantum affine algebras. However, the explicit formula of the q-character of a given representation has not been known so far. Frenkel and Mukhin proposed the iterative algorithm which generates the q-character of a given irreducible representation starting from its highest weight monomial. The algorithm is known to work for various classes of representations. In this note, however, we give an example in which the algorithm fails to generate the q-character. 1. Background 1.1. Finite-dimensional representations of quantum affine algebras. Let g be a simple Lie algebra over C, and let Uq(ĝ) be the untwisted quantum affine algebra of g by Drinfeld and Jimbo [D1, D2, J]. The following are the most basic facts on the finite-dimensional representations of Uq(ĝ), due to Chari-Pressley [CP1, CP2]: (i) The isomorphism classes of the irreducible finite-dimensional representations of Uq(ĝ) are parametrized by an n-tuple of polynomials of constant term 1, P = (Pi(u))i∈I , where I = {1, . . . , n} and n = rank g. The polynomials P are often called the Drinfeld polynomials because an analogous result for Yangian was obtained earlier by Drinfeld [D2]. (ii) For given Drinfeld polynomials P, let V (P) denote the corresponding irreducible representation. For a pair of Drinfeld polynomials P = (Pi(u))i∈I and Q = (Qi(u))i∈I , let PQ := (Pi(u)Qi(u))i∈I . Then, V (PQ) is a subquotient of V (P)⊗ V (Q). (iii) A representation V (P) is called the ith fundamental representation and denoted by Vωi(a) if Pi(u) = 1 − au and Pj(u) = 1 for any j 6= i. Suppose that Drinfeld polynomials P are in the form