Combinatorics of \(q\)-characters of finite-dimensional representations of quantum affine algebras.

*(English)*Zbl 1051.17013Let \(\mathfrak g\) be a Kac-Moody Lie algebra and let \(U_q\) be the corresponding Drinfel’d-Jimbo quantized enveloping algebra. If \(\mathfrak g\) is an affine Lie algebra then \(U_q\) is known as a quantum affine algebra.

The \(q\)-character homomorphism can be regarded as a \(q\)-analogue of the ordinary character homomorphism. It is an injective homomorphism from the representation ring of \(U_q\) to the ring of Laurent polynomials in infinitely many variables. In this paper the authors prove two of the conjectures from E. Frenkel and N. Reshetikhin [Contemp. Math. 248, 163–205 (2000; Zbl 0973.17015)] concerning the \(q\)-characters of quantum affine algebras. The first one is the \(q\)-analogue of the statement that the character of any irreducible \(U_q\)-module \(V\) of highest weight \(\lambda\) equals the sum of terms which correspond to weights of form \(\lambda\) minus a nonnegative linear combination of the simple roots. The second gives an explicit description of the image of the \(q\)-character homomorphism, generalizing the well-known fact that the image of the usual character homomorphism is the subring of the Laurent polynomial ring invariant under the Weyl group action.

The authors also consider the consequences of these results. In particular, they are able to show that the tensor product of fundamental representations is reducible if and only if at least one of the pairwise normalized \(R\)-matrices has a pole. The authors also state that in the course of writing the paper they were informed by H. Nakajima that he obtained an independent proof of Conjecture 1 from the paper of Frenkel and Reshetikhin in the ADE case using a geometric approach.

The \(q\)-character homomorphism can be regarded as a \(q\)-analogue of the ordinary character homomorphism. It is an injective homomorphism from the representation ring of \(U_q\) to the ring of Laurent polynomials in infinitely many variables. In this paper the authors prove two of the conjectures from E. Frenkel and N. Reshetikhin [Contemp. Math. 248, 163–205 (2000; Zbl 0973.17015)] concerning the \(q\)-characters of quantum affine algebras. The first one is the \(q\)-analogue of the statement that the character of any irreducible \(U_q\)-module \(V\) of highest weight \(\lambda\) equals the sum of terms which correspond to weights of form \(\lambda\) minus a nonnegative linear combination of the simple roots. The second gives an explicit description of the image of the \(q\)-character homomorphism, generalizing the well-known fact that the image of the usual character homomorphism is the subring of the Laurent polynomial ring invariant under the Weyl group action.

The authors also consider the consequences of these results. In particular, they are able to show that the tensor product of fundamental representations is reducible if and only if at least one of the pairwise normalized \(R\)-matrices has a pole. The authors also state that in the course of writing the paper they were informed by H. Nakajima that he obtained an independent proof of Conjecture 1 from the paper of Frenkel and Reshetikhin in the ADE case using a geometric approach.

Reviewer: Robert Marsh (Leicester)

##### MSC:

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |