The original KR character formula for nonexceptional quantum affine algebras

Let XN(r)A2n(2)X_N^{(r)}\neq A_{2n}^{(2)}, let Δ+g0\Delta_+^{\mathfrak{g}_0} be the positive roots of g0\mathfrak{g}_0, and let Qν(y)\mathcal{Q}^{\nu}(y) and KD,Gν(y)\mathcal{K}^{\nu}_{D,G}(y) be the normalized KR-character product and associated power series. Let Wm(a)(ζm(a))W_m^{(a)}(\zeta_m^{(a)}) be KR modules with arbitrary parameters. Original KR conjecture. The formula

Qν(y)=KD,Gν(y)αΔ+g0(1eα)\mathcal{Q}^{\nu}(y)=\frac{\mathcal{K}^{\nu}_{D,G}(y)}{\displaystyle\prod_{\alpha\in\Delta_+^{\mathfrak{g}_0}}(1-e^{-\alpha})}

should hold; consequently, KD,Gν(y)\mathcal{K}^{\nu}_{D,G}(y) should be a polynomial whose coefficients are the multiplicities of the g0\mathfrak{g}_0-irreducible components of (a,m)HWm(a)(ζm(a))νm(a)\bigotimes_{(a,m)\in H}W_m^{(a)}(\zeta_m^{(a)})^{\otimes\nu_m^{(a)}}. This is the original formulation for the stated nonexceptional case and remains open in the supplied text.

Sources & referencesView supporting material

Primary source

Atsuo Kuniba, Tomoki Nakanishi and Zengo Tsuboi, “The canonical solutions of the Q-systems and the Kirillov-Reshetikhin conjecture”, arXiv:math/0105145 (2002).

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