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We compute t-analogs of q-characters of all l-fundamental representations of the quantum affine algebras of type \({E}_{6}^{(1)}\), \({E}_{7}^{(1)}\), \({E}_{8}^{(1)}\) by a supercomputer. (Here l- stands for the loop.) In particular, we prove the fermionic formula for Kirillov–Reshetikhin modules conjectured by Hatayama et al.[Remarks on fermionic formula (1999)] for these classes of representations. We also give explicitly the monomial realization of the crystal of the corresponding fundamental representations of the quantum enveloping algebras associated with finite dimensional Lie algebras of types E 6, E 7, E 8. These are computations of Betti numbers of graded quiver varieties, quiver varieties and determination of all irreducible components of the lagrangian subvarieties of quiver varieties of types E 6, E 7, E 8, respectively.

t-Analogs of q-Characters of Quantum Affine Algebras of Type E6, E7, E8