Abstract
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.
Mathematics Subject Classifications (2000): Primary 17B37; Secondary 14D21, 14L30, 16G20
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
V.Chari and A.Pressley, Fundamental representations of Yangians and singularities of R-matrices, J. reine ange. Math. 417 (1991), 87–128
V.Chari and A.Pressley, Quantum affine algebras and their representations, Representations of groups (Banff, AB, 1994), Amer. Math. Soc., Providence, RI, 1995, pp.59–78
V.G.Drinfel’d, A new realization of Yangians and quantized affine algebras, Soviet math. Dokl. 32 (1988), 212–216
E.Frenkel and E.Mukhin, Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras, Comm. Math. Phys. 216 (2001), 23–57
E.Frenkel and N.Reshetikhin, The q-characters of representations of quantum affine algebras and deformations of \(\mathcal{W}\) -algebras, Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998), Contemp. Math., 248, Amer. Math. Soc., Providence, RI, 1999, pp.163–205
G.Hatayama, A.Kuniba, M.Okado, T.Takagi and Y.Yamada, Remarks on fermionic formula, Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998), Contemp. Math., 248, Amer. Math. Soc., Providence, RI, 1999, pp.243–291
D.Hernandez, The Kirillov–Reshetikhin conjecture and solutions of T-systems, J. Reine Angew. Math. 596 (2006), 63–87
D.Hernandez and H.Nakajima, Level 0 monomial crystals, Nagoya Math. J. 184 (2006), 85–153
S.J.Kang, J.A.Kim and D.U.Shin, Crystal bases of quantum classical algebras and Nakajima’s monomials, Publ. RIMS, Kyoto Univ. 40, 757–791 (2004)
A.N.Kirillov and N.Reshetikhin, Representation of Yangians and multiplicity of occurrence of the irreducible components of the tensor product of representations of simple Lie algebras, J. Sov. Math. 52 (1990), 3156–3164
M.Kleber, Combinatorial structure of finite-dimensional representations of Yangians: the simply-laced case., Internat. Math. Res. Notices 4 (1997), 187–201
H.Knight, Spectra of tensor products of finite-dimensional representations of Yangians, J. Algebra 174 (1995)(1), 187–196
G.Lusztig, Fermionic form and Betti numbers, preprint, arXiv:math.QA/0005010
H.Nakajima, t-analogue of the q-characters of finite dimensional representations of quantum affine algebras, in Physics and Combinatorics, Proceedings of the Nagoya 2000 International Workshop, World Scientific, 2001, 195–218
H.Nakajima, t-analogs of q-characters of Kirillov–Reshetikhin modules of quantum affine algebras, Represent. Theory (elect.) 7 (2003), 259–274
H.Nakajima, t-analogs of q-characters of quantum affine algebras of type A n , D n, in Combinatorial and geometric representation theory (Seoul, 2001), 141–160, Contemp. Math., 325, Amer. Math. Soc., Providence, RI, 2003
H.Nakajima, Quiver varieties and t-analogs of q-characters of quantum affine algebras, Ann. of Math. 160 (2004), 1057–1097
M.Varagnolo and E.Vasserot, Perverse sheaves and quantum Grothendieck rings, Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000), 345–365, Progr. Math., 210, Birkhäuser Boston, Boston, MA, 2003
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Nakajima, H. (2010). t-Analogs of q-Characters of Quantum Affine Algebras of Type E6, E7, E8. In: Gyoja, A., Nakajima, H., Shinoda, Ki., Shoji, T., Tanisaki, T. (eds) Representation Theory of Algebraic Groups and Quantum Groups. Progress in Mathematics, vol 284. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4697-4_10
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4697-4_10
Published:
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4696-7
Online ISBN: 978-0-8176-4697-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)