Abstract
Given a finite-dimensional module V for a finite-dimensional, complex semi-simple Lie algebra \(\mathcal {g}\), and a positive integer m, we construct a family of graded modules for the current algebra \(\mathcal {g}[t]\) indexed by simple C \(\mathcal {S}_{m}\)-modules. These modules are free of finite rank for the ring of symmetric polynomials and so can be localized to give finite-dimensional graded \(\mathcal {g}[t]\)-modules. We determine the graded characters of these modules and show that these graded characters admit a curious duality.
Similar content being viewed by others
References
Bennett, M., Chari, V.: Tilting modules for the current algebra of a simple Lie algebra, Proceedings of Symposia in Pure Mathematics (86): Recent Developments in Lie Algebras, Groups and Representation Theory 2012, 75–97
Bennett, M., Chari, V.: Character Formulae and Realization of Tilting Modules for \(\mathcal {sl}_{2}[t]\), arXiv:1409.4464
Bennett, M., Chari, V., Greenstein, J., Manning, N.: On homomorphisms between global Weyl modules. Represent. Theory 15, 733–752 (2011)
Chari, V.: On the fermionic formula and the Kirillov-Reshetikhin conjecture. Int. Math. Res. Not. 12, 629–654 (2001)
Chevalley, C.: Invariants of finite groups generated by reflections. Amer. J. Math. 77, 778–782 (1955)
Chari, V.: B Ion BGG reciprocity for the current algebra. Compos. Math. 151, 1265–1287 (2015)
Chari, V., Fourier, G., Khandai, T.: A categorical approach to Weyl modules. Transform. Groups 15(3), 517–549 (2010)
Chari, V., Pressley, A.: Weyl modules for classical and quantum affine algebras Represent. Theory 5, 191–223 (2001)
Fulton, W., Harris, J.: Representation theory, 1991, Graduate Texts in Mathematics. Springer-Verlag, New York
Humphreys, J.E.: Reflection groups and coxeter groups. Cambridge University Press (1990)
Feigin, B.: Multi-dimensional Weyl modules and symmetric functions. Comm. Math. Phys. 251(3), 427–445 (2004)
Kkoroshkin, A.: Highest weight categories and Macdonald Polynomials, arXiv:http://arXiv.org/abs/1312.7053
Kleshchev, A.: Affine highest weight categories and affine quasihereditary algebras. Proc. Lond. Math. Soc. 100(4), 841–882 (2014)
Moura, A.: Restricted limits of minimal affinizations. Pac. J. M. 244, 359–397 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by Vyjayanthi Chari.
Rights and permissions
About this article
Cite this article
Bennet, M., Jenkins, R. On Some Families of Modules for the Current Algebra. Algebr Represent Theor 20, 197–208 (2017). https://doi.org/10.1007/s10468-016-9637-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-016-9637-0