Abstract
The rings of symmetric polynomials form an inverse system whose limit, the ring of symmetric functions, is the model for the bosonic Fock space representation of the affine Lie algebra. We show that this limit naturally carries an action of the affine Lie algebra (in the sense of Rouquier), thereby obtaining a family of categorifications of the bosonic Fock space representation.
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Bernstein, J., Frenkel, I., Khovanov, M.: A categorification of the Temperley-Lieb algebra and Schur quotients of U(sl 2) via projective and Zuckerman functors. Selecta Math. 2(5), 199–241 (1999)
Brundan, J., Kleshchev, A.: Modular Littlewood-Richardson coefficients. Math. Z. 232(2), 287–320 (1999)
Brundan, J., Kleshchev, A.: On translation functors for general linear and symmetric groups. Proc. Lond. Math. Soc. (3) 80(1), 75–106 (2000)
Chuang, J., Rouquier, R.: Derived equivalences for symmetric groups and sl 2-categorification. Ann. Math. (2) 167(1), 245–298 (2008)
Friedlander, E.M.: Lectures on the cohomology of finite group schemes. http://www.math.northwestern.edu/~eric/lectures/nantes/nantes-final.pdf Accessed 15 June 2012
Donkin, S.: On Schur algebras and related algebras. II. J. Algebra 111(2), 354–364 (1987)
Friedlander, E.M., Suslin, A.: Cohomology of finite group schemes over a field. Invent. Math 127(2), 209–270 (1997)
Green, J.A.: Polynomial representations of GL n . Second corrected and augmented edition. With an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, Green and M. Schocker. In: Lecture Notes in Mathematics, vol. 830. Springer, Berlin (2007)
Goodman, R., Wallach, N.R.: Symmetry, representations, and invariants. In: Graduate Texts in Mathematics, vol. 255. Springer, Dordrecht (2009)
Hong, J., Touzé, A., Yacobi, O.: Polynomial functors and categorification of Fock space. www.math.toronto.edu/oyacobi/wallachpaper.pdf Accessed 15 June 2012
Hong, J., Yacobi, O.: Polynomial functors and categorifications of Fock space II www.math.toronto.edu/oyacobi/HYII.pdf (submitted)
Jantzen, J.C.: Representations of algebraic groups. Second edition. In: Mathematical Surveys and Monographs, vol. 107. American Mathematical Society, Providence, RI (2003)
Kac, V.: Infinite-Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)
Kleshchev, A.: Linear and projective representations of symmetric groups. In: Cambridge Tracts in Mathematics, vol. 163. Cambridge University Press, Cambridge (2005)
Lascoux, A., Leclerc, B., Thibon, J.: Hecke algebras at roots of unity and crystal bases of quantum affine algebras. Commun. Math. Phys. 181(1), 205–263 (1996)
Macdonald, I.G.: Symmetric functions and Hall polynomials, 2nd. With contributions by A. Zelevinsky. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1995)
Martin, S.: Schur algebras and representation theory. In: Cambridge Tracts in Mathematics, vol. 112. Cambridge University Press, Cambridge (1993)
Misra, K.C., Miwa, T.: Crystal base of the basic representation of \(U_{q}(\hat{\mathfrak{sl}}_n)\). Commun. Math. Phys. 134, 79–88 (1990)
Rouquier, R.: 2-Kac-Moody algebras. arXiv:0812.5023 [math.RT] (preprint)
Shan, P.: Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras. arXiv:0811.4549 [math.RT] (preprint)
Stroppel, C., Webster, B.: Quiver Schur algebras and q-Fock space. arXiv:1110.1115 [math.RA] (preprint)
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Hong, J., Yacobi, O. Polynomial Representations and Categorifications of Fock Space. Algebr Represent Theor 16, 1273–1311 (2013). https://doi.org/10.1007/s10468-012-9356-0
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DOI: https://doi.org/10.1007/s10468-012-9356-0