Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165



Symmetric polynomials and $U_q(\widehat{sl}_2)$

Author: Naihuan Jing
Journal: Represent. Theory 4 (2000), 46-63
MSC (2000): Primary 17B; Secondary 5E
Published electronically: February 7, 2000
MathSciNet review: 1740180
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information


We study the explicit formula of Lusztig's integral forms of the level one quantum affine algebra $U_q(\widehat{sl}_2)$ in the endomorphism ring of symmetric functions in infinitely many variables tensored with the group algebra of $\mathbb Z$. Schur functions are realized as certain orthonormal basis vectors in the vertex representation associated to the standard Heisenberg algebra. In this picture the Littlewood-Richardson rule is expressed by integral formulas, and is used to define the action of Lusztig's $\mathbb Z[q, q^{-1}]$-form of $U_q(\widehat{sl}_2)$ on Schur polynomials. As a result the $\mathbb Z[q, q^{-1}]$-lattice of Schur functions tensored with the group algebra contains Lusztig's integral lattice.

References [Enhancements On Off] (What's this?)

  • [BCP] J. Beck, V. Chari and A. Pressley, An algebraic characterization of the affine canonical basis, Duke Math. J. 99 (1999), 455-487. CMP 2000:01
  • [BFJ] J. Beck, I. B. Frenkel and N. Jing, Canonical basis and Macdonald polynomials, Adv. in Math. 140 (1998), 95-127. CMP 99:05
  • [CJ] V. Chari and N. Jing, Realization of level one representations of $U_q(\hat{\mathfrak g})$ at a root of unity, math.QA/9909118.
  • [CT] C. Carré and J.-Y. Thibon, Plethysm and vertex operators, Adv. in Appl. Math. 13 (1992), 390-403. MR 94c:05070
  • [CP] V. Chari and A. Pressley, Quantum affine algebras at roots of unity, Representation Theory 1 (1997), 280-328. MR 98e:17018
  • [DJKM] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, Transformation groups for soliton equations. Nonlinear integrable systems--classical theory and quantum theory (Kyoto, 1981), pp. 39-119, World Sci. Publishing, Singapore, 1983. MR 86a:58093
  • [F1] I. B. Frenkel, Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory, J. Funct. Anal. 44 (1981), 259-327. MR 83b:17012
  • [F2] I. B. Frenkel, Lectures at Yale University, 1986.
  • [FJ] I. B. Frenkel and N. Jing, Vertex representations of quantum affine algebras, Proc. Natl. Acad. Sci. USA 85 (1988), 9373-9377. MR 90e:17028
  • [G] H. Garland, The arithmetic theory of loop algebras, J. Algebra 53 (1978), 480-551; Erratum: The arithmetic theory of loop algebras, J. Algebra 63 (1980), 285. MR 80a:17012; MR 81d:17008
  • [Ga] A. M. Garsia, Orthogonality of Milne's polynomials and raising operators, Discrete Math. 99 (1992), 247-264. MR 93m:05201
  • [H] T. Hayashi, Q-analogues of Clifford and Weyl algebras-spinor and oscillator representations of quantum enveloping algebras, Commun. Math. Phys. 127 (1990), 129-144. MR 91a:17015
  • [JK] G. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Math. and its Appl. 16, Addison-Wesley, Reading, MA, 1981. MR 83k:20003
  • [J1] N. Jing, Vertex operators, symmetric functions and the spin groups $\Gamma_n$, J. Algebra 138 (1991), 340-398. MR 92e:17033
  • [J2] N. Jing, Vertex operators and Hall-Littlewood functions, Adv. in Math. 87 (1991), 226-248. MR 93c:17039
  • [J3] N. Jing, Vertex operators and generalized symmetric functions, in: Proc. of Conf. on Quantum Topology (KSU, March 1993), ed. D. Yetter, World Scientific, Singapore, 1994, pp. 111-126. MR 96e:17062
  • [J4] N. Jing, Boson-fermion correspondence for Hall-Littlewood polynomials, J. Math. Phys. 36 (1995), 7073-7080. MR 96m:17049
  • [J5] N. Jing, Quantum Kac-Moody algebras and vertex representations, Lett. Math. Phys. 44 (1998), no. 4, 261-271. MR 99j:17043
  • [K] M. Kashiwara, Global crystal bases of quantum groups, Duke Math. J. 73 (1993), 383-413.
  • [Ka] V. G. Kac, Infinite dimensional Lie algebras, 3rd. ed., Cambridge Univ. Press, Cambridge, 1990. MR 92k:17038
  • [LLT] A. Lascoux, B. Leclerc, J.-Y. Thibon, Ribbon tableaux, Hall-Littlewood functions, quantum affine algebras and unipotent varieties, Sém. Lothar. Combin. 34 (1995), 23 pp. MR 98m:05195
  • [LP] J. Lepowsky and P. Primc, Structure of the standard modules for affine Lie algebra $A_1^{(1)}$, Contemp. Math. 46, Amer. Math. Soc., Providence, RI, 1985. MR 87g:17021
  • [L] G. Lusztig, Introduction to quantum groups, Progress in Mathematics 110, Birkhäuser, Boston, 1993. MR 94m:17016
  • [M] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford University Press, New York, 1995. MR 96h:05207
  • [MM] K. C. Misra and T. Miwa, Crystal bases for the basic representations of $U_q(\hat{sl}(n))$, Commun. Math. Phys. 134 (1990), 79-88. MR 91j:17021
  • [Z] A. Zelevinsky, Representations of finite classical groups, A Hopf algebra approach. Lecture Notes in Mathematics, 869, Springer-Verlag, Berlin-New York, 1981. MR 83k:20017

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2000): 17B, 5E

Retrieve articles in all journals with MSC (2000): 17B, 5E

Additional Information

Naihuan Jing
Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205

Keywords: Symmetric functions, vertex operators, quantum affine algebras, Littlewood-Richardson rule
Received by editor(s): February 17, 1999
Received by editor(s) in revised form: December 10, 1999
Published electronically: February 7, 2000
Additional Notes: Research supported in part by NSA grant MDA904-97-1-0062 and Mathematical Sciences Research Institute.
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society