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

HTML articles powered by AMS MathViewer

- by Naihuan Jing
- Represent. Theory
**4**(2000), 46-63 - DOI: https://doi.org/10.1090/S1088-4165-00-00065-0
- Published electronically: February 7, 2000
- PDF | Request permission

## Abstract:

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

- J. Beck, V. Chari and A. Pressley, An algebraic characterization of the affine canonical basis, Duke Math. J.
**99**(1999), 455-487. - J. Beck, I. B. Frenkel and N. Jing, Canonical basis and Macdonald polynomials, Adv. in Math.
**140**(1998), 95-127. - V. Chari and N. Jing, Realization of level one representations of $U_q(\hat {\mathfrak g})$ at a root of unity, math.QA/9909118.
- Christophe Carré and Jean-Yves Thibon,
*Plethysm and vertex operators*, Adv. in Appl. Math.**13**(1992), no. 4, 390–403. MR**1190119**, DOI 10.1016/0196-8858(92)90018-R - Vyjayanthi Chari and Andrew Pressley,
*Quantum affine algebras at roots of unity*, Represent. Theory**1**(1997), 280–328. MR**1463925**, DOI 10.1090/S1088-4165-97-00030-7 - Etsur\B{o} Date, Masaki Kashiwara, Michio Jimbo, and Tetsuji Miwa,
*Transformation groups for soliton equations*, Nonlinear integrable systems—classical theory and quantum theory (Kyoto, 1981) World Sci. Publishing, Singapore, 1983, pp. 39–119. MR**725700** - I. B. Frenkel,
*Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory*, J. Functional Analysis**44**(1981), no. 3, 259–327. MR**643037**, DOI 10.1016/0022-1236(81)90012-4 - I. B. Frenkel, Lectures at Yale University, 1986.
- Igor B. Frenkel and Nai Huan Jing,
*Vertex representations of quantum affine algebras*, Proc. Nat. Acad. Sci. U.S.A.**85**(1988), no. 24, 9373–9377. MR**973376**, DOI 10.1073/pnas.85.24.9373 - Howard Garland,
*The arithmetic theory of loop algebras*, J. Algebra**53**(1978), no. 2, 480–551. MR**502647**, DOI 10.1016/0021-8693(78)90294-6 - Adriano M. Garsia,
*Orthogonality of Milne’s polynomials and raising operators*, Discrete Math.**99**(1992), no. 1-3, 247–264. MR**1158790**, DOI 10.1016/0012-365X(92)90375-P - Takahiro Hayashi,
*$q$-analogues of Clifford and Weyl algebras—spinor and oscillator representations of quantum enveloping algebras*, Comm. Math. Phys.**127**(1990), no. 1, 129–144. MR**1036118**, DOI 10.1007/BF02096497 - Gordon James and Adalbert Kerber,
*The representation theory of the symmetric group*, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by P. M. Cohn; With an introduction by Gilbert de B. Robinson. MR**644144** - Nai Huan Jing,
*Vertex operators, symmetric functions, and the spin group $\Gamma _n$*, J. Algebra**138**(1991), no. 2, 340–398. MR**1102815**, DOI 10.1016/0021-8693(91)90177-A - Nai Huan Jing,
*Vertex operators and Hall-Littlewood symmetric functions*, Adv. Math.**87**(1991), no. 2, 226–248. MR**1112626**, DOI 10.1016/0001-8708(91)90072-F - Nai Huan Jing,
*Vertex operators and generalized symmetric functions*, Proceedings of the Conference on Quantum Topology (Manhattan, KS, 1993) World Sci. Publ., River Edge, NJ, 1994, pp. 111–126. MR**1309930** - Nai Huan Jing,
*Boson-fermion correspondence for Hall-Littlewood polynomials*, J. Math. Phys.**36**(1995), no. 12, 7073–7080. MR**1359680**, DOI 10.1063/1.531207 - Naihuan Jing,
*Quantum Kac-Moody algebras and vertex representations*, Lett. Math. Phys.**44**(1998), no. 4, 261–271. MR**1627867**, DOI 10.1023/A:1007493921464 - M. Kashiwara, Global crystal bases of quantum groups, Duke Math. J.
**73**(1993), 383–413. - Victor G. Kac,
*Infinite-dimensional Lie algebras*, 3rd ed., Cambridge University Press, Cambridge, 1990. MR**1104219**, DOI 10.1017/CBO9780511626234 - Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon,
*Ribbon tableaux, Hall-Littlewood functions and unipotent varieties*, Sém. Lothar. Combin.**34**(1995), Art. B34g, approx. 23. MR**1399754** - James Lepowsky and Mirko Primc,
*Structure of the standard modules for the affine Lie algebra $A^{(1)}_1$*, Contemporary Mathematics, vol. 46, American Mathematical Society, Providence, RI, 1985. MR**814303**, DOI 10.1090/conm/046 - George Lusztig,
*Introduction to quantum groups*, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR**1227098** - I. G. Macdonald,
*Symmetric functions and Hall polynomials*, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR**1354144** - Kailash Misra and Tetsuji Miwa,
*Crystal base for the basic representation of $U_q(\mathfrak {s}\mathfrak {l}(n))$*, Comm. Math. Phys.**134**(1990), no. 1, 79–88. MR**1079801**, DOI 10.1007/BF02102090 - Andrey V. Zelevinsky,
*Representations of finite classical groups*, Lecture Notes in Mathematics, vol. 869, Springer-Verlag, Berlin-New York, 1981. A Hopf algebra approach. MR**643482**, DOI 10.1007/BFb0090287

## Bibliographic Information

**Naihuan Jing**- Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205
- MR Author ID: 232836
- Email: jing@math.ncsu.edu
- 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.
- © Copyright 2000 American Mathematical Society
- Journal: Represent. Theory
**4**(2000), 46-63 - MSC (2000): Primary 17B; Secondary 5E
- DOI: https://doi.org/10.1090/S1088-4165-00-00065-0
- MathSciNet review: 1740180