Symmetric polynomials and 𝑈_{𝑞}(̂𝑠𝑙₂)

Jing, Naihuan

doi:10.1090/S1088-4165-00-00065-0

Symmetric polynomials and $U_q(\widehat {sl}_2)$
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Represent. Theory 4 (2000), 46-63 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