Some closed formulas for canonical bases of Fock spaces

Leclerc, Bernard; Miyachi, Hyohe

doi:10.1090/S1088-4165-02-00136-X

Some closed formulas for canonical bases of Fock spaces
HTML articles powered by AMS MathViewer

by Bernard Leclerc and Hyohe Miyachi

Represent. Theory 6 (2002), 290-312

DOI: https://doi.org/10.1090/S1088-4165-02-00136-X

Published electronically: September 19, 2002

PDF | Request permission

Abstract:

We give some closed formulas for certain vectors of the canonical bases of the Fock space representation of $U_v(\mathfrak {sl}_n)$. As a result, a combinatorial description of certain parabolic Kazhdan-Lusztig polynomials for affine type $A$ is obtained.

References

Susumu Ariki, On the decomposition numbers of the Hecke algebra of $G(m,1,n)$, J. Math. Kyoto Univ. 36 (1996), no. 4, 789–808. MR 1443748, DOI 10.1215/kjm/1250518452

J. Chuang, R. Kessar

Symmetric groups, wreath products, Morita equivalences, and Broué’s abelian defect group conjecture

34

J. Chuang, K. M. Tan

Some canonical basis vectors in the basic $U_q(\mathfrak {sl}_n)$-module

248

Richard Dipper and Gordon James, The $q$-Schur algebra, Proc. London Math. Soc. (3) 59 (1989), no. 1, 23–50. MR 997250, DOI 10.1112/plms/s3-59.1.23
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

A. Hida, H. Miyachi

Module correspondences in some blocks of finite general linear groups

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
Gordon James and Andrew Mathas, Hecke algebras of type $\textbf {A}$ with $q=-1$, J. Algebra 184 (1996), no. 1, 102–158. MR 1402572, DOI 10.1006/jabr.1996.0251
Michio Jimbo and Tetsuji Miwa, Solitons and infinite-dimensional Lie algebras, Publ. Res. Inst. Math. Sci. 19 (1983), no. 3, 943–1001. MR 723457, DOI 10.2977/prims/1195182017
Thomas Jost, Morita equivalence for blocks of Hecke algebras of symmetric groups, J. Algebra 194 (1997), no. 1, 201–223. MR 1461487, DOI 10.1006/jabr.1996.6988
Thomas Jost, Morita equivalence for blocks of finite general linear groups, Manuscripta Math. 91 (1996), no. 1, 121–144. MR 1404422, DOI 10.1007/BF02567944
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
Masaki Kashiwara, Crystalizing the $q$-analogue of universal enveloping algebras, Comm. Math. Phys. 133 (1990), no. 2, 249–260. MR 1090425, DOI 10.1007/BF02097367
M. Kashiwara, On crystal bases of the $Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465–516. MR 1115118, DOI 10.1215/S0012-7094-91-06321-0
Masaki Kashiwara, Global crystal bases of quantum groups, Duke Math. J. 69 (1993), no. 2, 455–485. MR 1203234, DOI 10.1215/S0012-7094-93-06920-7
M. Kashiwara, T. Miwa, and E. Stern, Decomposition of $q$-deformed Fock spaces, Selecta Math. (N.S.) 1 (1995), no. 4, 787–805. MR 1383585, DOI 10.1007/BF01587910

M. Kashiwara, T. Tanisaki

Parabolic Kazhdan-Lusztig polynomials and Schubert varieties

249

Donald Knutson, $\lambda$-rings and the representation theory of the symmetric group, Lecture Notes in Mathematics, Vol. 308, Springer-Verlag, Berlin-New York, 1973. MR 0364425, DOI 10.1007/BFb0069217
Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), no. 1, 205–263. MR 1410572, DOI 10.1007/BF02101678
Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon, Ribbon tableaux, Hall-Littlewood functions, quantum affine algebras, and unipotent varieties, J. Math. Phys. 38 (1997), no. 2, 1041–1068. MR 1434225, DOI 10.1063/1.531807
Bernard Leclerc and Séverine Leidwanger, Schur functions and affine Lie algebras, J. Algebra 210 (1998), no. 1, 103–144. MR 1656417, DOI 10.1006/jabr.1998.7453
Bernard Leclerc and Jean-Yves Thibon, Canonical bases of $q$-deformed Fock spaces, Internat. Math. Res. Notices 9 (1996), 447–456. MR 1399410, DOI 10.1155/S1073792896000293

B. Leclerc, J.-Y. Thibon

Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials

Combinatorial methods in representation theory

28

Séverine Leidwanger, Basic representations of $A^{(1)}_{n-1}$ and $A^{(2)}_{2n}$ and the combinatorics of partitions, Adv. Math. 141 (1999), no. 1, 119–154. MR 1667148, DOI 10.1006/aima.1998.1781
G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498. MR 1035415, DOI 10.1090/S0894-0347-1990-1035415-6
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

H. Miyachi

Some remarks on James’ conjecture

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

R. Rouquier

Représentations et catégories dérivées

Joanna Scopes, Cartan matrices and Morita equivalence for blocks of the symmetric groups, J. Algebra 142 (1991), no. 2, 441–455. MR 1127075, DOI 10.1016/0021-8693(91)90319-4
Dennis W. Stanton and Dennis E. White, A Schensted algorithm for rim hook tableaux, J. Combin. Theory Ser. A 40 (1985), no. 2, 211–247. MR 814412, DOI 10.1016/0097-3165(85)90088-3
Eugene Stern, Semi-infinite wedges and vertex operators, Internat. Math. Res. Notices 4 (1995), 201–219. MR 1326065, DOI 10.1155/S107379289500016X
Michela Varagnolo and Eric Vasserot, On the decomposition matrices of the quantized Schur algebra, Duke Math. J. 100 (1999), no. 2, 267–297. MR 1722955, DOI 10.1215/S0012-7094-99-10010-X