Kleshchev’s decomposition numbers and branching coefficients in the Fock space
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- by Joseph Chuang, Hyohe Miyachi and Kai Meng Tan PDF
- Trans. Amer. Math. Soc. 360 (2008), 1179-1191 Request permission
Abstract:
We give combinatorial descriptions of some coefficients of the canonical basis of the $q$-deformed Fock space representation of $U_q(\widehat {\mathfrak {sl}}_e)$ and of some matrix entries for the action of the Chevalley generators $f_r$ with respect to the canonical basis. These are $q$-analogues of results of Kleshchev on decomposition numbers and branching coefficients for symmetric groups and Schur algebras.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
- Joseph Chuang, Hyohe Miyachi, and Kai Meng Tan, Row and column removal in the $q$-deformed Fock space, J. Algebra 254 (2002), no. 1, 84–91. MR 1927432, DOI 10.1016/S0021-8693(02)00062-5
- 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
- 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
- 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
- Alexander Kleshchev, On decomposition numbers and branching coefficients for symmetric and special linear groups, Proc. London Math. Soc. (3) 75 (1997), no. 3, 497–558. MR 1466660, DOI 10.1112/S0024611597000427
- 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
- Bernard Leclerc, Symmetric functions and the Fock space, Symmetric functions 2001: surveys of developments and perspectives, NATO Sci. Ser. II Math. Phys. Chem., vol. 74, Kluwer Acad. Publ., Dordrecht, 2002, pp. 153–177. MR 2059362, DOI 10.1007/978-94-010-0524-1_{4}
- 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
- 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
- G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), no. 2, 365–421. MR 1088333, DOI 10.1090/S0894-0347-1991-1088333-2
- George Lusztig, Canonical bases and Hall algebras, Representation theories and algebraic geometry (Montreal, PQ, 1997) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 514, Kluwer Acad. Publ., Dordrecht, 1998, pp. 365–399. MR 1653038
- Andrew Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, University Lecture Series, vol. 15, American Mathematical Society, Providence, RI, 1999. MR 1711316, DOI 10.1090/ulect/015
- 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
- Olivier Schiffmann, The Hall algebra of a cyclic quiver and canonical bases of Fock spaces, Internat. Math. Res. Notices 8 (2000), 413–440. MR 1753691, DOI 10.1155/S1073792800000234
- 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
Additional Information
- Joseph Chuang
- Affiliation: Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom
- Email: joseph.chuang@bris.ac.uk
- Hyohe Miyachi
- Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan
- MR Author ID: 649846
- Email: miyachi@math.nagoya-u.ac.jp
- Kai Meng Tan
- Affiliation: Department of Mathematics, National University of Singapore, 2, Science Drive 2, Singapore 117543
- MR Author ID: 656415
- Email: tankm@nus.edu.sg
- Received by editor(s): July 23, 2005
- Published electronically: October 2, 2007
- © Copyright 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 360 (2008), 1179-1191
- MSC (2000): Primary 17B37; Secondary 20C08
- DOI: https://doi.org/10.1090/S0002-9947-07-04202-X
- MathSciNet review: 2357693