## Sign sequences and decomposition numbers

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- by Kai Meng Tan and Wei Hao Teo PDF
- Trans. Amer. Math. Soc.
**365**(2013), 6385-6401 Request permission

## Abstract:

We obtain a closed formula for the $v$-decomposition numbers $d_{\lambda \mu }(v)$ arising from the canonical basis of the Fock space representation of $U_v(\widehat {\mathfrak {sl}}_e)$, where the partition $\lambda$ is obtained from $\mu$ by moving some nodes in its Young diagram, all of which have the same $e$-residue. We also show that when these $v$-decomposition numbers are evaluated at $v=1$, we obtain the corresponding decomposition numbers for the Schur algebras and symmetric groups.## References

- Joseph Chuang, Hyohe Miyachi, and Kai Meng Tan,
*Row and column removal in the $q$-deformed Fock space*, J. Algebra**254**(2002), no.ย 1, 84โ91. MR**1927432**, DOI 10.1016/S0021-8693(02)00062-5 - Joseph Chuang, Hyohe Miyachi, and Kai Meng Tan,
*Kleshchevโs decomposition numbers and branching coefficients in the Fock space*, Trans. Amer. Math. Soc.**360**(2008), no.ย 3, 1179โ1191. MR**2357693**, DOI 10.1090/S0002-9947-07-04202-X - Richard Dipper and Gordon James,
*The $q$-Schur algebra*, Proc. London Math. Soc. (3)**59**(1989), no.ย 1, 23โ50. MR**997250**, DOI 10.1112/plms/s3-59.1.23 - Gordon James,
*The decomposition matrices of $\textrm {GL}_n(q)$ for $n\le 10$*, Proc. London Math. Soc. (3)**60**(1990), no.ย 2, 225โ265. MR**1031453**, DOI 10.1112/plms/s3-60.2.225 - 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 - Alexander Kleshchev,
*On decomposition numbers and branching coefficients for symmetric and special linear groups*, Proc. London Math. Soc. (3)**75**(1997), no.ย 3, 497โ558. MR**1466660**, DOI 10.1112/S0024611597000427 - 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.ย 153โ177. 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), 447โ456. MR**1399410**, DOI 10.1155/S1073792896000293 - Michela Varagnolo and Eric Vasserot,
*On the decomposition matrices of the quantized Schur algebra*, Duke Math. J.**100**(1999), no.ย 2, 267โ297. MR**1722955**, DOI 10.1215/S0012-7094-99-10010-X

## Additional Information

**Kai Meng Tan**- Affiliation: Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076
- MR Author ID: 656415
- Email: tankm@nus.edu.sg
**Wei Hao Teo**- Affiliation: Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076
- Email: tweihao@dso.org.sg
- Received by editor(s): January 27, 2012
- Received by editor(s) in revised form: April 11, 2012
- Published electronically: August 19, 2013
- Additional Notes: This research was supported by MOE Academic Research Fund R-146-000-135-112. The authors thank Joseph Chuang for many helpful discussions resulting in this article.
- © Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**365**(2013), 6385-6401 - MSC (2010): Primary 17B37, 20C08, 20C30, 20G43
- DOI: https://doi.org/10.1090/S0002-9947-2013-05860-6
- MathSciNet review: 3105756