## The rational Schur algebra

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- by Richard Dipper and Stephen Doty
- Represent. Theory
**12**(2008), 58-82 - DOI: https://doi.org/10.1090/S1088-4165-08-00303-8
- Published electronically: February 12, 2008
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## Abstract:

We extend the family of classical Schur algebras in type $A$, which determine the polynomial representation theory of general linear groups over an infinite field, to a larger family, the rational Schur algebras, which determine the rational representation theory of general linear groups over an infinite field. This makes it possible to study the rational representation theory of such general linear groups directly through finite dimensional algebras. We show that rational Schur algebras are quasihereditary over any field, and thus have finite global dimension.

We obtain explicit cellular bases of a rational Schur algebra by a descent from a certain ordinary Schur algebra. We also obtain a description, by generators and relations, of the rational Schur algebras in characteristic zero.

## References

- A. A. Beilinson, G. Lusztig, and R. MacPherson,
*A geometric setting for the quantum deformation of $\textrm {GL}_n$*, Duke Math. J.**61**(1990), no. 2, 655–677. MR**1074310**, DOI 10.1215/S0012-7094-90-06124-1 - Georgia Benkart, Manish Chakrabarti, Thomas Halverson, Robert Leduc, Chanyoung Lee, and Jeffrey Stroomer,
*Tensor product representations of general linear groups and their connections with Brauer algebras*, J. Algebra**166**(1994), no. 3, 529–567. MR**1280591**, DOI 10.1006/jabr.1994.1166 - Richard Brauer,
*On algebras which are connected with the semisimple continuous groups*, Ann. of Math. (2)**38**(1937), no. 4, 857–872. MR**1503378**, DOI 10.2307/1968843 - E. Cline, B. Parshall, and L. Scott,
*Finite-dimensional algebras and highest weight categories*, J. Reine Angew. Math.**391**(1988), 85–99. MR**961165** - Richard Dipper, Stephen Doty, and Jun Hu,
*Brauer algebras, symplectic Schur algebras and Schur-Weyl duality*, Trans. Amer. Math. Soc.**360**(2008), no. 1, 189–213. MR**2342000**, DOI 10.1090/S0002-9947-07-04179-7 - S. Donkin,
*On Schur algebras and related algebras. I*, J. Algebra**104**(1986), no. 2, 310–328. MR**866778**, DOI 10.1016/0021-8693(86)90218-8 - Stephen Doty,
*Presenting generalized $q$-Schur algebras*, Represent. Theory**7**(2003), 196–213. MR**1990659**, DOI 10.1090/S1088-4165-03-00176-6 - S.R. Doty, New versions of Schur-Weyl duality, in
*Finite Groups 2003*, Proceedings of the Gainesville Conference on Finite Groups, de Gruyter, Berlin, New York 2004. - Stephen Doty, Karin Erdmann, and Anne Henke,
*A generic algebra associated to certain Hecke algebras*, J. Algebra**278**(2004), no. 2, 502–531. MR**2071650**, DOI 10.1016/j.jalgebra.2004.04.007 - Stephen Doty and Anthony Giaquinto,
*Presenting Schur algebras as quotients of the universal enveloping algebra of $\mathfrak {g}\mathfrak {l}_2$*, Algebr. Represent. Theory**7**(2004), no. 1, 1–17. MR**2046950**, DOI 10.1023/B:ALGE.0000019386.04383.f9 - Stephen Doty and Anthony Giaquinto,
*Presenting Schur algebras*, Int. Math. Res. Not.**36**(2002), 1907–1944. MR**1920169**, DOI 10.1155/S1073792802201026 - Jie Du,
*Kazhdan-Lusztig bases and isomorphism theorems for $q$-Schur algebras*, Kazhdan-Lusztig theory and related topics (Chicago, IL, 1989) Contemp. Math., vol. 139, Amer. Math. Soc., Providence, RI, 1992, pp. 121–140. MR**1197832**, DOI 10.1090/conm/139/1197832 - Karin Erdmann,
*Decomposition numbers for symmetric groups and composition factors of Weyl modules*, J. Algebra**180**(1996), no. 1, 316–320. MR**1375581**, DOI 10.1006/jabr.1996.0067 - J. J. Graham and G. I. Lehrer,
*Cellular algebras*, Invent. Math.**123**(1996), no. 1, 1–34. MR**1376244**, DOI 10.1007/BF01232365 - J. A. Green,
*Locally finite representations*, J. Algebra**41**(1976), no. 1, 137–171. MR**412221**, DOI 10.1016/0021-8693(76)90173-3 - James A. Green,
*Polynomial representations of $\textrm {GL}_{n}$*, Lecture Notes in Mathematics, vol. 830, Springer-Verlag, Berlin-New York, 1980. MR**606556**, DOI 10.1007/BFb0092296 - J. A. Green,
*Combinatorics and the Schur algebra*, J. Pure Appl. Algebra**88**(1993), no. 1-3, 89–106. MR**1233316**, DOI 10.1016/0022-4049(93)90015-L - R. M. Green,
*A straightening formula for quantized codeterminants*, Comm. Algebra**24**(1996), no. 9, 2887–2913. MR**1396863**, DOI 10.1080/00927879608825720 - R.M. Green, ${q}$-Schur algebras and quantized enveloping algebras,
*Ph.D. thesis*, University of Warwick, 1995. - R. M. Green and P. P. Martin,
*Constructing cell data for diagram algebras*, J. Pure Appl. Algebra**209**(2007), no. 2, 551–569. MR**2293327**, DOI 10.1016/j.jpaa.2006.07.016 - Gordon D. James,
*The decomposition of tensors over fields of prime characteristic*, Math. Z.**172**(1980), no. 2, 161–178. MR**580858**, DOI 10.1007/BF01182401 - Jens Carsten Jantzen,
*Representations of algebraic groups*, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR**2015057** - David Kazhdan and George Lusztig,
*Representations of Coxeter groups and Hecke algebras*, Invent. Math.**53**(1979), no. 2, 165–184. MR**560412**, DOI 10.1007/BF01390031 - Kazuhiko Koike,
*On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters*, Adv. Math.**74**(1989), no. 1, 57–86. MR**991410**, DOI 10.1016/0001-8708(89)90004-2 - Steffen König and Changchang Xi,
*On the structure of cellular algebras*, Algebras and modules, II (Geiranger, 1996) CMS Conf. Proc., vol. 24, Amer. Math. Soc., Providence, RI, 1998, pp. 365–386. MR**1648638**, DOI 10.1007/s002200050458 - George Lusztig,
*Introduction to quantum groups*, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR**1227098** - G. E. Murphy,
*On the representation theory of the symmetric groups and associated Hecke algebras*, J. Algebra**152**(1992), no. 2, 492–513. MR**1194316**, DOI 10.1016/0021-8693(92)90045-N - G. E. Murphy,
*The representations of Hecke algebras of type $A_n$*, J. Algebra**173**(1995), no. 1, 97–121. MR**1327362**, DOI 10.1006/jabr.1995.1079 - John R. Stembridge,
*Rational tableaux and the tensor algebra of $\textrm {gl}_n$*, J. Combin. Theory Ser. A**46**(1987), no. 1, 79–120. MR**899903**, DOI 10.1016/0097-3165(87)90077-X - David J. Woodcock,
*Straightening codeterminants*, J. Pure Appl. Algebra**88**(1993), no. 1-3, 317–320. MR**1233333**, DOI 10.1016/0022-4049(93)90032-O

## Bibliographic Information

**Richard Dipper**- Affiliation: Mathematisches Institut B, Universität Stuttgart, Pfaffenwaldring 57, Stuttgart, 70569, Germany
- Email: Richard.Dipper@mathematik.uni-stuttgart.de
**Stephen Doty**- Affiliation: Mathematics and Statistics, Loyola University Chicago, Chicago, Illinois 60626
- MR Author ID: 59395
- ORCID: 0000-0003-3927-3009
- Email: doty@math.luc.edu
- Received by editor(s): November 28, 2005
- Received by editor(s) in revised form: October 23, 2007
- Published electronically: February 12, 2008
- Additional Notes: This work was partially supported by DFG project DI 531/5-2 and NSA grant DOD MDA904-03-1-00.
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory
**12**(2008), 58-82 - MSC (2000): Primary 16G99; Secondary 20G05
- DOI: https://doi.org/10.1090/S1088-4165-08-00303-8
- MathSciNet review: 2375596