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The rational Schur algebra

Author(s): Richard Dipper; Stephen Doty
Journal: Represent. Theory 12 (2008), 58-82.
MSC (2000): Primary 16G99; Secondary 20G05
Posted: 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:

[BLM]
A.A. Beĭlinson,, G. Lusztig, and R. MacPherson, A geometric setting for the quantum deformation of $ {\rm GL}\sb n$, Duke Math. J. 61 (1990), 655-677. MR 1074310 (91m:17012)

[B]
G. Benkart, M. Chakrabarti, T. Halverson, R. Leduc, C. Lee, and J. Stroomer, Tensor product representations of general linear groups and their connections with Brauer algebras, J. Algebra 166 (1994), 529-567. MR 1280591 (95d:20071)

[Br]
R. Brauer, On algebras which are connected with semisimple Lie groups, Annals Math. 38 (1937), 857-872. MR 1503378

[CPS]
E. Cline, B. Parshall, and L. Scott, Finite dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988), 85-99. MR 961165 (90d:18005)

[DDH]
R. Dipper, S.R. Doty, and Jun Hu, Brauer algebras, symplectic Schur algebras and Schur-Weyl duality, Trans. Amer. Math. Soc. 360 (2008), 189-213 MR 2342000

[Do]
S. Donkin, On Schur algebras and related algebras. I, J. Algebra 104 (1986), 310-328. MR 866778 (89b:20084a)

[D1]
S.R. Doty, Presenting generalized $ q$-Schur algebras, Represent. Theory 7 (2003), 196-213 (electronic). MR 1990659 (2004f:17020)

[D2]
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.

[DEH]
S.R. Doty, K. Erdmann, and A. Henke, A generic algebra associated to certain Hecke algebras, J. Algebra 278 (2004), 502-531. MR 2071650 (2005f:20007)

[DG1]
S.R. Doty and A. Giaquinto, Presenting Schur algebras as quotients of the universal enveloping algebra of $ \mathfrak{gl}_2$, Algebr. Represent. Theory 7 (2004), 1-17. MR 2046950 (2005a:20072)

[DG2]
S.R. Doty and A. Giaquinto, Presenting Schur algebras, Internat. Math. Res. Notices 2002, 1907-1944. MR 1920169 (2004e:16037)

[Du]
Jie Du, Kazhdan-Lusztig bases and isomorphism theorems for $ q$-Schur algebras, in Kazhdan-Lusztig theory and related topics (Chicago, IL 1989), 121-140, Contemp. Math. 139, Amer. Math. Soc., Providence, RI, 1992. MR 1197832 (94b:17019)

[E]
K. Erdmann, Decomposition numbers for symmetric groups and composition factors of Weyl modules, J. Algebra 180 (1996), 316-320. MR 1375581 (97a:20016)

[GL]
J.J. Graham and G.I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), 1-34. MR 1376244 (97h:20016)

[G1]
J.A. Green, Locally finite representations, J. Algebra 41 (1976), 137-171. MR 0412221 (54:348)

[G2]
J.A. Green, Polynomial Representations of $ \mathsf{GL}_n$, Lecture Notes in Math. 830, Springer-Verlag, New York 1980. MR 606556 (83j:20003)

[G3]
J.A. Green, Combinatorics and the Schur algebra, J. Pure Appl. Algebra 88 (1993), 89-106. MR 1233316 (94g:05100)

[RMG1]
R.M. Green, A straightening algorithm for quantized codeterminants, Comm. Algebra 24 (1996), 2887-2913. MR 1396863 (97k:20069)

[RMG2]
R.M. Green, $ {q}$-Schur algebras and quantized enveloping algebras, Ph.D. thesis, University of Warwick, 1995.

[GM]
R.M. Green and P.P. Martin, Constructing cell data for diagram algebras, J. Pure Appl. Algebra 209 (2007), 551-569. MR 2293327 (2007k:16029)

[Jam]
G.D. James, The decomposition of tensors over fields of prime characteristic, Math. Zeit. 172 (1980), 161-178. MR 580858 (81h:20015)

[Jan]
J.C. Jantzen, Representations of Algebraic Groups, 2nd ed., Math. Surveys and Monographs 107, Amer. Math. Society 2003. MR 2015057 (2004h:20061)

[KL]
D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. MR 560412 (81j:20066)

[K]
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), 57-86. MR 991410 (90j:22014)

[KX]
S. König and C.C. Xi, On the structure of cellular algebras, Algebras and Modules. II, CMS Conference Proceedings 24 (1998), 365-386. MR 1648638 (2000a:16011)

[Lu]
G. Lusztig, Introduction to Quantum Groups, Birkhäuser Boston 1993. MR 1227098 (94m:17016)

[M1]
G.E. Murphy, On the representation theory of the symmetric groups and associated Hecke algebras, J. Algebra 152 (1992), 492-513. MR 1194316 (94c:17031)

[M2]
G.E. Murphy, The representations of Hecke algebras of type $ A_n$, J. Algebra 173 (1995), 97-121. MR 1327362 (96b:20013)

[St]
J.R. Stembridge, Rational tableaux and the tensor algebra of $ \mathfrak{gl}\sb n$, J. Combin. Theory Ser. A 46 (1987), 79-120. MR 899903 (89a:05012)

[Wo]
D.J. Woodcock, Straightening codeterminants, J. Pure Appl. Algebra 88 (1993), 317-320. MR 1233333 (94h:05096)

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Additional 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
Email: doty@math.luc.edu

DOI: 10.1090/S1088-4165-08-00303-8
PII: S 1088-4165(08)00303-8
Keywords: Schur algebras, finite dimensional algebras, quasihereditary algebras, general linear groups
Received by editor(s): November 28, 2005
Received by editor(s) in revised form: October 23, 2007
Posted: 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 of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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