A characteristic free approach to Brauer algebras

Authors:
Steffen König and Changchang Xi

Journal:
Trans. Amer. Math. Soc. **353** (2001), 1489-1505

MSC (1991):
Primary 16D25, 16G30, 20G0; Secondary 57M25, 81R05

DOI:
https://doi.org/10.1090/S0002-9947-00-02724-0

Published electronically:
December 18, 2000

MathSciNet review:
1806731

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Abstract | References | Similar Articles | Additional Information

Brauer algebras arise in representation theory of orthogonal or symplectic groups. These algebras are shown to be iterated inflations of group algebras of symmetric groups. In particular, they are cellular (as had been shown before by Graham and Lehrer). This gives some information about block decomposition of Brauer algebras.

**[Alp]**J.L.ALPERIN, Local representation theory, Cambridge University Press (1986). MR**87i:20002****[Br]**R.BRAUER, On algebras which are connected with the semisimple continous groups. Annals of Math. 38, 854-872 (1937).**[Brow1]**W.P.BROWN, Generalized matrix algebras. Canad. J. Math. 7, 188-190 (1955). MR**16:789b****[Brow2]**W.P.BROWN, The semisimplicity of . Annals of Math. (2) 63, 324-335 (1956). MR**17:821g****[Brow3]**W.P.BROWN, An algebra related to the orthogonal group. Michigan J. Math. 3, 1-22 (1955-56). MR**17:232a****[GHJ]**F.M.GOODMAN, P. DE LA HARPE AND V.F.R.JONES, Coxeter graphs and towers of algebras. MSRI Publ. 14, Springer (1989). MR**91c:46082****[GL]**J.J.GRAHAM AND G.I.LEHRER, Cellular algebras. Invent. Math. 123, 1-34 (1996). MR**97h:20016****[HW1]**P.HANLON AND D.WALES, On the decomposition of Brauer's centralizer algebras. J. Alg. 121, 409-445 (1989). MR**91a:20041a****[HW2]**P.HANLON AND D.WALES, Eigenvalues connected with Brauer's centralizer algebras. J. Alg. 121, 446-476 (1989). MR**91a:20041b****[HW3]**P.HANLON AND D.WALES, A tower construction for the radical in Brauer's centralizer algebras. J. Alg. 164, 773-830 (1994). MR**95f:20070****[J]**G.JAMES, The representation theory of the symmetric groups. Springer LNM 682 (1978). MR**80g:20019****[JK]**G.JAMES AND A.KERBER, The representation theory of the symmetric group. Encyclopedia of Math. and its Appl. 16. Addison-Wesley (1981). MR**83k:20003****[Ker]**S.V.KEROV, Realizations of representations of the Brauer semigroup. Zap. Nauchn. Sem. Len. (LOMI) 164, 188-193 (1987) (also: J. Soviet Math. 47, 2503-2507 (1989)). MR**89j:22029****[K1]**S.K¨ONIG, Cyclotomic Schur algebras and blocks of cyclic defect, Canad. Math. Bull. 43, 79-86 (2000). CMP**2000:10****[KX1]**S.K¨ONIG AND C.C.XI, On the structure of cellular algebras. In: Algebras and modules II (Geiranger 1996), CMS Conf. Proc. 24, Amer. Math. Soc., 365-386 (1998). MR**2000a:16011****[KX2]**S.K¨ONIG AND C.C.XI, Cellular algebras: inflations and Morita equivalences. Journal London Math. Soc. 60, 700-722 (1999). CMP**2000:11****[Ram]**A.RAM, Characters of Brauer's centralizer algebras. Pacific J. Math. 169, 173-200 (1995). MR**96k:20020****[We1]**H.WENZL, On the structure of Brauer's centralizer algebras. Annals of Math. 128, 173-193 (1988). MR**89h:20059****[We2]**H.WENZL, Quantum groups and subfactors of type , , and . Comm. Math. Physics 133, 383-432 (1990). MR**92k:17032**

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Additional Information

**Steffen König**

Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany

Address at time of publication:
Department of Mathematics and Computer Science, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom

Email:
sck5@mcs.le.ac.ak

**Changchang Xi**

Affiliation:
Department of Mathematics, Beijing Normal University, 100875 Beijing, People’s Republic of China

Email:
xicc@bnu.edu.cn

DOI:
https://doi.org/10.1090/S0002-9947-00-02724-0

Keywords:
Brauer algebras,
orthogonal groups,
symplectic groups,
cellular algebras

Received by editor(s):
January 26, 1998

Received by editor(s) in revised form:
January 13, 2000

Published electronically:
December 18, 2000

Additional Notes:
Both authors have obtained support from the Volkswagen Foundation (Research in Pairs Programme of the Mathematical Research Institute Oberwolfach). S. König also obtained support from Beijing Normal University during his stay in Beijing in May 1997, when most of this paper has been written. C. C. Xi also obtained support from the Young Teacher Foundation of Chinese Educational Committee and from NSF of China (Grant No. 19831070).

Article copyright:
© Copyright 2000
American Mathematical Society