A characteristic free approach to Brauer algebras

König, Steffen; Xi, Changchang

doi:10.1090/S0002-9947-00-02724-0

A characteristic free approach to Brauer algebras
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by Steffen König and Changchang Xi PDF

Trans. Amer. Math. Soc. 353 (2001), 1489-1505 Request permission

Abstract:

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.

References

J. L. Alperin, Local representation theory, Cambridge Studies in Advanced Mathematics, vol. 11, Cambridge University Press, Cambridge, 1986. Modular representations as an introduction to the local representation theory of finite groups. MR 860771, DOI 10.1017/CBO9780511623592
R.Brauer, On algebras which are connected with the semisimple continous groups. Annals of Math. 38, 854–872 (1937).
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Frederick M. Goodman, Pierre de la Harpe, and Vaughan F. R. Jones, Coxeter graphs and towers of algebras, Mathematical Sciences Research Institute Publications, vol. 14, Springer-Verlag, New York, 1989. MR 999799, DOI 10.1007/978-1-4613-9641-3
J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), no. 1, 1–34. MR 1376244, DOI 10.1007/BF01232365
Phil Hanlon and David Wales, On the decomposition of Brauer’s centralizer algebras, J. Algebra 121 (1989), no. 2, 409–445. MR 992775, DOI 10.1016/0021-8693(89)90076-8
Phil Hanlon and David Wales, On the decomposition of Brauer’s centralizer algebras, J. Algebra 121 (1989), no. 2, 409–445. MR 992775, DOI 10.1016/0021-8693(89)90076-8
Phil Hanlon and David Wales, A tower construction for the radical in Brauer’s centralizer algebras, J. Algebra 164 (1994), no. 3, 773–830. MR 1272114, DOI 10.1006/jabr.1994.1089
G. D. James, The representation theory of the symmetric groups, Lecture Notes in Mathematics, vol. 682, Springer, Berlin, 1978. MR 513828
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
S. V. Kerov, Realizations of representations of the Brauer semigroup, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 164 (1987), no. Differentsial′naya Geom. Gruppy Li i Mekh. IX, 188–193, 199 (Russian, with English summary); English transl., J. Soviet Math. 47 (1989), no. 2, 2503–2507. MR 947338, DOI 10.1007/BF01840432
S.König, Cyclotomic Schur algebras and blocks of cyclic defect, Canad. Math. Bull. 43, 79–86 (2000).
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
S.König and C.C.Xi, Cellular algebras: inflations and Morita equivalences. Journal London Math. Soc. 60, 700–722 (1999).
Arun Ram, Characters of Brauer’s centralizer algebras, Pacific J. Math. 169 (1995), no. 1, 173–200. MR 1346252
Hans Wenzl, On the structure of Brauer’s centralizer algebras, Ann. of Math. (2) 128 (1988), no. 1, 173–193. MR 951511, DOI 10.2307/1971466
Hans Wenzl, Quantum groups and subfactors of type $B$, $C$, and $D$, Comm. Math. Phys. 133 (1990), no. 2, 383–432. MR 1090432