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The blocks of the Brauer algebra in characteristic zero

Author(s): Anton Cox; Maud De Visscher; Paul Martin
Journal: Represent. Theory 13 (2009), 272-308.
MSC (2000): Primary 20G05
Posted: July 7, 2009
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Abstract | References | Similar articles | Additional information

Abstract: We determine the blocks of the Brauer algebra in characteristic zero. We also give information on the submodule structure of standard modules for this algebra.

References:

[Bra37]
R. Brauer, On algebras which are connected with the semisimple continuous groups, Ann. of Math. 38 (1937), 857-872. MR 1503378

[Bro55]
W. Brown, An algebra related to the orthogonal group, Michigan Math. J. 3 (1955), 1-22. MR 0072122 (17:232a)

[CDM]
A. G. Cox, M. De Visscher, and P. P. Martin, A geometric characterisation of the blocks of the Brauer algebra, J. London Math. Soc, to appear.

[CMPX06]
A. G. Cox, P. P. Martin, A. E. Parker, and C. Xi, Representation theory of towers of recollement: theory, notes, and examples, J. Algebra 302 (2006), 340-360. MR 2236606 (2007b:16028)

[DDH08]
R. Dipper, S. Doty, and J. Hu, Brauer's centralizer algebras, symplectic Schur algebras and Schur-Weyl duality, Trans. Amer. Math. Soc. 360 (2008), 189-213. MR 2342000 (2008g:16032)

[DH09]
S. Doty, and J. Hu, Schur-Weyl duality for orthogonal groups, Proc. London Math. Soc., 98 (2009), 679-713.

[Dia88]
P. Diaconis, Group representations in probability and statistics, Institute of Mathematical Statistics Lecture Notes, Monograph Series 11, Institute of Mathematical Statistics, 1988. MR 964069 (90a:60001)

[Don98]
S. Donkin, The $ q$-Schur algebra, London Math. Soc. Lecture Notes Series, vol. 253, Cambridge University Press, 1998. MR 1707336 (2001h:20072)

[Dot98]
S. Doty, Polynomial representations, algebraic monoids, and Schur algebras of classical type, J. Pure Appl. Algebra 123 (1998), 165-199. MR 1492900 (98j:20057)

[DT]
S. Donkin and R. Tange, The Brauer algebra and the symplectic Schur algebra, Math. Z., to appear.

[DWH99]
W. F. Doran, D. B. Wales, and P. J. Hanlon, On the semisimplicity of the Brauer centralizer algebras, J. Algebra 211 (1999), 647-685. MR 1666664 (99k:16034)

[Ful97]
W. Fulton, Young tableaux, London Math. Soc. Student Texts, vol. 35, Cambridge, 1997. MR 1464693 (99f:05119)

[GL96]
J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), 1-34. MR 1376244 (97h:20016)

[Gre80]
J. A. Green, Polynomial representations of GL$ _n$, Lecture Notes in Mathematics 830, Springer, 1980. MR 606556 (83j:20003)

[HP06]
R. Hartmann and R. Paget, Young modules and filtration multiplicities for Brauer algebras, Math. Z. 254 (2006), 333-357. MR 2262706 (2008j:20009)

[HW89a]
P. J. Hanlon and D. B. Wales, Eigenvalues connected with Brauer's centralizer algebras, J. Algebra 121 (1989), 446-476. MR 992776 (91a:20041b)

[HW89b]
-, On the decomposition of Brauer's centralizer algebras, J. Algebra 121 (1989), 409-445. MR 992775 (91a:20041a)

[HW90]
-, Computing the discriminants of Brauer's centralizer algebras, Math. Comp. 54 (1990), 771-796. MR 1010599 (90k:20076)

[HW94]
-, A tower construction for the radical in Brauer's centralizer algebras, J. Algebra 164 (1994), 773-830. MR 1272114 (95f:20070)

[Jam78]
G. D. James, The representation theory of the symmetric groups, Lecture Notes in Mathematics, vol. 682, Springer, 1978. MR 513828 (80g:20019)

[JK81]
G. D. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley, 1981.

[KX99]
S. König and C. Xi, Cellular algebras: Inflations and Morita equivalences, J. London Math. Soc. 60 (1999), 700-722. MR 1753809 (2001a:16020)

[KX01]
S. König and C. Xi, A characteristic free approach to Brauer algebras, Trans. Amer. Math. Soc. 353 (2001), 1489-1505. MR 1806731 (2002f:16039)

[Mar91]
P. P. Martin, Potts models and related problems in statistical mechanics, World Scientific, Singapore, 1991. MR 1103994 (92m:82030)

[Mar96]
-, The structure of the Partition algebras, J. Algebra 183 (1996), 319-358. MR 1399030 (98g:05152)

[Mat99]
A. Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric groups, University lecture series, vol. 15, American Mathematical Society, 1999. MR 1711316 (2001g:20006)

[MRH04]
P. P. Martin and S. Ryom-Hansen, Virtual algebraic Lie theory: Tilting modules and Ringel duals for blob algebras, Proc. London Math. Soc. 89 (2004), 655-675. MR 2107010 (2005i:20007)

[MW03]
P. P. Martin and D. Woodcock, Generalized blob algebras and alcove geometry, London Math. Soc. J. of Comp. and Math. 6 (2003), 249-296. MR 2051586 (2005f:20013)

[Naz96]
M. Nazarov, Young's orthogonal form for Brauer's centralizer algebra, J. Algebra 182 (1996), 664-693. MR 1398116 (97m:20057)

[Oeh01]
S. Oehms, Centralizer coalgebras, FRT-construction and symplectic monoids, J. Algebra 244 (2001), 19-44. MR 1856529 (2002h:16062)

[OR07]
R. Orellana and A. Ram, Affine braids, Markov traces and the category $ \mathcal O$, in Algebraic groups and homogeneous spaces, Tata Inst. Fund. Res. Stud. Math. (2007), 423-473. MR 2348913 (2008m:17034)

[Rui05]
H. Rui, A criterion on the semisimple Brauer algebras, J. Comb. Theory Ser. A 111 (2005), 78-88. MR 2144855 (2006g:16042)

[Wen88]
H. Wenzl, On the structure of Brauer's centralizer algebra, Ann. Math. 128 (1988), 173-193. MR 951511 (89h:20059)

[Wey46]
H. Weyl, The classical groups, their invariants and representations, Princeton University Press, 1946. MR 1488158 (98k:01049)

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Additional Information:

Anton Cox
Affiliation: Centre for Mathematical Science, City University, Northampton Square, London, EC1V 0HB, England.
Email: A.G.Cox@city.ac.uk

Maud De Visscher
Affiliation: Centre for Mathematical Science, City University, Northampton Square, London, EC1V 0HB, England
Email: M.Devisscher@city.ac.uk

Paul Martin
Affiliation: Centre for Mathematical Science, City University, Northampton Square, London, EC1V 0HB, England
Address at time of publication: Department of Pure Mathematics, University at Leeds, Leeds, LS2 9JT, England
Email: ppmartin@maths.leeds.ac.uk

DOI: 10.1090/S1088-4165-09-00305-7
PII: S 1088-4165(09)00305-7
Received by editor(s): December 22,2005
Posted: July 7, 2009
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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