The blocks of the Brauer algebra in characteristic zero

Cox, Anton; De Visscher, Maud; Martin, Paul

doi:10.1090/S1088-4165-09-00305-7

The blocks of the Brauer algebra in characteristic zero
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by Anton Cox, Maud De Visscher and Paul Martin PDF

Represent. Theory 13 (2009), 272-308 Request permission

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

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
Wm. P. Brown, An algebra related to the orthogonal group, Michigan Math. J. 3 (1955), 1–22. MR 72122
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.
Anton Cox, Paul Martin, Alison Parker, and Changchang Xi, Representation theory of towers of recollement: theory, notes, and examples, J. Algebra 302 (2006), no. 1, 340–360. MR 2236606, DOI 10.1016/j.jalgebra.2006.01.009
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. Doty, and J. Hu, Schur-Weyl duality for orthogonal groups, Proc. London Math. Soc., 98 (2009), 679–713.
Persi Diaconis, Group representations in probability and statistics, Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 11, Institute of Mathematical Statistics, Hayward, CA, 1988. MR 964069, DOI 10.1007/BFb0086177
S. Donkin, The $q$-Schur algebra, London Mathematical Society Lecture Note Series, vol. 253, Cambridge University Press, Cambridge, 1998. MR 1707336, DOI 10.1017/CBO9780511600708
Stephen Doty, Polynomial representations, algebraic monoids, and Schur algebras of classical type, J. Pure Appl. Algebra 123 (1998), no. 1-3, 165–199. MR 1492900, DOI 10.1016/S0022-4049(96)00082-5
S. Donkin and R. Tange, The Brauer algebra and the symplectic Schur algebra, Math. Z., to appear.
William F. Doran IV, David B. Wales, and Philip J. Hanlon, On the semisimplicity of the Brauer centralizer algebras, J. Algebra 211 (1999), no. 2, 647–685. MR 1666664, DOI 10.1006/jabr.1998.7592
William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR 1464693
J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), no. 1, 1–34. MR 1376244, DOI 10.1007/BF01232365
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
Robert Hartmann and Rowena Paget, Young modules and filtration multiplicities for Brauer algebras, Math. Z. 254 (2006), no. 2, 333–357. MR 2262706, DOI 10.1007/s00209-006-0950-x
Phil Hanlon and David Wales, Eigenvalues connected with Brauer’s centralizer algebras, J. Algebra 121 (1989), no. 2, 446–476. MR 992776, DOI 10.1016/0021-8693(89)90077-X
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, Computing the discriminants of Brauer’s centralizer algebras, Math. Comp. 54 (1990), no. 190, 771–796. MR 1010599, DOI 10.1090/S0025-5718-1990-1010599-4
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, DOI 10.1007/BFb0067708
G. D. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley, 1981.
Steffen König and Changchang Xi, Cellular algebras: inflations and Morita equivalences, J. London Math. Soc. (2) 60 (1999), no. 3, 700–722. MR 1753809, DOI 10.1112/S0024610799008212
Steffen König and Changchang Xi, A characteristic free approach to Brauer algebras, Trans. Amer. Math. Soc. 353 (2001), no. 4, 1489–1505. MR 1806731, DOI 10.1090/S0002-9947-00-02724-0
Paul Martin, Potts models and related problems in statistical mechanics, Series on Advances in Statistical Mechanics, vol. 5, World Scientific Publishing Co., Inc., Teaneck, NJ, 1991. MR 1103994, DOI 10.1142/0983
Paul Martin, The structure of the partition algebras, J. Algebra 183 (1996), no. 2, 319–358. MR 1399030, DOI 10.1006/jabr.1996.0223
Andrew Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, University Lecture Series, vol. 15, American Mathematical Society, Providence, RI, 1999. MR 1711316, DOI 10.1090/ulect/015
P. P. Martin and S. Ryom-Hansen, Virtual algebraic Lie theory: tilting modules and Ringel duals for blob algebras, Proc. London Math. Soc. (3) 89 (2004), no. 3, 655–675. MR 2107010, DOI 10.1112/S0024611504014789
Paul P. Martin and David Woodcock, Generalized blob algebras and alcove geometry, LMS J. Comput. Math. 6 (2003), 249–296. MR 2051586, DOI 10.1112/S1461157000000450
Maxim Nazarov, Young’s orthogonal form for Brauer’s centralizer algebra, J. Algebra 182 (1996), no. 3, 664–693. MR 1398116, DOI 10.1006/jabr.1996.0195
Sebastian Oehms, Centralizer coalgebras, FRT-construction, and symplectic monoids, J. Algebra 244 (2001), no. 1, 19–44. MR 1856529, DOI 10.1006/jabr.2001.8909
Rosa Orellana and Arun Ram, Affine braids, Markov traces and the category $\scr O$, Algebraic groups and homogeneous spaces, Tata Inst. Fund. Res. Stud. Math., vol. 19, Tata Inst. Fund. Res., Mumbai, 2007, pp. 423–473. MR 2348913
Hebing Rui, A criterion on the semisimple Brauer algebras, J. Combin. Theory Ser. A 111 (2005), no. 1, 78–88. MR 2144855, DOI 10.1016/j.jcta.2004.11.009
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
Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR 1488158