Abstract
Let k be an algebraically closed field of characteristic p > 0, let m, r be integers with m ≥ 1, r ≥ 0 and m ≥ r and let S 0(2m, r) be the symplectic Schur algebra over k as introduced by the first author. We introduce the symplectic Schur functor, derive some basic properties of it and relate this to work of Hartmann and Paget. We do the same for the orthogonal Schur algebra. We give a modified Jantzen sum formula and a block result for the symplectic Schur algebra under the assumption that r and the residue of 2m mod p are small relative to p. From this we deduce a block result for the orthogonal Schur algebra under similar assumptions. We also show that, in general, the block relations of the Brauer algebra and the symplectic and orthogonal Schur algebra are the same. Finally, we deduce from the previous results a new proof of the description of the blocks of the Brauer algebra in characteristic 0 as obtained by Cox, De Visscher and Martin.
Similar content being viewed by others
References
Adamovich A.M., Rybnikov G.L.: Tilting modules for classical groups and Howe duality in positive characteristic. Transform. Groups 1(1–2), 1–34 (1996)
Auslander M., Reiten I., Smalø S.: Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge University Press, Cambridge (1995)
Benson D.J., Carlson J.F.: Nilpotent elements in the Green ring. J. Algebra 104(2), 329–350 (1986)
Brauer R.: On algebras which are connected with the semisimple continuous groups. Ann. Math. (2) 38(4), 857–872 (1937)
Brown W.P.: An algebra related to the orthogonal group. Mich. Math. J. 3, 1–22 (1955)
Brown W.P.: The semisimplicity of \({\omega_f^n}\) . Ann. Math. (2) 63, 324–335 (1956)
Brundan, J.: Dense Orbits and Double Cosets. Algebraic Groups and their Representations (Cambridge, 1997), pp. 259–274. Kluwer, Dordrecht
Cliff, G.: A basis of bideterminants for the coordinate ring of the orthogonal group (preprint)
Cox, A., De Visscher, M., Martin, P.: The blocks of the Brauer algebra in characteristic zero (preprint)
Cox, A., De Visscher, M., Martin, P.: A geometric characterisation of the blocks of the Brauer algebra (preprint)
De Concini C., Procesi C.: A characteristic free approach to invariant theory. Adv. Math. 21(3), 330–354 (1976)
Dipper R., Doty S., Hu J.: Brauer algbras, symplectic Schur algebras and Schur–Weyl duality. Trans. Am. Math. Soc. 360(1), 189–213 (2008)
Donkin S.: On Schur algebras and related algebras, I. J. Algebra 104(2), 310–328 (1986)
Donkin S.: On Schur algebras and related algebras, II. J. Algebra 111(2), 354–364 (1987)
Donkin, S.: Good filtrations of rational modules for reductive groups. In: The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986). American Mathematical Society, Providence (1987)
Donkin S.: Representations of symplectic groups and the symplectic tableaux of R. C. King. Linear Multilinear Algebra 29(2), 113–124 (1991)
Donkin, S.: On tilting modules and invariants for algebraic groups. In: Finite-Dimensional Algebras and Related Topics (Ottawa, ON, 1992), pp. 59–77. Kluwer, Dordrecht (1994)
Donkin S.: On tilting modules for algebraic groups. Math. Z. 212(1), 39–60 (1993)
Donkin S.: The q-Schur Algebra, LMS Lecture Note Series, vol. 253. Cambridge University Press, Cambridge (1998)
Donkin, S.: Tilting modules for algebraic groups and finite dimensional algebras. In: Handbook of Tilting Theory, pp. 215–257. LMS Lecture Note Series, vol. 332. Cambridge University Press, Cambridge.
Doran W.F., Hanlon P., Wales D.: On the semisimplicity of the Brauer centralizer algebras. J. Algebra 211(2), 647–685 (1999)
Doty S.: Polynomial representations, algebraic monoids, and Schur algebras of classical type. J. Pure Appl. Algebra 123(1–3), 165–199 (1998)
Doty, S., Hu, J.: Schur–Weyl duality for orthogonal groups (preprint)
Erdmann K., Sáenz C.: On standardly stratified algebras. Comm. Algebra 31(7), 3429–3446 (2003)
Green J.A.: Polynomial Representations of GL n , Lecture Notes in Mathematics, vol. 830. Springer, Berlin (1980)
Hanlon P., Wales D.: On the decomposition of Brauer’s centralizer algebras. J. Algebra 121(2), 409–445 (1989)
Hartmann R., Paget R.: Young modules and filtration multiplicities for Brauer algebras. Math. Z. 254(2), 333–357 (2006)
James G.D.: The Representation Theory of the Symmetric Groups. Lecture Notes in Mathematics, vol. 682. Springer, Berlin (1978)
Jantzen J.C.: Representations of Algebraic Groups. Pure and Applied Mathematics, vol. 131. Academic Press, Boston (1987)
King, R.C.: Weight multiplicities for the classical groups. In: Group Theoretical Methods in Physics (Fourth Internat. Colloq., Nijmegen, 1975). Lecture Notes in Physics, vol. 50, pp. 490–499. Springer, Berlin (1976)
King R.C., Welsh T.A.: Construction of orthogonal group modules using tableaux. Linear Multilinear Algebra 33(3–4), 251–283 (1993)
König S., Xi C.: A characteristic free approach to Brauer algebras. Trans. Am. Math. Soc. 353(4), 1489–1505 (2001)
Martin P., Woodcock D.: The partition algebras and a new deformation of the Schur algebras. J. Algebra 203(1), 91–124 (1998)
Martin S.: Schur Algebras and Representation Theory, Cambridge Tracts in Mathematics, vol. 112. Cambridge University Press, Cambridge (1993)
Oehms S.: Centralizer coalgebras, FRT-construction, and symplectic monoids. J. Algebra 244(1), 19–44 (2001)
Rotman J.J.: An Introduction to Homological Algebra, Pure and Applied Mathematics, vol. 85. Academic Press, New York (1979)
Serre J.-P.: Représentations Linéaires des Groupes Finis. Hermann, Paris (1967)
Serre J.-P.: Semisimplicity and tensor products of group representations: converse theorems, with an appendix by Walter Feit. J. Algebra 194(2), 496–520 (1997)
Tange R.H.: The symplectic ideal and a double centraliser theorem. J. Lond. Math. Soc. 77(3), 687–699 (2008)
Wenzl H.: On the structure of Brauer’s centralizer algebras. Ann. Math. (2) 128(1), 173–193 (1988)
Weyl H.: The Classical Groups, Their Invariants and Representations, 2nd edn. Princeton University Press, Princeton (1946)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Donkin, S., Tange, R. The Brauer algebra and the symplectic Schur algebra. Math. Z. 265, 187–219 (2010). https://doi.org/10.1007/s00209-009-0510-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-009-0510-2