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The second fundamental theorem of invariant theory for the orthogonal group. (English) Zbl 1263.20043

Let \(V\) be a complex, finite-dimensional vector space of dimension \(n\) with an orthogonal form and let \(O(V)\) be the corresponding orthogonal group. There is an associated algebra \(B_r(n)\) called the \(r\)-string Brauer algebra, and a surjective homomorphism \(\nu\colon B_r(n)\to\text{End}_{O(V)}(V^{\otimes r})\) by a result of R. Brauer [Ann. Math. (2) 38, 857-872 (1937; Zbl 0017.39105)]. In the present paper, the authors show that the kernel of \(\nu\) is generated by a single idempotent element \(E\in B_r(n)\) which they describe explicitly.

MSC:

20G05 Representation theory for linear algebraic groups
14L24 Geometric invariant theory
20C08 Hecke algebras and their representations
16G20 Representations of quivers and partially ordered sets
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[1] M. Atiyah, R. Bott, and V. K. Patodi, ”On the heat equation and the index theorem,” Invent. Math., vol. 19, pp. 279-330, 1973. · Zbl 0257.58008 · doi:10.1007/BF01425417
[2] J. S. Birman and H. Wenzl, ”Braids, link polynomials and a new algebra,” Trans. Amer. Math. Soc., vol. 313, iss. 1, pp. 249-273, 1989. · Zbl 0684.57004 · doi:10.2307/2001074
[3] R. Brauer, ”On algebras which are connected with the semisimple continuous groups,” Ann. of Math., vol. 38, iss. 4, pp. 857-872, 1937. · Zbl 0017.39105 · doi:10.2307/1968843
[4] C. de Concini and C. Procesi, ”A characteristic free approach to invariant theory,” Advances in Math., vol. 21, iss. 3, pp. 330-354, 1976. · Zbl 0347.20025 · doi:10.1016/S0001-8708(76)80003-5
[5] W. F. Doran IV, D. B. Wales, and P. J. Hanlon, ”On the semisimplicity of the Brauer centralizer algebras,” J. Algebra, vol. 211, iss. 2, pp. 647-685, 1999. · Zbl 0944.16002 · doi:10.1006/jabr.1998.7592
[6] J. J. Graham and G. I. Lehrer, ”Cellular algebras,” Invent. Math., vol. 123, iss. 1, pp. 1-34, 1996. · Zbl 0853.20029 · doi:10.1007/BF01232365
[7] J. J. Graham and G. I. Lehrer, ”Diagram algebras, Hecke algebras and decomposition numbers at roots of unity,” Ann. Sci. École Norm. Sup., vol. 36, iss. 4, pp. 479-524, 2003. · Zbl 1062.20003 · doi:10.1016/S0012-9593(03)00020-X
[8] J. J. Graham and G. I. Lehrer, ”Cellular algebras and diagram algebras in representation theory,” in Representation Theory of Algebraic Groups and Quantum Groups, Tokyo: Math. Soc. Japan, 2004, vol. 40, pp. 141-173. · Zbl 1135.20302
[9] J. Hu and Z. Xiao, ”On tensor spaces for Birman-Murakami-Wenzl algebras,” J. Algebra, vol. 324, iss. 10, pp. 2893-2922, 2010. · Zbl 1272.17019 · doi:10.1016/j.jalgebra.2010.08.017
[10] V. F. R. Jones, ”Hecke algebra representations of braid groups and link polynomials,” Ann. of Math., vol. 126, iss. 2, pp. 335-388, 1987. · Zbl 0631.57005 · doi:10.2307/1971403
[11] J. Loday, Cyclic Homology, Second ed., New York: Springer-Verlag, 1998, vol. 301. · Zbl 0928.19001
[12] G. I. Lehrer and R. B. Zhang, ”Strongly multiplicity free modules for Lie algebras and quantum groups,” J. Algebra, vol. 306, iss. 1, pp. 138-174, 2006. · Zbl 1169.17003 · doi:10.1016/j.jalgebra.2006.03.043
[13] G. I. Lehrer and R. B. Zhang, ”A Temperley-Lieb analogue for the BMV algebra,” in Representation Theory of Algebraic Groups and Quantum Groups, New York: Springer-Verlag, 2010, vol. 284, pp. 155-190. · Zbl 1260.16030 · doi:10.1007/978-0-8176-4697-4_7
[14] G. I. Lehrer and R. Zhang, ”On endomorphisms of quantum tensor space,” Lett. Math. Phys., vol. 86, iss. 2-3, pp. 209-227, 2008. · Zbl 1213.17015 · doi:10.1007/s11005-008-0284-1
[15] G. I. Lehrer, H. Zhang, and R. B. Zhang, ”A quantum analogue of the first fundamental theorem of classical invariant theory,” Comm. Math. Phys., vol. 301, iss. 1, pp. 131-174, 2011. · Zbl 1262.17008 · doi:10.1007/s00220-010-1143-3
[16] G. Lusztig, Introduction to Quantum Groups, Boston, MA: Birkhäuser, 1993, vol. 110. · Zbl 0788.17010
[17] C. Procesi, ”150 years of invariant theory,” in The Heritage of Emmy Noether, Ramat Gan: Bar-Ilan Univ., 1999, vol. 12, pp. 5-21. · Zbl 0931.15021
[18] C. Procesi, Lie Groups. An Approach Through Invariants and Representations, New York: Springer-Verlag, 2007. · Zbl 1154.22001
[19] C. Procesi, ”The invariant theory of \(n\times n\) matrices,” Advances in Math., vol. 19, iss. 3, pp. 306-381, 1976. · Zbl 0331.15021 · doi:10.1016/0001-8708(76)90027-X
[20] D. R. Richman, ”The fundamental theorems of vector invariants,” Adv. in Math., vol. 73, iss. 1, pp. 43-78, 1989. · Zbl 0668.15016 · doi:10.1016/0001-8708(89)90059-5
[21] H. Rui and M. Si, ”A criterion on the semisimple Brauer algebras. II,” J. Combin. Theory Ser. A, vol. 113, iss. 6, pp. 1199-1203, 2006. · Zbl 1108.16009 · doi:10.1016/j.jcta.2005.09.005
[22] H. Weyl, The Classical Groups, Princeton, NJ: Princeton Univ. Press, 1997. · Zbl 1024.20501
[23] C. Xi, ”On the quasi-heredity of Birman-Wenzl algebras,” Adv. Math., vol. 154, iss. 2, pp. 280-298, 2000. · Zbl 0971.16008 · doi:10.1006/aima.2000.1919
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