Quotients of representation rings

Wenzl, Hans

doi:10.1090/S1088-4165-2011-00401-5

Quotients of representation rings
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by Hans Wenzl

Represent. Theory 15 (2011), 385-406

DOI: https://doi.org/10.1090/S1088-4165-2011-00401-5

Published electronically: May 3, 2011

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Abstract:

We give a proof using so-called fusion rings and $q$-deformations of Brauer algebras that the representation ring of an orthogonal or symplectic group can be obtained as a quotient of a ring $Gr(O(\infty ))$. This is obtained here as a limiting case for analogous quotient maps for fusion categories, with the level going to $\infty$. This in turn allows a detailed description of the quotient map in terms of a reflection group. As an application, one obtains a general description of the branching rules for the restriction of representations of $Gl(N)$ to $O(N)$ and $Sp(N)$ as well as detailed information about the structure of the $q$-Brauer algebras in the nonsemisimple case for certain specializations.

References

Henning Haahr Andersen, Tensor products of quantized tilting modules, Comm. Math. Phys. 149 (1992), no. 1, 149–159. MR 1182414, DOI 10.1007/BF02096627
Henning Haahr Andersen and Jan Paradowski, Fusion categories arising from semisimple Lie algebras, Comm. Math. Phys. 169 (1995), no. 3, 563–588. MR 1328736, DOI 10.1007/BF02099312
Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, Graduate Texts in Mathematics, Vol. 13, Springer-Verlag, New York-Heidelberg, 1974. MR 0417223, DOI 10.1007/978-1-4684-9913-1
Christian Blanchet, Hecke algebras, modular categories and $3$-manifolds quantum invariants, Topology 39 (2000), no. 1, 193–223. MR 1710999, DOI 10.1016/S0040-9383(98)00066-4
Anna Beliakova and Christian Blanchet, Modular categories of types $B$, $C$ and $D$, Comment. Math. Helv. 76 (2001), no. 3, 467–500. MR 1854694, DOI 10.1007/PL00000385
Brauer, R., On algebras which are connected with the semisimple continuous groups, Ann. of Math. 63 (1937), 854-872.
Pierre Deligne, La série exceptionnelle de groupes de Lie, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 4, 321–326 (French, with English and French summaries). MR 1378507
Thomas J. Enright and Jeb F. Willenbring, Hilbert series, Howe duality and branching for classical groups, Ann. of Math. (2) 159 (2004), no. 1, 337–375. MR 2052357, DOI 10.4007/annals.2004.159.337
Cox, A., De Visscher, M., Martin, P., Alcove geometry and a translation principle for the Brauer algebra. arXiv:0807.3892
Philippe Di Francesco, Pierre Mathieu, and David Sénéchal, Conformal field theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York, 1997. MR 1424041, DOI 10.1007/978-1-4612-2256-9
J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), no. 1, 1–34. MR 1376244, DOI 10.1007/BF01232365
Frederick M. Goodman and Holly Hauschild Mosley, Cyclotomic Birman-Wenzl-Murakami algebras. I. Freeness and realization as tangle algebras, J. Knot Theory Ramifications 18 (2009), no. 8, 1089–1127. MR 2554337, DOI 10.1142/S0218216509007397
Frederick M. Goodman and Hans Wenzl, A path algorithm for affine Kazhdan-Lusztig polynomials, Math. Z. 237 (2001), no. 2, 235–249. MR 1838309, DOI 10.1007/PL00004866
Hu, J., BMW algebra, quantized coordinate algebra and type $C$ Schur-Weyl duality, Representation Theory 15 (2011), 1–62.
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras. I, II, J. Amer. Math. Soc. 6 (1993), no. 4, 905–947, 949–1011. MR 1186962, DOI 10.1090/S0894-0347-1993-99999-X
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
Victor G. Kac and Dale H. Peterson, Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. in Math. 53 (1984), no. 2, 125–264. MR 750341, DOI 10.1016/0001-8708(84)90032-X
Christian Kassel, Quantum groups, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995. MR 1321145, DOI 10.1007/978-1-4612-0783-2
R. C. King, Modification rules and products of irreducible representations of the unitary, orthogonal, and symplectic groups, J. Mathematical Phys. 12 (1971), 1588–1598. MR 287816, DOI 10.1063/1.1665778
Kazuhiko Koike and Itaru Terada, Young-diagrammatic methods for the representation theory of the classical groups of type $B_n,\;C_n,\;D_n$, J. Algebra 107 (1987), no. 2, 466–511. MR 885807, DOI 10.1016/0021-8693(87)90099-8
Robert Leduc and Arun Ram, A ribbon Hopf algebra approach to the irreducible representations of centralizer algebras: the Brauer, Birman-Wenzl, and type A Iwahori-Hecke algebras, Adv. Math. 125 (1997), no. 1, 1–94. MR 1427801, DOI 10.1006/aima.1997.1602
D. E. Littlewood, On invariant theory under restricted groups, Philos. Trans. Roy. Soc. London Ser. A 239 (1944), 387–417. MR 12299, DOI 10.1098/rsta.1944.0003
I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
Martin, P., The decomposition matrices of the Brauer algebra over the complex field. arXiv:0908.1500
Morrison, S., Peters, E., Snyder, N., Knot polynomial identities and quantum group coincidences, arXiv:1003.0022
Arun Ram, A “second orthogonality relation” for characters of Brauer algebras, European J. Combin. 18 (1997), no. 6, 685–706. MR 1468338, DOI 10.1006/eujc.1996.0132
Arun Ram and Hans Wenzl, Matrix units for centralizer algebras, J. Algebra 145 (1992), no. 2, 378–395. MR 1144939, DOI 10.1016/0021-8693(92)90109-Y
Wolfgang Soergel, Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting modules, Represent. Theory 1 (1997), 83–114. MR 1444322, DOI 10.1090/S1088-4165-97-00021-6
Wolfgang Soergel, Charakterformeln für Kipp-Moduln über Kac-Moody-Algebren, Represent. Theory 1 (1997), 115–132. MR 1445716, DOI 10.1090/S1088-4165-97-00017-4
Sheila Sundaram, Tableaux in the representation theory of the classical Lie groups, Invariant theory and tableaux (Minneapolis, MN, 1988) IMA Vol. Math. Appl., vol. 19, Springer, New York, 1990, pp. 191–225. MR 1035496
Imre Tuba and Hans Wenzl, On braided tensor categories of type $BCD$, J. Reine Angew. Math. 581 (2005), 31–69. MR 2132671, DOI 10.1515/crll.2005.2005.581.31
V. G. Turaev, The Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988), no. 3, 527–553. MR 939474, DOI 10.1007/BF01393746
Turaev, V., Quantum invariants, DeGruyter.
V. Turaev and H. Wenzl, Quantum invariants of $3$-manifolds associated with classical simple Lie algebras, Internat. J. Math. 4 (1993), no. 2, 323–358. MR 1217386, DOI 10.1142/S0129167X93000170
Vladimir Turaev and Hans Wenzl, Semisimple and modular categories from link invariants, Math. Ann. 309 (1997), no. 3, 411–461. MR 1474200, DOI 10.1007/s002080050120
Antony Wassermann, Operator algebras and conformal field theory. III. Fusion of positive energy representations of $\textrm {LSU}(N)$ using bounded operators, Invent. Math. 133 (1998), no. 3, 467–538. MR 1645078, DOI 10.1007/s002220050253
Hans Wenzl, Hecke algebras of type $A_n$ and subfactors, Invent. Math. 92 (1988), no. 2, 349–383. MR 936086, DOI 10.1007/BF01404457
Hans Wenzl, Quantum groups and subfactors of type $B$, $C$, and $D$, Comm. Math. Phys. 133 (1990), no. 2, 383–432. MR 1090432, DOI 10.1007/BF02097374
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, Braids and invariants of $3$-manifolds, Invent. Math. 114 (1993), no. 2, 235–275. MR 1240638, DOI 10.1007/BF01232670
Hans Wenzl, $C^*$ tensor categories from quantum groups, J. Amer. Math. Soc. 11 (1998), no. 2, 261–282. MR 1470857, DOI 10.1090/S0894-0347-98-00253-7
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
Changchang Xi, On the quasi-heredity of Birman-Wenzl algebras, Adv. Math. 154 (2000), no. 2, 280–298. MR 1784677, DOI 10.1006/aima.2000.1919