Combinatorial extension of stable branching rules for classical groups
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Abstract:
We give new combinatorial formulas for decomposition of the tensor product of integrable highest weight modules over the classical Lie algebras of types $B, C, D$, and the branching decomposition of an integrable highest weight module with respect to a maximal Levi subalgebra of type $A$. This formula is based on a combinatorial model of classical crystals called spinor model. We show that our formulas extend in a bijective way various stable branching rules for classical groups to arbitrary highest weights, including the Littlewood restriction rules.References
- A. D. Berenstein and A. V. Zelevinsky, Tensor product multiplicities and convex polytopes in partition space, J. Geom. Phys. 5 (1988), no. 3, 453–472. MR 1048510, DOI 10.1016/0393-0440(88)90033-2
- Arkady Berenstein and Andrei Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math. 143 (2001), no. 1, 77–128. MR 1802793, DOI 10.1007/s002220000102
- Shun-Jen Cheng and Jae-Hoon Kwon, Howe duality and Kostant homology formula for infinite-dimensional Lie superalgebras, Int. Math. Res. Not. IMRN , posted on (2008), Art. ID rnn 085, 52. MR 2439556, DOI 10.1093/imrn/rnn085
- Shun-Jen Cheng, Jae-Hoon Kwon, and Weiqiang Wang, Kostant homology formulas for oscillator modules of Lie superalgebras, Adv. Math. 224 (2010), no. 4, 1548–1588. MR 2646304, DOI 10.1016/j.aim.2010.01.002
- Shun-Jen Cheng, Ngau Lam, and Weiqiang Wang, Super duality and irreducible characters of ortho-symplectic Lie superalgebras, Invent. Math. 183 (2011), no. 1, 189–224. MR 2755062, DOI 10.1007/s00222-010-0277-4
- Elizabeth Dan-Cohen, Ivan Penkov, and Vera Serganova, A Koszul category of representations of finitary Lie algebras, Adv. Math. 289 (2016), 250–278. MR 3439686, DOI 10.1016/j.aim.2015.10.023
- 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
- Roe Goodman and Nolan R. Wallach, Representations and invariants of the classical groups, Encyclopedia of Mathematics and its Applications, vol. 68, Cambridge University Press, Cambridge, 1998. MR 1606831
- Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, vol. 42, American Mathematical Society, Providence, RI, 2002. MR 1881971, DOI 10.1090/gsm/042
- Takahiro Hayashi, $q$-analogues of Clifford and Weyl algebras—spinor and oscillator representations of quantum enveloping algebras, Comm. Math. Phys. 127 (1990), no. 1, 129–144. MR 1036118, DOI 10.1007/BF02096497
- Roger Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), no. 2, 539–570. MR 986027, DOI 10.1090/S0002-9947-1989-0986027-X
- Roger Howe, Eng-Chye Tan, and Jeb F. Willenbring, Stable branching rules for classical symmetric pairs, Trans. Amer. Math. Soc. 357 (2005), no. 4, 1601–1626. MR 2115378, DOI 10.1090/S0002-9947-04-03722-5
- 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
- Kyeonghoon Jeong, Crystal bases for Kac-Moody superalgebras, J. Algebra 237 (2001), no. 2, 562–590. MR 1816704, DOI 10.1006/jabr.2000.8590
- V. G. Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96. MR 486011, DOI 10.1016/0001-8708(77)90017-2
- Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
- Joel Kamnitzer, The crystal structure on the set of Mirković-Vilonen polytopes, Adv. Math. 215 (2007), no. 1, 66–93. MR 2354986, DOI 10.1016/j.aim.2007.03.012
- M. Kashiwara, On crystal bases of the $Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465–516. MR 1115118, DOI 10.1215/S0012-7094-91-06321-0
- Masaki Kashiwara, On crystal bases, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 155–197. MR 1357199
- Masaki Kashiwara and Toshiki Nakashima, Crystal graphs for representations of the $q$-analogue of classical Lie algebras, J. Algebra 165 (1994), no. 2, 295–345. MR 1273277, DOI 10.1006/jabr.1994.1114
- 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, On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters, Adv. Math. 74 (1989), no. 1, 57–86. MR 991410, DOI 10.1016/0001-8708(89)90004-2
- 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
- Kazuhiko Koike and Itaru Terada, Young diagrammatic methods for the restriction of representations of complex classical Lie groups to reductive subgroups of maximal rank, Adv. Math. 79 (1990), no. 1, 104–135. MR 1031827, DOI 10.1016/0001-8708(90)90059-V
- Jae-Hoon Kwon, Littlewood identity and crystal bases, Adv. Math. 230 (2012), no. 2, 699–745. MR 2914963, DOI 10.1016/j.aim.2012.02.010
- Jae-Hoon Kwon, Super duality and crystal bases for quantum ortho-symplectic superalgebras, Int. Math. Res. Not. IMRN 23 (2015), 12620–12677. MR 3431632, DOI 10.1093/imrn/rnv076
- Jae-Hoon Kwon, Super duality and crystal bases for quantum ortho-symplectic superalgebras II, J. Algebraic Combin. 43 (2016), no. 3, 553–588. MR 3482440, DOI 10.1007/s10801-015-0646-6
- Cédric Lecouvey, Quantization of branching coefficients for classical Lie groups, J. Algebra 308 (2007), no. 1, 383–413. MR 2290928, DOI 10.1016/j.jalgebra.2006.08.028
- Cédric Lecouvey, Crystal bases and combinatorics of infinite rank quantum groups, Trans. Amer. Math. Soc. 361 (2009), no. 1, 297–329. MR 2439408, DOI 10.1090/S0002-9947-08-04480-2
- Cédric Lecouvey, Masato Okado, and Mark Shimozono, Affine crystals, one-dimensional sums and parabolic Lusztig $q$-analogues, Math. Z. 271 (2012), no. 3-4, 819–865. MR 2945586, DOI 10.1007/s00209-011-0892-9
- Peter Littelmann, Paths and root operators in representation theory, Ann. of Math. (2) 142 (1995), no. 3, 499–525. MR 1356780, DOI 10.2307/2118553
- 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
- Dudley E. Littlewood, Theory of Group Characters, Clarendon Press, Oxford, 1945.
- 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
- Toshiki Nakashima, Crystal base and a generalization of the Littlewood-Richardson rule for the classical Lie algebras, Comm. Math. Phys. 154 (1993), no. 2, 215–243. MR 1224078, DOI 10.1007/BF02096996
- M. J. Newell, Modification rules for the orthogonal and symplectic groups, Proc. Roy. Irish Acad. Sect. A 54 (1951), 153–163. MR 0043093
- Ivan Penkov and Konstantin Styrkas, Tensor representations of classical locally finite Lie algebras, Developments and trends in infinite-dimensional Lie theory, Progr. Math., vol. 288, Birkhäuser Boston, Boston, MA, 2011, pp. 127–150. MR 2743762, DOI 10.1007/978-0-8176-4741-4_{4}
- Steven V. Sam and Andrew Snowden, Stability patterns in representation theory, Forum Math. Sigma 3 (2015), Paper No. e11, 108. MR 3376738, DOI 10.1017/fms.2015.10
- 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
- Weiqiang Wang, Duality in infinite-dimensional Fock representations, Commun. Contemp. Math. 1 (1999), no. 2, 155–199. MR 1696098, DOI 10.1142/S0219199799000080
- Hans Wenzl, Quotients of representation rings, Represent. Theory 15 (2011), 385–406. MR 2801174, DOI 10.1090/S1088-4165-2011-00401-5
Additional Information
- Jae-Hoon Kwon
- Affiliation: Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea
- MR Author ID: 618315
- Email: jaehoonkw@snu.ac.kr
- Received by editor(s): December 12, 2015
- Received by editor(s) in revised form: October 1, 2016, and October 16, 2016
- Published electronically: February 1, 2018
- Additional Notes: This work was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1501-01.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 6125-6152
- MSC (2010): Primary 17B37, 22E46, 05E10
- DOI: https://doi.org/10.1090/tran/7104
- MathSciNet review: 3814326