Stable branching rules for classical symmetric pairs

Transactions of the American Mathematical Society

Volumn 357, Issue 4, 2005, Pages 1601-1626

  Howe, Roger ^a   Tan, Eng Chye ^b   Willenbring, Jeb F ^c  

^a YALE UNIVERSITY   (United States)

^b NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

^c UNIVERSITY OF WISCONSIN MILWAUKEE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 16244382424     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-04-03722-5     Document Type: Article

References (51)

1. [Ab] J. Abramsky, Ph.D. thesis, The University of Southampton, England, 1969.
2. [BKW] G. Black, R. King, B. Wybourne, Kronecker products for compact semisimple Lie groups, J. Phys. A 16 (1983), no. 8, 1555-1589. MR0708193 (85e:22020)
3. [CGR] Y. Chen, A. Garsia and J. Remmel, Algorithms for plethysm in combinatorics and algebra, Contemp. Math. 24, AMS, Providence, R.I. (1984), 109-153. MR0777698 (86f:05010)
4. [EHW] T. Enright, R. Howe, and N. Wallach, Classification of unitary highest weight modules, in Representation Theory of Reductive Groups (P. Trombi, Ed.), pp. 97-144, Birkhäser, Boston, 1983. MR0733809 (86c:22028)
5. [EW1] T. Enright and J. Willenbring, Hilbert series, Howe duality, and branching rules, Proc. Nat. Acad. Sciences 100 (2003), No. 2, 434-437. MR1950645
6. [EW2] T. Enright and J. Willenbring, Hilbert series, Howe duality, and branching rules for classical groups, Ann. of Math. 159 (2004), No. 1, 337-375. MR2052357
7. [Fu] W. Fulton, Young Tableaux, Cambridge University Press, Cambridge, UK, 1997. MR1464693 (99f:05119)
8. [GW] R. Goodman and N. Wallach, Representations and Invariants of the Classical Groups, Cambridge U. Press, 1998. MR1606831 (99b:20073)
9. [GK] K. Gross and R. Kunze, Finite-dimensional induction and new results on invariants for classical groups, I, Amer. J. Math. 106 (1984), 893-974. MR0749261 (86h:22027)
10. [HC] Harish-Chandra, Representations of semisimple Lie groups IV/V/VI, Amer J. Math. 77 (1955), 743-777; Amer. J. Math. 78 (1956), 1-41; Amer J. Math. 78 (1956), 564-628. MR0072427 (17:282c); MR0082055 (18:490c); MR0082056 (18:490d)
11. [HC] Harish-Chandra, Representations of semisimple Lie groups IV/V/VI, Amer J. Math. 77 (1955), 743-777; Amer. J. Math. 78 (1956), 1-41; Amer J. Math. 78 (1956), 564-628. MR0072427 (17:282c); MR0082055 (18:490c); MR0082056 (18:490d)
12. [HC] Harish-Chandra, Representations of semisimple Lie groups IV/V/VI, Amer J. Math. 77 (1955), 743-777; Amer. J. Math. 78 (1956), 1-41; Amer J. Math. 78 (1956), 564-628. MR0072427 (17:282c); MR0082055 (18:490c); MR0082056 (18:490d)
13. [Ho1] R. Howe, Reciprocity laws in the theory of dual pairs, in Representation Theory of Reductive Groups, Prog. in Math. 40, P. Trombi, ed., Birkhäuser, Boston (1983), 159-175. MR0733812 (85k:22033)
14. [Ho2] R. Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), 539-570. MR0986027 (90h:22015a)
15. [Ho3] R. Howe, Transcending classical invariant theory, J. Amer. Math. Soc. 2 (1989), 535-552. MR0985172 (90k:22016)
16. [Ho4] R. Howe, Perspectives on Invariant Theory, The Schur Lectures, I. Piatetski-Shapiro and S. Gelbart (eds.), Israel Mathematical Conference Proceedings, 1995, 1-182. MR1321638 (96e: 13006)
17. [HTW] R. Howe, E-C. Tan and J. Willenbring, Reciprocity algebras and branching for classical symmetric pairs, in preparation.
18. [JK] G. James and A. Kerber, The Representation Theory of the Symmetric Group, Enc. of Math. and App. 16, Addison-Wesley, Reading, MA., 1981. MR0644144 (83k:20003)
19. [KV] M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Inv. Math. 44 (1978), 1-47. MR0463359 (57:3311)
20. [Ki1] R. King, Generalized Young tableaux and the general linear group, J. Math. Phys. 11 (1970), 280-293. MR0251972 (40:5197)
21. [Ki2] R. King, Modification rules and products of irreducible representations for the unitary, orthogonal, and symplectic groups, J. Math. Phys. 12 (1971), 1588-1598. MR0287816 (44:5019)
22. [Ki3] R. King, Branching rules for classical Lie groups using tensor and apinor methods, J. Phys. A 8 (1975), 429-449. MR0411400 (53:15136)
23. [Ki4] R. King, S-Functions and characters of Lie algebras and superalgebras, in Invariant Theory and Tableaux (D. Stanton, Ed.), IMA Vol. Math. Appl. 19, Springer, New York, 1990, 227-261. MR1035497 (90k:17014)
24. [KTW] R. King, F. Toumazet and B. G. Wybourne, Products and symmetrized powers of irreducible representations of SO*(2n), J. Phys. A 31 (1998), 6691-6705. MR1643812 (2000i:81043)
25. [KW] R. King and B. Wybourne, Holomorphic discrete series and harmonic series unitary representations of non-compact Lie groups: Sp(2n, ℝ), U(p, q) and SO*(2n), J. Phys. A 18 (1985), 3113-3139. MR0812427 (87b:22032)
26. [Kn1] A. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, Princeton, NJ, 1986. MR0855239 (87j:22022)
27. [Kn2] A. Knapp, Branching theorems for compact symmetric spaces, Represent. Theory 5 (2001), 404-436. MR1870596 (20021:20065)
28. [Kn3] A. Knapp, Lie Groups Beyond An Introduction, Second edition, Progress in Mathematics 140, Birkhäuser Boston, Inc., Boston, MA, 2002. MR1920389 (2003c:22001)
29. [Kn4] A. Knapp, Geometric interpretations of two branching theorems of D. E. Littlewood., J. Algebra 270 (2003), No. 2, 728-754. MR2019637
30. [Ko] Kazuhiko Koike, On the decomposition of tensor products of the representations of the classical groups, Adv. Math. 74 (1989), No. 1, 57-86. MR0991410 (90j:22014)
31. [KT] K. Koike and I. 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. MR1031827 (91a:22013)
32. [Ku] S. Kudla, Seesaw dual reductive pairs, in Automorphic Forms of Several Variables, Taniguchi Symposium, Katata, 1983, Birkhäuser, Boston (1983), 244-268. MR0763017 (86b:22032)
33. [Li1] D. Littlewood, On invariant theory under restricted groups, Philos. Trans. Roy. Soc. London. Ser. A., 239 (1944), 387-417. MR0012299 (7:6e)
34. [Li2] D. Littlewood, Theory of Group Characters, Clarendon Press, Oxford, 1945.
35. [Li3] D. Littlewood, Products and plethysms of characters with orthogonal, symplectic and symmetric groups, Can. Jour. Math. 10 (1958), 17-32. MR0095209 (20:1715)
36. [Ma] I. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1979. MR0553598 (84g:05003)
37. [Ne] M. Newell, Modification rules for the orthogonal and symplectic groups, Proc. Roy. Irish Acad. Sect. A. 54 (1951), 153-163. MR0043093 (13:204e)
38. [OZ] B. Orsted and G. Zhang, Tensor products of analytic continuations of holomorphic discrete series, Can. J. Math. 49 (1997), No. 6, 1224-1241. MR1611656 (2000b:22012)
39. [Pr] R. Proctor, Young tableaux, Gelfand patterns and branching rules for classical groups, J. Alg., Vol. 164 (1994), 299-360. MR1271242 (96e:05180)
40. [Re] J. Repka, Tensor product of holomorphic discrete series representations, Can. J. Math. 31, No. 4 (1979), 863-844. MR0540911 (82c:22017)
41. [RWB] D. Rowe, B. Wybourne and P. Butler, Unitary representations, branching rules and matrix elements for the non-compact symplectic group, J. Phys. A 18 (1985), 939-953. MR0792219 (87b:22044)
42. [Sa] B. Sagan, The Symmetric Group, Wadsworth and Cole, Pacific Grove, CA, 1991. MR1093239 (93f:05102)
43. [Sc1] W. Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Inv. Math. 9 (1969), 61-80. MR0259164 (41:3806)
44. [Sc2] W. Schmid, On the characters of the discrete series (the Hermitian symmetric case), Inv. Math. 30 (1975), 47-199. MR0396854 (53:714)
45. [St1] R. Stanley, The stable behaviour of some characters of SL(n,ℂ), Lin. and Mult. Alg., Vol. 16 (1984), 3-27. MR0T68993 (86e:22025)
46. [St2] R. Stanley, Enumerative Combinatorics, Vol. 1, Wadsworth and Cole, Monterey, CA, 1986. MR0847717 (87j:05003)
47. [St3] R. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, Cambridge, UK, 1999. MR1676282 (2000k:05026)
48. [Su] S. Sundaram, Tableaux in the representation theory of the classical Lie groups, in Invariant Theory and Tableaux (D. Stanton, Ed.), IMA Vol. Math. Appl. 19, Springer, New York, 1990, 191-225. MR1035496 (91e:22022)
49. [TTW] J-Y Thibon, F. Toumazet and B. Wybourne, Products and plethysms for the fundamental harmonic series representations of U(p, q), J. Phys. A 30 (1997), 4851-4856. MR1479810 (98k:81093)
50. [We] H. Weyl, The Classical Groups, Princeton Univ. Press, Princeton, 1946. MR0000255 (1:42c)
51. [Zh] D. Zhelobenko, Compact Lie Groups and their Representations, Transl. of Math. Mono. 40, AMS, Providence, R.I., 1973. MR0473098 (57:12776b)