The structure of r-fold tensor products of irreducible tame representations of U(∞)=lim lim U(n) are described, versions of contragredient representations and invariants are realized, and methods of calculating multiplicities, Clebsch–Gordan, and Racah coefficients are given using invariant theory on Bargmann–Segal–Fock spaces.

1.
I.
Segal
, “
The structure of a class of representations of the unitary group on a Hilbert space
,”
Proc. Am. Math. Soc.
8
,
197
203
(
1957
).
2.
A.
Kirillov
, “
Representations of an infinite dimensional unitary group
,”
Dodkl. Akad. Nauk SSSR
14
,
1355
1358
(
1973
).
3.
S. Stratila and D. Voiculescu, “Representations of AF Algebras and the Group U(∞),” Lecture Notes in Mathematics Vol. 486 (Springer, New York, 1975).
4.
D.
Pickrell
, “
Decompositions of regular representations of U(H),
Pacific J. Math.
128
,
319
332
(
1987
).
5.
G. I. Ol’shanski, “Representations of infinite-dimensional classical groups, limits of enveloping algebras, and Yangians,” in Topics in Representation Theory, Advances in Soviet Mathematics, edited by A. A. Kirillov (American Mathematical Society, Providence, RI, 1991), Vol. 2, pp. 1–66.
6.
G. I.
Ol’shanski
, “
The method of holomorphic extensions in the theory on unitary representations of infinite dimensional classical groups
,”
Funct. Anal. Appl.
22
,
273
285
(
1989
).
7.
G. I. Ol’shanski, “Unitary representations of infinite-dimensional pairs (G, K) and the formalism of R. Howe,” in Representations of Lie Groups and Related Topics, edited by A. M. Vershik and D. P. Zelobenko (Gordon and Breach, New York, 1991).
8.
I. M. Gelfand and M. I. Graev, “Principal representations of the group U(∞),” Ref. 7.
9.
V. Kac, “Highest weight representations of infinite dimensional Lie algebras,” in Proceedings of the International Congress of Mathematicians, Helsinki, 1978, pp. 1355–1358.
10.
W. H.
Klink
and
T.
Ton-That
, “
Multiplicity, invariants, and tensor products of compact groups
,”
J. Math. Phys.
37
,
6468
6485
(
1996
).
11.
T. Ton-That, “Reciprocity theorems for holomorphic representations of some infinite dimensional groups,” Helv. Phys. Acta (to be published).
12.
T. Ton-That, “Invariant theory for tame representations of infinite dimensional classical groups,” AMS Abstract of the Special Session on Invariant Theory, AMS Meeting No. 924, Montreal, Canada, September 1997 (unpublished).
13.
D. P. Zhelobenko, Compact Lie Groups and their Representations, Translations of Mathematical Monographs, Vol. 40 (American Mathematical Society, Providence, RI, 1973).
14.
H. Weyl, The Classical Groups, Their Invariants and Representations, 2nd ed. (Princeton University Press, Princeton, NJ, 1946).
15.
R. M.
Howe
, “
Dual representations of gl(∞) and decomposition of Fock spaces
,”
J. Phys. A
30
,
2757
2781
(
1997
).
16.
I. M. Gelfand and G. E. Shilov, Generalized Functions (Academic, New York, 1988), Vol. 4.
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