Abstract
In the theory of finite-dimensional representations of complex reductive algebraic groups, the group GL n (ℂ) is singled out by the fact that besides the usual language of weight lattices, roots, and characters, there exists an additional important combinatorial tool: the Young tableaux. For example, the sum over the weights of all tableaux of a fixed shape is the character of the corresponding representation, and the Littlewood-Richardson rule describes the decomposition of tensor products of GL n (ℂ)-modules purely in terms of the combinatoric of these Young tableaux. The advantage of this type of formula is (for example compared to Steinberg’s formula to decompose tensor products) that there is no cancellation of terms. This makes it much easier (and sometimes even possible) to prove for example that certain representations occur in a given tensor product.
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References
A. Berele, A Schensted-type correspondence for the symplectic group, J. Combin. Theory Ser. A, 43 (1986), 320–328.
A. D. Berenstein and A. V. Zelevinsky, Triple multiplicities for sl r +1 and the spectrum of the exterior algebra of the adjoint representation, J. Alg. Combinatorics, 1 (1992), 7–22.
W. Fulton and J. Harris, Representation theory, Graduate Texts in Math., Springer Verlag, Berlin and New York, (1991).
I. M. Gelfand and M. L. Zetlin, Finite dimensional representations of the group of unimodular matrices, Dokl. Akad. Nauk. USSR, 71 (1950), 825–828.
T. Joseph, Quantum Groups and Their Primitive Ideals, Springer Verlag, Berlin and New York, to appear.
M. Kashiwara, Crystalizing the q-analogue of Universal Enveloping algebras, Commun. Math. Phys., 133 (1990), 249–260.
M. Kashiwara, Crystal bases of modified quantized enveloping algebras, RIMS, 917 (1993).
R. C. King, Weight multiplicities for the classical groups, Group Theoretical methods in Physics, Lecture Notes in Phys., (Janner, Jannssen, and Boon, eds.), Springer Verlag, Berlin and New York, 50 (1976).
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. in Math., 79 (1990), 104–135.
S. Kumar, Proof of the Parthasarathy-Ranga-Rao-Varadarajan Conjecture, Invent. Math., 93 (1988), 117–130.
V. Lakshrnibai and C. S. Seshadri, Geometry of G/P V, J. Algebra, 100 (1986), 462–557.
V. Lakshmibai and C. S. Seshadri, Standard monomial theory, in: Proceedings of the Hyderabad Conference on algebraic groups, Manoj Prakashan, Madras, (1991), 279–323.
P. Littelmann, A generalization of the Littlewood-Richardson rule, J. Algebra, 130 (1990), 328–368.
P. Littelmann, A Littlewood-Richardson type rule for symmetrizable Kac-Moody algebras, Invent. Math., 116 (1994), 329–346.
P. Littelmann, Paths and root operators in representation theory, preprint, to appear in: Ann. of Math. (2) (1993).
D. E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups, Oxford University Press, London, (1950).
D. E. Littlewood and A. R. Richardson, Group characters and algebra, Philos. Trans. Roy. Soc. London Ser. A, 233 (1934), 99–141.
G. Lusztig, Canonical bases arising from quantized enveloping algebras II, Progr. Theoret. Phys., 102 (1990), 175–201.
G. Lusztig, Introduction to quantum groups, Birkhäuser Verlag, Boston, Progr. Math., 110 (1993).
I. G. Macdonald, Symmetric Functions and Hall polynomials, Oxford University Press, London, (1979).
O. Mathieu, Construction d’un groupe de Kac-Moody et applications, Compos. Math., 69 (1989), 37–60.
T. Nakashima, Crystal base and a generalization of the Littlewood-Richardson rule for the classical Lie algebras, preprint, RIMS, 783 (1991).
R. A. Proctor, A Schensted algorithm which models tensor representations of the orthogonal group, Canad. J. Math., 42 (1990), 28–49.
G. B. Robinson, On the representation of the symmetric group, Amer. J. Math., 69 (1938), 745–760.
M. P. Schutzenberger, La correspendance de Robinson, Combinatoire et représentation du groupe symmetrique, Lecture Notes in Math., Springer Verlag, Berlin and New York, 579 (1976).
S. Sundaram, Orthogonal tableaux and an insertion algorithm for SO(2n + 1), J. Combin. Theory Ser. A, 53 (1990), 239–256.
H. Weyl, The classical groups, their invariants and representations, Princeton University Press, Princeton, (1946).
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© 1995 Birkhäser Verlag, Basel, Switzerland
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Littelmann, P. (1995). The Path Model for Representations of Symmetrizable Kac-Moody Algebras. In: Chatterji, S.D. (eds) Proceedings of the International Congress of Mathematicians. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-9078-6_23
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DOI: https://doi.org/10.1007/978-3-0348-9078-6_23
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