On the Various Realizations of the Basic Representation of A n - 1(1) and the Combinatorics of Partitions
Séverine Leidwanger
DOI: 10.1023/A:1011985828993
Abstract
The infinite dimensional Lie algebra 
l n = A n-1 (1) can be realized in several ways as an algebra of differential operators. The aim of this note is to show that the intertwining operators between the realizations of l n corresponding to all partitions of n can be described very simply by using combinatorial constructions.
l n = A n-1 (1) can be realized in several ways as an algebra of differential operators. The aim of this note is to show that the intertwining operators between the realizations of l n corresponding to all partitions of n can be described very simply by using combinatorial constructions.Pages: 133–144
Keywords: Lie algebras; representation theory; combinatorics of partitions; symmetric functions
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References
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2. G. James and A. Kerber, The Representation Theory of the Symmetric Group, Addison Wesley, Reading, MA, 1981.
3. V.G. Kac, Infinite Dimensional Lie Algebras, 3rd ed., Cambridge, UK, 1990.
4. F. ten Kroode and J. Van de Leur, “Bosonic and fermionic realizations of the affine algebra \hat gln,” Commun. Math. Phys. 137 (1991), 67-107.
5. V.G. Kac and D.H. Peterson, “112 Constructions of the basic representation of the loop group of E8,” Proceedings of the Conference “Anomalies, geometry, topology” Argonne,
1985. World Scientific, Singapore, 1985, pp. 276-298.
6. V.G. Kac, D.A. Kazhdan, J. Lepowsky, and R.L. Wilson, “Realization of the basic representations of the euclidean Lie algebras,” Adv. Math. 42 (1981), 83-112.
7. V.G. Kac and A.K. Raina, “Bombay lectures on highest weight representations of infinite dimensional Lie algebras,” World Scientific, Singapore, 1987. ( (
8. S. Leidwanger, “Basic representations of A 1) and A 2) and the combinatorics of partitions,” Adv. Math. 141 n - 1 2n (1999), 119-154.
9. I.G. Macdonald, Symmetric Functions and Hall polynomials, 2nd ed., Oxford, UK, 1995.
10. T. Muir, A Treatise on the Theory of Determinants, Macmillan, London, 1882.
11. J.B. Olsson, “Combinatorics and Representations of Finite Groups,” Vorlesungen aus dem Fachbereich Mathematik der Universitat GH Essen Heft 20 (1993).