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The Polynomial Behavior of Weight Multiplicities for the Affine Kac–Moody Algebras A(1)r

Published online by Cambridge University Press:  04 December 2007

Georgia Benkart ,
Seok-Jin Kang ,
Hyeonmi Lee ,
Kailash C. Misra  and
Dong-Uy Shin
Georgia Benkart
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706-1388, U.S.A. E-mail: [email protected]
Seok-Jin Kang
Affiliation:
Department of Mathematics, Seoul National University, Seoul 151-742, Korea. E-mail: [email protected]
Hyeonmi Lee
Affiliation:
Department of Mathematics, Seoul National University, Seoul 151-742, Korea. E-mail: [email protected]
Kailash C. Misra
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, U.S.A. E-mail: [email protected]
Dong-Uy Shin
Affiliation:
Department of Mathematics, Seoul National University, Seoul 151-742, Korea. E-mail: [email protected]
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Abstract

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We prove that the multiplicity of an arbitrary dominant weight for an irreducible highest weight representation of the affine Kac–Moody algebra A(1)r is a polynomial in the rank r. In the process we show that the degree of this polynomial is less than or equal to the depth of the weight with respect to the highest weight. These results allow weight multiplicity information for small ranks to be transferred to arbitrary ranks.

Keywords

affine Kac–Moody algebrahighest weight representationsweight multiplicities
Type
Research Article
Information
Compositio Mathematica , Volume 126 , Issue 1 , March 2001 , pp. 91 - 111
Copyright
© 2001 Kluwer Academic Publishers