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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: benkart@math.wisc.edu
Seok-Jin Kang
Affiliation:
Department of Mathematics, Seoul National University, Seoul 151-742, Korea. E-mail: sjkang@math.snu.ac.kr
Hyeonmi Lee
Affiliation:
Department of Mathematics, Seoul National University, Seoul 151-742, Korea. E-mail: hmlee@math.snu.ac.kr
Kailash C. Misra
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, U.S.A. E-mail: misra@math.ncsu.edu
Dong-Uy Shin
Affiliation:
Department of Mathematics, Seoul National University, Seoul 151-742, Korea. E-mail: dushin@math.snu.ac.kr
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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