A relationship between Gelfand-Tsetlin bases and Chari-Loktev bases for irreducible finite dimensional representations of special linear Lie algebras

Abstract

We consider two bases for an arbitrary finite dimensional irreducible representation of a complex special linear Lie algebra: the classical Gelfand-Tsetlin basis and the relatively new Chari-Loktev basis. Both are parametrized by the set of (integral Gelfand-Tsetlin) patterns with a fixed bounding sequence determined by the highest weight of the representation. We define the "row-wise dominance" partial order on this set of patterns, and prove that the transition matrix between the two bases is triangular with respect to this partial order. We write down explicit expressions for the diagonal elements of the transition matrix.

8 pages; 1 figure; to appear in Proceedings of the International Conf. on Algebra, Discrete Mathematics and Applications held at Aurangabad, Maharashtra, India, during 9--11 Dec 2017; the last section of arXiv:math/1606.01191v1 (but not its journal variant) contains a preliminary version of this article; the present posting incorporates a substantial reworking; version 2: corrected a reference