Abstract
One of the most important classes of Lie algebras is sl_n, which are the n×n matrices with trace 0. The representation theory for sl_n has been an interesting research area for the past hundred years and in it, the simple finite-dimensional modules have become very important. They were classified and Gelfand and Tsetlin actually gave an explicit construction of a basis for every simple finite-dimensional module. This paper extends their work by providing theorems and proofs and constructs monomial bases of the simple module.
Highlights
Let be a Lie algebra of all matrices of order
We work with finite-dimensional modules and finite-dimensional representation of
Choose integers, such that the inequality is satisfied. These partitions are quite important because they appear to be the core in constructing representations. These chosen integers are used to construct some index set
Summary
Let be a Lie algebra of all matrices of order. In this paper, we work with finite-dimensional modules and finite-dimensional representation of. Choose integers , , , such that the inequality is satisfied These partitions are quite important because they appear to be the core in constructing representations. L. Tsetlin gave an explicit construction of a basis for every simple finitedimensional module of. Tsetlin gave an explicit construction of a basis for every simple finitedimensional module of In their work, they gave all the irreducible representations of general linear algebra ( ). E. Ramirez provided a classification and explicit bases of tableaux of all irreducible generic Gelfand-Tsetlin modules for the Lie algebra [2]. This paper will show that the Gelfand-Tsetlin constructions given in the year [1] forms all the irreducible representations of special linear algebra by providing proofs to results.
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