Supersymmetric Schur Polynomials And Representation Of Lie Superalgebras

This thesis is a review of the representations and character formulas of the Lie superalgebra gl(m,n),the Schur-Weyl duality,and super Schur polynomials.The main arrangements of this article are as follows:In the first two chapters,we briefly review basic definitions and some examples of classical Lie superalgebras and recall some developments of the representations and character theories of Lie superalgebras.In the third chapter,we mainly discuss the Schur-Weyl duality of Lie superalgebra g[(m,n).Classical Schur-Weyl duality reveals the double commutant properties of the general linear Lie algebras gl(n)and symmetric groups Sl on vector sapces(Cn)(?)l,and provides a deep connection between the representations of symmetric groups Sl and polynomial representations of the Lie algebra gl(n).Then we can use the RobinsonSchensted-Knuth correspondence to index a basis for the irreducible polynomial representation of the general linear Lie superalgebras.In the last chapter,we introduce the super Schur polynomials to obtain a character formula for the finite-dimensional irreducible representation of the general linear Lie superalgebras.Super Schur polynomials can be defined in three different ways:as a quotient of altemants,using the Jacobi-Trudi identity and using semistandard tableaux.With Schur-Weyl duality,we can interpret the super Schur polynomials as the characters of polynomial representations of Lie superalgebra gl(m,n).