A_N Multiplicity Rules And Schur Functions

Abstract

We show that a specialization in Weyl character formula can be carried out in such a way that its right-hand side becomes simply a Schur Function. For this, we need the use of fundamental weights. In the generic definition, an Elementary Schur Function $S_Q(x_1,x_2,..,x_Q)$ of degree Q is known to be defined by some polynomial of Q indeterminates $ x_1,x_2,..,x_Q $. It is also known that definition of Elementary Schur Functions can be generalized in such a way that for any partition $(Q_k)$ of weight Q and length k one has a Generalized Schur Function $S_{(Q_k)}(x_1,x_2,..,x_Q)$. When they are considered for $A_{N-1}$ Lie algebras, a kind of degeneration occurs for these generic definitions. This is mainly due to the fact that, for an $A_{N-1}$ Lie algebra, only a finite number of indeterminates, namely (N-1), can be independent. This leads us to define {\bf Degenerated Schur Functions} by taking, for $Q > N-1$, all the indeterminates $x_Q$ to be non-linearly dependent on first (N-1) indeterminates $x_1,x_2,..,x_{N-1}$. With this in mind, we show that for each and every dominant weight of $A_{N-1}$ we always have a (Degenerated) Schur Function which provides the right-hand side of Weyl character formula. Generalized Schur Functions are known to be expressed by determinants of some matrices of Elementary Schur Functions. We would like to call these expressions {\bf multiplicity rules}. This is mainly due to the fact that, to calculate weight multiplicities, these rules give us an efficient method which works equally well no matter how big is the rank of algebras or the dimensions of representations.