An Multiplicity Rules and Schur Functions

In applications of Weyl character formula for AN Lie algebras, it can be shown that the followings are valid by the use of some properly chosen system of weights which we call fundamental weights. Characters can be attributed conveniently to Weyl orbits rather than representations. The classsical Schur function SN (x1, x2, .., xN ) of degree N can be defined to be character of the representation for completely symmetric tensor with N indices. Generalized Schur Functions Sq1,q2,..,qM (x1, x2, .., xN ) of the same degree are then defined by all partitions (q1, q2, .., qM ) with length N (=q1 + q2 + ..+ qM , N ≥ M). Weight multiplicities can be calculated from Weyl character formula by the aid of some reduction rules governing these Generalized Schur Functions. They are therefore called the multiplicity rules. This turns the problem of calculating weight multiplicities to a problem of solving linear system of equations so that the method works equally simple whatever the rank of algebra or the dimension of representation is big. It is therefore seen that the existence of multiplicity rules brings an ultimate solution to the problem of calculating weight multiplicities for AN Lie algebras and with some additional remarks the same will also be shown to be true for other finite Lie algebras.