Last update: 11 September 2013
This section gives a brief summary of the classical symmetric function theory. See [Mac] Chapter 1 for a complete treatment.
Fix a positive integer A partition is a sequence of nonnegative integers. Let denote the set of partitions. We have the following sequence of inclusions There is a partial ordering, the dominance ordering, on elements of given by
Let denote the symmetric group. The sign of a permutation is the determinant of the corresponding permutation matrix. acts on elements of by permuting the positions.
For each the raising operator is the operator which acts on elements of by
Let be commuting variables and for each define Define an action of on monomials by The ring is the ring of symmetric functions.
Bases of symmetric functions
For each partition define the monomial symmetric function by where the sum runs over all in the orbit of i.e., over all distinct permutations of
The Schur functions are given by where
The elementary symmetric functions are given by defining for each positive integer and for all partitions
The homogeneous symmetric functions are given by defining for each positive integer and for all sequences
Define integers by One has the following (nontrivial) facts:
|(a)||The are nonnegative integers.|
Each of the sets forms a of
There is an inner product on the ring of symmetric functions given by making the Schur functions orthonormal,
The homogeneous symmetric functions are the dual basis to the basis of monomial symmetric functions, A consequence of this is that
One has the following "Jacobi-Trudi" formula for the Schur functions in terms of the homogeneous symmetric functions, There is also a formula for the Schur function in terms of raising operators and the homogeneous symmetric function.
This is an excerpt of a paper entitled Weyl Group Symmetric Functions and the Representation Theory of Lie Algebras, written by Arun Ram.