Last update: 11 September 2013
Each of the simple root systems is determined by a Cartan matrix A list of the Cartan matrices for simple root systems can be found in [Bou1981] p. 250-258. We shall denote the entry of the Cartan matrix by so that Let be the dimension of the Cartan matrix.
Let be basis vectors in a vector space. Define where denotes the nonnegative integers. The elements of and are called the weights, the integral weights, and the dominant integral weights, respectively. The are called the fundamental weights. We have the following sequence of inclusions
Let be an element of We shall use the notation for the integer The simple roots are given in terms of the entries of the Cartan matrix, There is a partial ordering on the weight lattice given by for nonnegative integers We say that in dominance.
Define linear operators by The Weyl group is the group generated by the The sign of an element is where is the smallest nonnegative integer such that there exists an expression We will need the following proposition, see [Bou1981] Ch. 6, §1 Thm. 2.
|(a)||Every Weyl group orbit contains a unique element in|
|(b)||If and then, for|
is a root if for some and simple root Let be the set of roots and let and where the ordering is as in (2.2). It is true that The elements of and are called positive and negative roots respectively. The raising operator associated to a positive root is the operator which acts on elements of by
Corresponding to each we write, formally, so that In particular if then (If one finds this "exponential" notation unsettling one can substitute for and write instead of Define an action of the Weyl group by for each and Define
For each define the orbit sum, or monomial symmetric function, by
For each define the Weyl character by where
The elementary, or fundamental, symmetric functions are given by defining for each positive integer and for all elements in
Define integers by the identity It is true that
|(a)||The are nonnegative integers.|
Each of the sets forms a of The fact that the form a basis of follows immediately from (2.3). One can show by elementary techniques and without using representation theory, see [Bou1981] Ch. 6, §3, that the form a basis of This fact also follows from the facts about the numbers above. Assuming that the form a of it follows that the form a of simply by expanding in terms of One can also obtain this result in a different fashion by using representation theory.
Let where If then define Let denote taking the coefficient of the identity, in Then define where is the order of the Weyl group.
([Mac1991]) The inner product defined above satisfies
Since This is zero if because, (2.3), the orbits and do not intersect. If then Thus
If one prefers one may simply define the inner product by making the Weyl characters orthonormal.
Homogeneous symmetric functions
Let and define Since is always finite is always finite dimensional.
Define an inner product on by defining for all Then define the homogeneous symmetric functions to be the dual basis to the monomial symmetric functions, Using the integers defined in (2.10), the homogeneous symmetric functions are given in terms of the Weyl characters by for all The form a basis of
Each of the sets forms a of To see this choose some total ordering of the elements of which is a refinement of the dominance partial order. Then, by (2.10a-c), the matrix, with rows and columns indexed by elements of having as the entry is upper unitriangular with nonnegative integer entries. This implies that it is invertible as a matrix with integer entries. The fact that the are a basis of is by definition. The other two statements now follow from (2.10) and (2.13).
Fix Define for all and all so that is defined for all One has the following "Jacobi-Trudi" type identity for the Weyl characters in terms of the
Let Then for each
We show that elements are the dual basis to the basis Expanding by (2.8) and clearing denominators we have that Substitute to get Compare coefficients of for on each side of this equation. Since we know by (2.3) that is not an element of for any except the identity. Thus we know that if then
Recall that raising operators act on elements of We allow the raising operators to act upon the by defining for each sequence We use the convention that if (Note: It is important to keep in mind that raising operators act on elements of and not on symmetric functions.)
A sketch of the proof is as follows. Evaluating the right hand side of the above we get where An element in is called regular if for all The sets and are equal. This is proved by expressing in the form and using [Bou1981] Ch. 6, §1 Cor. 2. Under this bijection The terms arising from the subsets for which is not regular cancel with each other. This can be shown by showing that is skew-symmetric with respect to and that if is not regular then These arguments show that
The proof of Cor. (2.15) was motivated by the proof of the Weyl denominator formula given in [Mac1991].
The above definition defines an analogue of homogeneous symmetric functions for the spaces One would like to say that in some sense the are well defined on all of With this in mind we introduce the following.
For each pair such that define a linear map by It is clear that
|2)||For each is the identity on|
This is an excerpt of a paper entitled Weyl Group Symmetric Functions and the Representation Theory of Lie Algebras, written by Arun Ram.