Last update: 11 September 2013
Fix a Cartan matrix and define to be the associative algebra (over with generated by with relations (Serre relations) Here and
Let Let be a module. A vector is called a weight vector if, for each for some constant We associate to the weight
If is a weight vector of weight then is a weight vector of weight This is the motivation for the definition of the raising operators This also gives the motivation for the definition of the dominance partial order as if and only if for some sequence of raising operators
The dominant weights appear as a result of the following fact.
(see [Hum1972]) There is a unique finite dimensional irreducible representation of corresponding to each dominant weight This irreducible representation is characterized by the fact that it contains a unique vector, up to scalar multiples, which is a weight vector of weight
(Weyl) Every finite dimensional representation of (corresponding to a simple Cartan matrix) is completely decomposable as a direct sum of irreducible representations
The motivation for Weyl group symmetric functions is that they are the generating functions of the weights in these representations. Let be a finite dimensional Let be a basis of such that each element of is a weight vector. Then the character of is the generating function of the weights of
The irreducibles combinatorially.
In this section we shall define the vector spaces affording the irreducible representations in a combinatorial fashion to motivate the Weyl group, the Weyl characters and the monomial symmetric functions.
Recall the notations and Let be letters and suppose that they satisfy the relations (We should not be surprised by the appearance of the values as it is clear that our basic object, the Cartan matrix, must come into the picture in a crucial way.) We shall study words in the letters Let be a dummy letter marking the end of a word. Fix a dominant integral weight Define to be the vector space which is a linear span of words in the ending with with the additional relations
Given that the vectors of the form span it is possible to choose a basis of of vectors of this form. Define the weight of a word to be where the are simple roots. Define the Weyl character to be the generating function of the weights of the basis (This is analogous to the combinatorial definition of the Schur functions in terms of column strict tableaux.) The definition of the Weyl characters is motivated by the following theorem.
(Weyl character formula) For each where
If we define for the generators of then we can use the defining relations for to rewrite the expressions of the form and as elements of In this way we get an action of on is an irreducible
Let and define Define It is clear from the definitions that
|(a)||The are nonnegative integers.|
(see [Hum1972]) for all
To show this one uses the action on to define a map on by One can show that is a bijection from to for each Then if is an element of the Weyl group, is a bijection between and This gives that
From (3.3) and (2.3) we have that
Note that Since the orbits are finite and the integers are finite we see that is finite dimensional.
Let and let and be as given above so that and Then the vector space is The weight of a composite word is given by Now let be a basis of of vectors of the form and let be a basis of of vectors of the form Then the words form a basis of Then the character of the generating function of the basis is In this way the multiplication of elements of has a representation theoretical significance.
is also a If is one of the generators or then we define This defines a action on
The elementary symmetric functions are the characters of the tensor product By Weyl's theorem (3.2) we know that this tensor product can be decomposed as a direct sum of irreducible representations. In fact, for each for nonnegative integers This is the motivation for the definition of the elementary symmetric functions. Since the are the basic generators of the weight lattice the are the most fundamental Weyl characters. (3.8) shows that these do indeed generate
This is an excerpt of a paper entitled Weyl Group Symmetric Functions and the Representation Theory of Lie Algebras, written by Arun Ram.