Weyl Group Symmetric Functions and the Representation Theory of Lie Algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 11 September 2013

Representation theory

Fix a Cartan matrix C=(αi,αj) and define 𝔘 to be the associative algebra (over ) with 1 generated by xi, yi, hi, 1in with relations (Serre relations) [hi,hj]=0 (1i,jn), (S1) [xi,yi]= hi, [xi,yj]= 0ifij, (S2) [hi,xj]= αj,αi xj, [hi,yj]=- αj,αi yj, (S3) (adxi) -αj,αi+1 (xj)=0(ij), (Sij+) (adyi) -αj,αi+1 (yj)=0(ij). (Sij-) Here [a,b]=ab-ba and (ada)k(b)=[a,[a,[a,,[a,b]]].

Let 𝔥*=iωi. Let V be a 𝔘 module. A vector vV is called a weight vector if, for each i, hiv=λiv, for some constant λi. We associate to v the weight λ=iλiωi.

If v is a weight vector of weight γ then xiv is a weight vector of weight γ+αi. This is the motivation for the definition of the raising operators Rα. This also gives the motivation for the definition of the dominance partial order as λμ if and only if μ=Rλ for some sequence of raising operators R=Rβ1Rβ2Rβk.

The dominant weights appear as a result of the following fact.

(see [Hum1972]) There is a unique finite dimensional irreducible representation Vλ of 𝔘 corresponding to each dominant weight λP+. 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 λ.

(Weyl) Every finite dimensional representation V of 𝔘 (corresponding to a simple Cartan matrix) is completely decomposable as a direct sum of irreducible representations Vλ, λP+.

The motivation for Weyl group symmetric functions is that they are the generating functions of the weights in these representations. Let V be a finite dimensional 𝔘-module. Let B be a basis of 𝔘 such that each element of B is a weight vector. Then the character of V is the generating function of the weights of B, χV=bB ewt(b).

The irreducibles Vλ, combinatorially.

In this section we shall define the vector spaces Vλ affording the irreducible 𝔘 representations in a combinatorial fashion to motivate the Weyl group, the Weyl characters and the monomial symmetric functions.

Recall the notations [a,b]=ab-ba and (ada)k(b)=[a,[a,[a,,[a,b]]]. Let y1,y2,,yn be letters and suppose that they satisfy the relations (adyi) -αj,αi+1 (yj)=0, (ij). (We should not be surprised by the appearance of the values αi,αj 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 yi. Let v+ be a dummy letter marking the end of a word. Fix a dominant integral weight λP+. Define Vλ to be the vector space which is a linear span of words in the yi ending with v+, Vλ= span {yi1yi2yikv+}, with the additional relations yjλj+1 v+=0.

Given that the vectors of the form yi1yikv+ span Vλ it is possible to choose a basis B of Vλ of vectors of this form. Define the weight of a word v=yi1yikv+ to be wt(v)=λ- j=1kαij, where the αij are simple roots. Define the Weyl character χλ to be the generating function of the weights of the basis B, χλ=bB ewt(b). (3.3) (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 λP+, χλ= wWε(w) ew(λ+ρ) wWε(w) ewρ , where ρ=iωi.

If we define hiv+ = λiv+, xiv+ = 0, for the generators xi, hi, 1in of 𝔘 then we can use the defining relations for 𝔘 to rewrite the expressions of the form xiyi1yi2yikv+ and hiyi1yikv+ as elements of V+. In this way we get an action of 𝔘 on Vλ. Vλ is an irreducible 𝔘-module.

Weight multiplicities

Let μP and define (Vλ)μ=span { v=yi1yikv+ |wt(v)=μ } . Define Kλμ=dim ((Vλ)μ). (3.5) It is clear from the definitions that

(a) The Kλμ are nonnegative integers.
(b) Kλλ=1 for all λP+.
(c) Kλμ=0 if μλ.
The key fact motivating the Weyl group is the following.

(see [Hum1972]) Kλμ= Kλ,wμ, for all wW.

To show this one uses the 𝔘 action on Vλ to define a map si on Vλ by siv=exp (xi)exp (-yi)exp (xi)v. One can show that si is a bijection from (Vλ)μ to (Vλ)siμ for each μP. Then if w=si1sip is an element of the Weyl group, si1sip is a bijection between (Vλ)μ and (Vλ)wμ. This gives that Kλμ=Kλ,wμ.

From (3.3) and (2.3) we have that χλ = μP Kλμeμ = νP+ Kλν μWν eμ = νP+ Kλνmν. (3.7)

Note that dimVλ= μP+ Wμ Kλμ. Since the orbits Wμ are finite and the integers Kλμ are finite we see that Vλ is finite dimensional.

Tensor products

Let λ,μP+ and let Vλ and Vμ be as given above so that Vλ=span{yi1yikv+}, and Vμ=span{yi1yikv+}. Then the vector space VλVμ is VλVμ=span { yi1yirv+ yj1yjsv+ } . The weight of a composite word is given by wt ( yi1yirv+ yj1yjsv+ ) = wt(yi1yirv+)+ wt(yj1yjsv+). Now let B be a basis of Vλ of vectors of the form y1yirv+ and let B be a basis of Vμ of vectors of the form yj1yjsv+. Then the words bb, bB, bB form a basis of VλVμ. Then the character of VλVμ, the generating function of the basis BB, is bb ewt(bb) =bBbB ewt(b) ewt(b) =χλχμ. In this way the multiplication of elements of AW has a representation theoretical significance.

VλVμ is also a 𝔘-module. If g is one of the generators xi, yi or hi then we define g ( yi1yirv+ yj1yjsv+ ) = gyi1yirv+ yj1yjsv+ +yi1yirv+ gyj1yjsv+. This defines a 𝔘 action on VλVμ.

The elementary symmetric functions eλ are the characters of the tensor product (Vω1)λ1 (Vωn)λn . 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 λ, eλ=χλ+ μ<λ Kλμ χμ, (3.8) for nonnegative integers Kλμ. This is the motivation for the definition of the elementary symmetric functions. Since the ωr are the basic generators of the weight lattice P, the er=χωr are the most fundamental Weyl characters. (3.8) shows that these do indeed generate AW.

Notes and references

This is an excerpt of a paper entitled Weyl Group Symmetric Functions and the Representation Theory of Lie Algebras, written by Arun Ram.

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