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

Weyl group symmetric functions

Each of the simple root systems is determined by a Cartan matrix C. A list of the Cartan matrices for simple root systems can be found in [Bou1981] p. 250-258. We shall denote the (i,j) entry of the Cartan matrix by αi,αj so that C=(αi,αj). Let n be the dimension of the Cartan matrix.

Let ω1,ω2,,ωn be basis vectors in a vector space. Define 𝔥*=iωi, P=iωi, P+=iωi, where denotes the nonnegative integers. The elements of 𝔥*, P, and P+ are called the weights, the integral weights, and the dominant integral weights, respectively. The ωi are called the fundamental weights. We have the following sequence of inclusions P+P𝔥*. (2.1)

Let γ=iγiωi be an element of P. We shall use the notation γ,αi for the integer γi. The simple roots αi are given in terms of the entries of the Cartan matrix, αi=j αi,αj ωj. There is a partial ordering on the weight lattice given by γκifκ=γ- kiαi, (2.2) for nonnegative integers ki. We say that γκ in dominance.

Define linear operators si:PP by siγ=γ- γ,αi αi. The Weyl group is the group generated by the si: W=s1,s2,,sn. The sign of an element wW is ε(w)=(-1)p, where p is the smallest nonnegative integer such that there exists an expression si1si2sip=w. We will need the following proposition, see [Bou1981] Ch. 6, §1 Thm. 2.

(a) Every Weyl group orbit Wγ, γP+ contains a unique element in P+.
(b) If λ,μP+ and ρ=iωi then, for v,wW, w(λ+ρ)= v(μ+ρ) v=w.

αP is a root if α=wαi for some wW and simple root αi. Let Φ be the set of roots and let Φ+={αΦ|α>0} and Φ-={αΦ|α<0} where the ordering is as in (2.2). It is true that Φ=Φ+Φ-. The elements of Φ+ and Φ- are called positive and negative roots respectively. The raising operator Rα associated to a positive root α is the operator which acts on elements of P by Rαγ=γ+α. (2.4)

Corresponding to each λP we write, formally, eλ so that eλeμ= eλ+μ. In particular if λ=iλiωi then eλ = eλ1ω1 eλ2ω2 eλnωn = (eω1)λ1 (eω2)λ2 (eωn)λn. (2.5) (If one finds this "exponential" notation unsettling one can substitute zi for eωi and write zλ=z1λ1z2λ2znλn instead of eλ.) Define an action of the Weyl group by weλ= ewλ, (2.6) for each wW and λP. Define AW= [ eω1, e-ω1,, eωn, e-ωn ] W .

Bases of AW

For each λP+ define the orbit sum, or monomial symmetric function, by mλ= νWλ eν. (2.7)

For each λP+ define the Weyl character by χλ= wW ε(w) ew(λ+ρ) wW ε(w) ewρ (2.8) where ρ=iωi.

The elementary, or fundamental, symmetric functions are given by defining e0 = 1, er = χωr, for each positive integer r, and eλ= e1λ1 e2λ2 enλn, (2.9) for all elements λ=iλiωi in P+.

Define integers Kλμ by the identity χλ=μP+ Kλμmμ. (2.10) It is true that

(a) The Kλμ are nonnegative integers.
(b) Kλλ=1 for all λP+.
(c) Kλμ=0 if μλ.
All of these facts follow from representation theory see §3 (3.5). I know of no easy way to prove these results without using representation theory.

Each of the sets {mλ}λP+, {χλ}λP+, {eλ}λP+, forms a -basis of AW. The fact that the mλ form a basis of AW follows immediately from (2.3). One can show by elementary techniques and without using representation theory, see [Bou1981] Ch. 6, §3, that the χλ, λP+ form a basis of AW. This fact also follows from the facts about the numbers Kλμ above. Assuming that the χλ form a -basis of AW it follows that the eλ, λP+ form a -basis of AW simply by expanding eλ in terms of eμ, μP+. One can also obtain this result in a different fashion by using representation theory.

Inner product

Let d=wWε(w) ewρ, where ρ=iωi. If f=νPfνeν then define f=νfνe-ν. Let [f]1 denote taking the coefficient of the identity, e0, in f. Then define f,g= 1|W| [fdgd]1, where |W| is the order of the Weyl group.

([Mac1991]) The inner product defined above satisfies χλ,χμ =δλμ.


Since χλ=d-1wWε(w)ew(λ+ρ), χλ,χμ =1|W|v,wW ε(v)ε(w) [ev(λ+ρ)e-w(μ+ρ)]1. This is zero if λμ, because, (2.3), the orbits W(λ+ρ) and W(μ+ρ) do not intersect. If λ=μ, then v(λ+ρ)=w(λ+ρ)v=w. Thus χλ,χλ=1|W|wW1=1.

If one prefers one may simply define the inner product by making the Weyl characters orthonormal.

Homogeneous symmetric functions

Let κP+ and define Γκ= {μP+|μκ}, andΛκ= span{χμ|μΓκ}. Since Γκ is always finite Λκ is always finite dimensional.

Define an inner product on Λκ by defining χλ,χμ =δλμ for all λ,μΓκ. Then define the homogeneous symmetric functions hλ, λΓκ to be the dual basis to the monomial symmetric functions, hλ,mμ =δλμ. (2.12) Using the integers Kλμ defined in (2.10), the homogeneous symmetric functions are given in terms of the Weyl characters by hμ=λΓκ χλKλμ, (2.13) for all μΓκ. The hμ, μΓκ form a basis of Λκ.

Each of the sets {mλ}λP+, {χλ}λP+, {eλ}λP+, {hλ}λP+, forms a -basis of Λκ. To see this choose some total ordering of the elements of Γκ which is a refinement of the dominance partial order. Then, by (2.10a-c), the matrix, with rows and columns indexed by elements of Γκ, having Kλν as the λ,ν entry is upper unitriangular with nonnegative integer entries. This implies that it is invertible as a matrix with integer entries. The fact that the χμ are a basis of Λκ is by definition. The other two statements now follow from (2.10) and (2.13).

"Jacobi-Trudi" formulas

Fix κP+. Define hwλ=hλ for all λΓκ and all wW so that hλ is defined for all λWΓκ. One has the following "Jacobi-Trudi" type identity for the Weyl characters in terms of the hλ.

Let ρ=iωi. Then for each λΓκ χλ=wW ε(w)hλ+ρ-wρ.


We show that elements wWε(w)hλ+ρ-wρ are the dual basis to the basis χμ, μΓκ. χλ=μWΓκ χλ,hμ eμ. Expanding χλ by (2.8) and clearing denominators we have that wW ε(w) ew(λ+ρ)-ρ = ( wWε(w) ewρ-ρ ) ( μWΓκ χλ,hμ eμ ) = μWΓκ wW ε(w) χλ,hμ eμ+wρ-ρ Substitute γ=μ+wρ-ρ to get wWε(w) ew(λ+ρ)-ρ= μWΓκ χλ,wWε(w)hγ+ρ-wρ eγ. Compare coefficients of eγ for γP+ on each side of this equation. Since λP+ we know by (2.3) that w(λ+ρ)-ρ is not an element of P+ for any wW except the identity. Thus we know that if μP+ then χλ,wWε(w)hμ+ρ-wρ =δλμ.

Recall that raising operators act on elements of P. We allow the raising operators to act upon the hλ by defining R(hλ)= hR(λ), for each sequence R=Rβ1Rβ2Rβk. We use the convention that hλ=0 if λWΓκ. (Note: It is important to keep in mind that raising operators act on elements of P and not on symmetric functions.)

For all λΓκ, χλ=α>0 (1-Rα)hλ.


A sketch of the proof is as follows. Evaluating the right hand side of the above we get α>0 (1-Rα) hλ+ρ-ρ= EΦ+ (-1)E hλ+ρ+(-ρ+σE), where σE=αEα. An element γ=iγiωi in P+ is called regular if γi>0 for all i. The sets {-ρ+σE|EΦ+,ρ+σEregular} and {-wρ|wW} are equal. This is proved by expressing ρ in the form ρ=αΦ+α and using [Bou1981] Ch. 6, §1 Cor. 2. Under this bijection (-1)|E|=ε(w). The terms arising from the subsets E for which -ρ+σE is not regular cancel with each other. This can be shown by showing that α>0(1-Rα)(-ρ) is skew-symmetric with respect to W and that if γP+ is not regular then wWε(w)wγ=0. These arguments show that α>0 (1-Rα) hλ=wW ε(w) hλ+ρ-wρ.

The proof of Cor. (2.15) was motivated by the proof of the Weyl denominator formula given in [Mac1991].

Direct limits

The above definition defines an analogue of homogeneous symmetric functions for the spaces Λk. One would like to say that in some sense the hλ are well defined on all of AW. With this in mind we introduce the following.

For each pair β,κP+ such that βκ define a linear map fβκ:ΛκΛβ by sλ { sλ, ifλβ; 0, ifλβ. It is clear that

1) If βγκ then fβκ=fβγfγκ,
2) For each βP+, fββ is the identity on Λβ.
Thus (Λβ,fβγ) form an inverse system of vector spaces, see Bourbaki Theory of Sets I §7, and Bourbaki Algebra I §10. Define Λ=lim (Λβ,fβγ). Then the homogeneous symmetric function hλ is a well defined element of Λ for all λP+ and is equal to hμ=λP+ sλKλμ. An alternate option is to view the homogeneous symmetric function as an element in the direct product of vector spaces λP+ χλ. Depending on what one would like to compute this can create problems with infinite sums. The direct limit approach allows one to control these problems by fixing an ordering on infinite sums.

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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