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
aram@unimelb.edu.au

Last update: 11 September 2013

Weyl group symmetric functions

Each of the simple root systems is determined by a Cartan matrix $C\text{.}$ 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 $⟨{\alpha }_i,{\alpha }_j⟩$ so that $$C=\left(⟨{\alpha }_i,{\alpha }_j⟩\right)\text{.}$$ Let $n$ be the dimension of the Cartan matrix.

Let ${\omega }_1,{\omega }_2,\dots ,{\omega }_n$ be basis vectors in a vector space. Define $${𝔥}^{*}=\sum _iℂ{\omega }_i,P=\sum _i\mathbb{Z}{\omega }_i,P^{+}=\sum _i\mathbb{N}{\omega }_i,$$ where $\mathbb{N}$ 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 ${\omega }_i$ are called the fundamental weights. We have the following sequence of inclusions $$\begin{array}{cc}{P}^{+}\subseteq P\subseteq {𝔥}^{*}\text{.}& \text{(2.1)}\end{array}$$

Let $\gamma ={\sum }_i{\gamma }_i{\omega }_i$ be an element of $P\text{.}$ We shall use the notation $⟨\gamma ,{\alpha }_i⟩$ for the integer ${\gamma }_i\text{.}$ The simple roots ${\alpha }_i$ are given in terms of the entries of the Cartan matrix, $${\alpha }_i=\sum _j⟨{\alpha }_i,{\alpha }_j⟩{\omega }_j\text{.}$$ There is a partial ordering on the weight lattice given by $$\begin{array}{cc}\gamma \ge \kappa \text{if}\kappa =\gamma -\sum k_i{\alpha }_i,& \text{(2.2)}\end{array}$$ for nonnegative integers $k_i\text{.}$ We say that $\gamma \ge \kappa$ in dominance.

Define linear operators $s_i:P\to P$ by $$s_i\gamma =\gamma -⟨\gamma ,{\alpha }_i⟩{\alpha }_i\text{.}$$ The Weyl group is the group generated by the $s_i\text{:}$ $W=⟨s_1,s_2,\dots ,s_n⟩\text{.}$ The sign of an element $w\in W$ is $\epsilon \left(w\right)={(-1)}^p,$ where $p$ is the smallest nonnegative integer such that there exists an expression $s_{i_1}{s}_{i_2}\cdots s_{i_p}=w\text{.}$ We will need the following proposition, see [Bou1981] Ch. 6, §1 Thm. 2.

(a)	Every Weyl group orbit $W\gamma ,$ $\gamma \in P^{+}$ contains a unique element in $P^{+}\text{.}$
(b)	If $\lambda ,\mu \in P^{+}$ and $\rho ={\sum }_i{\omega }_i$ then, for $v,w\in W,$ $$w(\lambda +\rho )=v(\mu +\rho )⟺v=w\text{.}$$

$\alpha \in P$ is a root if $\alpha =w{\alpha }_i$ for some $w\in W$ and simple root ${\alpha }_i\text{.}$ Let $\Phi$ be the set of roots and let ${\Phi }^{+}=\{\alpha \in \Phi \hspace{0.17em}|\hspace{0.17em}\alpha >0\}$ and ${\Phi }^{-}=\{\alpha \in \Phi \hspace{0.17em}|\hspace{0.17em}\alpha <0\}$ where the ordering is as in (2.2). It is true that $\Phi ={\Phi }^{+}\cup {\Phi }^{-}\text{.}$ The elements of ${\Phi }^{+}$ and ${\Phi }^{-}$ are called positive and negative roots respectively. The raising operator $R_{\alpha }$ associated to a positive root $\alpha$ is the operator which acts on elements of $P$ by $$\begin{array}{cc}{R}_{\alpha }\gamma =\gamma +\alpha \text{.}& \text{(2.4)}\end{array}$$

Corresponding to each $\lambda \in P$ we write, formally, $e^{\lambda }$ so that $$e^{\lambda }{e}^{\mu }=e^{\lambda +\mu }\text{.}$$ In particular if $\lambda ={\sum }_i{\lambda }_i{\omega }_i$ then $$\begin{array}{cc}\begin{array}{ccc}{e}^{\lambda }& =& e^{{\lambda }_1{\omega }_1}{e}^{{\lambda }_2{\omega }_2}\cdots e^{{\lambda }_n{\omega }_n}\\ & =& {\left(e^{{\omega }_1}\right)}^{{\lambda }_1}{\left(e^{{\omega }_2}\right)}^{{\lambda }_2}\cdots {\left(e^{{\omega }_n}\right)}^{{\lambda }_n}\text{.}\end{array}& \text{(2.5)}\end{array}$$ (If one finds this "exponential" notation unsettling one can substitute $z_i$ for $e^{{\omega }_i}$ and write $z^{\lambda }=z_1^{{\lambda }_1}{z}_2^{{\lambda }_2}\cdots z_n^{{\lambda }_n}$ instead of $e^{\lambda }\text{.)}$ Define an action of the Weyl group by $$\begin{array}{cc}w{e}^{\lambda }=e^{w\lambda },& \text{(2.6)}\end{array}$$ for each $w\in W$ and $\lambda \in P\text{.}$ Define $$A^W=\mathbb{Z}{[e^{{\omega }_1},e^{-{\omega }_1},\cdots ,e^{{\omega }_n},e^{-{\omega }_n}]}^W\text{.}$$

Bases of $A^W$

For each $\lambda \in P^{+}$ define the orbit sum, or monomial symmetric function, by $$\begin{array}{cc}{m}_{\lambda }=\sum _{\nu \in W\lambda }{e}^{\nu }\text{.}& \text{(2.7)}\end{array}$$

For each $\lambda \in P^{+}$ define the Weyl character by $$\begin{array}{cc}{\chi }^{\lambda }=\frac{{\sum }_{w\in W}\epsilon \left(w\right)e^{w(\lambda +\rho )}}{{\sum }_{w\in W}\epsilon \left(w\right)e^{w\rho }}& \text{(2.8)}\end{array}$$ where $\rho ={\sum }_i{\omega }_i\text{.}$

The elementary, or fundamental, symmetric functions are given by defining $$\begin{array}{ccc}{e}_0& =& 1,\\ e_r& =& {\chi }^{{\omega }_r},\end{array}$$ for each positive integer $r,$ and $$\begin{array}{cc}{e}_{\lambda }=e_1^{{\lambda }_1}{e}_2^{{\lambda }_2}\cdots e_n^{{\lambda }_n},& \text{(2.9)}\end{array}$$ for all elements $\lambda ={\sum }_i{\lambda }_i{\omega }_i$ in $P^{+}\text{.}$

Define integers $K_{\lambda \mu }$ by the identity $$\begin{array}{cc}{\chi }^{\lambda }=\sum _{\mu \in P^{+}}{K}_{\lambda \mu }{m}_{\mu }\text{.}& \text{(2.10)}\end{array}$$ It is true that

(a)	The $K_{\lambda \mu }$ are nonnegative integers.
(b)	$K_{\lambda \lambda }=1$ for all $\lambda \in P^{+}\text{.}$
(c)	$K_{\lambda \mu }=0$ if $\mu \nleqq \lambda \text{.}$

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 $$\begin{array}{c}{\left\{m_{\lambda }\right\}}_{\lambda \in P^{+}},\\ {\left\{{\chi }_{\lambda }\right\}}_{\lambda \in P^{+}},\\ {\left\{e_{\lambda }\right\}}_{\lambda \in P^{+}},\end{array}$$ forms a $\mathbb{Z}\text{-basis}$ of $A^W\text{.}$ The fact that the $m_{\lambda }$ form a $\mathbb{Z}$ basis of $A^W$ follows immediately from (2.3). One can show by elementary techniques and without using representation theory, see [Bou1981] Ch. 6, §3, that the ${\chi }^{\lambda },$ $\lambda \in P^{+}$ form a $\mathbb{Z}$ basis of $A^W\text{.}$ This fact also follows from the facts about the numbers $K_{\lambda \mu }$ above. Assuming that the ${\chi }^{\lambda }$ form a $\mathbb{Z}\text{-basis}$ of $A^W$ it follows that the $e_{\lambda },$ $\lambda \in P^{+}$ form a $\mathbb{Z}\text{-basis}$ of $A^W$ simply by expanding $e_{\lambda }$ in terms of $e^{\mu },$ $\mu \in P^{+}\text{.}$ One can also obtain this result in a different fashion by using representation theory.

Inner product

Let $$d=\sum _{w\in W}\epsilon \left(w\right)e^{w\rho },$$ where $\rho ={\sum }_i{\omega }_i\text{.}$ If $f={\sum }_{\nu \in P}{f}_{\nu }{e}^{\nu }$ then define $\bar{f}={\sum }_{\nu }{f}_{\nu }{e}^{-\nu }\text{.}$ Let ${\left[f\right]}_1$ denote taking the coefficient of the identity, $e^0,$ in $f\text{.}$ Then define $$⟨f,g⟩=\frac{1}{\left|W\right|}{\left[fd\bar{g}\bar{d}\right]}_1,$$ where $\left|W\right|$ is the order of the Weyl group.

([Mac1991]) The inner product defined above satisfies $$⟨{\chi }^{\lambda },{\chi }^{\mu }⟩={\delta }_{\lambda \mu }\text{.}$$

		Proof.
	Since ${\chi }^{\lambda }=d^{-1}{\sum }_{w\in W}\epsilon \left(w\right)e^{w(\lambda +\rho )},$ $$⟨{\chi }^{\lambda },{\chi }^{\mu }⟩=\frac{1}{\left|W\right|}\sum _{v,w\in W}\epsilon \left(v\right)\epsilon \left(w\right){\left[e^{v(\lambda +\rho )}{e}^{-w(\mu +\rho )}\right]}_1\text{.}$$ This is zero if $\lambda \ne \mu ,$ because, (2.3), the orbits $W(\lambda +\rho )$ and $W(\mu +\rho )$ do not intersect. If $\lambda =\mu ,$ then $v(\lambda +\rho )=w(\lambda +\rho )\iff v=w\text{.}$ Thus $⟨{\chi }^{\lambda },{\chi }^{\lambda }⟩=\frac{1}{\left|W\right|}{\sum }_{w\in W}1=1\text{.}$ $\square$

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

Homogeneous symmetric functions

Let $\kappa \in P^{+}$ and define $${\Gamma }_{\kappa }=\{\mu \in P^{+}\hspace{0.17em}|\hspace{0.17em}\mu \le \kappa \},\text{and}{\Lambda }_{\kappa }=\text{span}\left\{{\chi }^{\mu }\hspace{0.17em}\right|\hspace{0.17em}\mu \in {\Gamma }_{\kappa }\}\text{.}$$ Since ${\Gamma }_{\kappa }$ is always finite ${\Lambda }_{\kappa }$ is always finite dimensional.

Define an inner product on ${\Lambda }_{\kappa }$ by defining $$⟨{\chi }^{\lambda },{\chi }^{\mu }⟩={\delta }_{\lambda \mu }$$ for all $\lambda ,\mu \in {\Gamma }_{\kappa }\text{.}$ Then define the homogeneous symmetric functions $h_{\lambda },$ $\lambda \in {\Gamma }_{\kappa }$ to be the dual basis to the monomial symmetric functions, $$\begin{array}{cc}⟨h_{\lambda },m_{\mu }⟩={\delta }_{\lambda \mu }\text{.}& \text{(2.12)}\end{array}$$ Using the integers $K_{\lambda \mu }$ defined in (2.10), the homogeneous symmetric functions are given in terms of the Weyl characters by $$\begin{array}{cc}{h}_{\mu }=\sum _{\lambda \in {\Gamma }_{\kappa }}{\chi }^{\lambda }{K}_{\lambda \mu },& \text{(2.13)}\end{array}$$ for all $\mu \in {\Gamma }_{\kappa }\text{.}$ The $h_{\mu },$ $\mu \in {\Gamma }_{\kappa }$ form a basis of ${\Lambda }_{\kappa }\text{.}$

Each of the sets $$\begin{array}{c}{\left\{m_{\lambda }\right\}}_{\lambda \in P^{+}},\\ {\left\{{\chi }_{\lambda }\right\}}_{\lambda \in P^{+}},\\ {\left\{e_{\lambda }\right\}}_{\lambda \in P^{+}},\\ {\left\{h_{\lambda }\right\}}_{\lambda \in P^{+}},\end{array}$$ forms a $\mathbb{Z}\text{-basis}$ of ${\Lambda }_{\kappa }\text{.}$ To see this choose some total ordering of the elements of ${\Gamma }_{\kappa }$ which is a refinement of the dominance partial order. Then, by (2.10a-c), the matrix, with rows and columns indexed by elements of ${\Gamma }_{\kappa },$ having $K_{\lambda \nu }$ as the $\lambda ,\nu$ entry is upper unitriangular with nonnegative integer entries. This implies that it is invertible as a matrix with integer entries. The fact that the ${\chi }^{\mu }$ are a basis of ${\Lambda }_{\kappa }$ is by definition. The other two statements now follow from (2.10) and (2.13).

"Jacobi-Trudi" formulas

Fix $\kappa \in P^{+}\text{.}$ Define $$h_{w\lambda }=h_{\lambda }$$ for all $\lambda \in {\Gamma }_{\kappa }$ and all $w\in W$ so that $h_{\lambda }$ is defined for all $\lambda \in W{\Gamma }_{\kappa }\text{.}$ One has the following "Jacobi-Trudi" type identity for the Weyl characters in terms of the $h_{\lambda }\text{.}$

Let $\rho ={\sum }_i{\omega }_i\text{.}$ Then for each $\lambda \in {\Gamma }_{\kappa }$ $${\chi }^{\lambda }=\sum _{w\in W}\epsilon \left(w\right)h_{\lambda +\rho -w\rho }\text{.}$$

		Proof.
	We show that elements ${\sum }_{w\in W}\epsilon \left(w\right)h_{\lambda +\rho -w\rho }$ are the dual basis to the basis ${\chi }^{\mu },$ $\mu \in {\Gamma }_{\kappa }\text{.}$ $${\chi }^{\lambda }=\sum _{\mu \in W{\Gamma }_{\kappa }}⟨{\chi }^{\lambda },h_{\mu }⟩e^{\mu }\text{.}$$ Expanding ${\chi }^{\lambda }$ by (2.8) and clearing denominators we have that $$\begin{array}{ccc}\sum _{w\in W}\epsilon \left(w\right)e^{w(\lambda +\rho )-\rho }& =& \left(\sum _{w\in W}\epsilon \left(w\right)e^{w\rho -\rho }\right)\left(\sum _{\mu \in W{\Gamma }_{\kappa }}⟨{\chi }^{\lambda },h_{\mu }⟩e^{\mu }\right)\\ & =& \sum _{\mu \in W{\Gamma }_{\kappa }}\sum _{w\in W}\epsilon \left(w\right)⟨{\chi }^{\lambda },h_{\mu }⟩e^{\mu +w\rho -\rho }\end{array}$$ Substitute $\gamma =\mu +w\rho -\rho$ to get $$\sum _{w\in W}\epsilon \left(w\right)e^{w(\lambda +\rho )-\rho }=\sum _{\mu \in W{\Gamma }_{\kappa }}⟨{\chi }^{\lambda },\sum _{w\in W}\epsilon \left(w\right)h_{\gamma +\rho -w\rho }⟩e^{\gamma }\text{.}$$ Compare coefficients of $e^{\gamma }$ for $\gamma \in P^{+}$ on each side of this equation. Since $\lambda \in P^{+}$ we know by (2.3) that $w(\lambda +\rho )-\rho$ is not an element of $P^{+}$ for any $w\in W$ except the identity. Thus we know that if $\mu \in P^{+}$ then $$⟨{\chi }^{\lambda },\sum _{w\in W}\epsilon \left(w\right)h_{\mu +\rho -w\rho }⟩={\delta }_{\lambda \mu }\text{.}$$ $\square$

Recall that raising operators act on elements of $P\text{.}$ We allow the raising operators to act upon the $h_{\lambda }$ by defining $$R\left(h_{\lambda }\right)=h_{R\left(\lambda \right)},$$ for each sequence $R=R_{{\beta }_1}{R}_{{\beta }_2}\cdots R_{{\beta }_k}\text{.}$ We use the convention that $h_{\lambda }=0$ if $\lambda \notin W{\Gamma }_{\kappa }\text{.}$ (Note: It is important to keep in mind that raising operators act on elements of $P$ and not on symmetric functions.)

For all $\lambda \in {\Gamma }_{\kappa },$ $${\chi }^{\lambda }=\prod _{\alpha >0}(1-R_{\alpha })h_{\lambda }\text{.}$$

		Proof.
	A sketch of the proof is as follows. Evaluating the right hand side of the above we get $$\prod _{\alpha >0}(1-R_{\alpha })h_{\lambda +\rho -\rho }=\sum _{E\subseteq {\Phi }^{+}}{(-1)}^E{h}_{\lambda +\rho +(-\rho +{\sigma }_E)},$$ where ${\sigma }_E={\sum }_{\alpha \in E}\alpha \text{.}$ An element $\gamma ={\sum }_i{\gamma }_i{\omega }_i$ in $P^{+}$ is called regular if ${\gamma }_i>0$ for all $i\text{.}$ The sets $\{-\rho +{\sigma }_E\hspace{0.17em}|\hspace{0.17em}E\subseteq {\Phi }^{+},\rho +{\sigma }_E\hspace{0.17em}\text{regular}\}$ and $\{-w\rho \hspace{0.17em}|\hspace{0.17em}w\in W\}$ are equal. This is proved by expressing $\rho$ in the form $\rho ={\sum }_{\alpha \in {\Phi }^{+}}\alpha$ and using [Bou1981] Ch. 6, §1 Cor. 2. Under this bijection ${(-1)}^{\left|E\right|}=\epsilon \left(w\right)\text{.}$ The terms arising from the subsets $E$ for which $-\rho +{\sigma }_E$ is not regular cancel with each other. This can be shown by showing that ${\prod }_{\alpha >0}(1-R_{\alpha })(-\rho )$ is skew-symmetric with respect to $W$ and that if $\gamma \in P^{+}$ is not regular then ${\sum }_{w\in W}\epsilon \left(w\right)w\gamma =0\text{.}$ These arguments show that $$\prod _{\alpha >0}(1-R_{\alpha })h_{\lambda }=\sum _{w\in W}\epsilon \left(w\right)h_{\lambda +\rho -w\rho }\text{.}$$ $\square$

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 ${\Lambda }_k\text{.}$ One would like to say that in some sense the $h_{\lambda }$ are well defined on all of $A^W\text{.}$ With this in mind we introduce the following.

For each pair $\beta ,\kappa \in P^{+}$ such that $\beta \le \kappa$ define a linear map $f_{\beta \kappa }:{\Lambda }_{\kappa }\to {\Lambda }_{\beta }$ by $$s_{\lambda }⟼\{\begin{array}{cc}{s}_{\lambda },& \text{if}\hspace{0.17em}\lambda \le \beta \text{;}\\ 0,& \text{if}\hspace{0.17em}\lambda \nleqq \beta \text{.}\end{array}$$ It is clear that

1)	If $\beta \le \gamma \le \kappa$ then $f_{\beta \kappa }=f_{\beta \gamma }\circ f_{\gamma \kappa },$
2)	For each $\beta \in P^{+},$ $f_{\beta \beta }$ is the identity on ${\Lambda }_{\beta }\text{.}$

Thus $({\Lambda }_{\beta },f_{\beta \gamma })$ form an inverse system of vector spaces, see Bourbaki Theory of Sets I §7, and Bourbaki Algebra I §10. Define $$\Lambda =\underset{⟵}{\text{lim}}({\Lambda }_{\beta },f_{\beta \gamma })\text{.}$$ Then the homogeneous symmetric function $h_{\lambda }$ is a well defined element of $\Lambda$ for all $\lambda \in P^{+}$ and is equal to $$h_{\mu }=\sum _{\lambda \in P^{+}}{s}_{\lambda }{K}_{\lambda \mu }\text{.}$$ An alternate option is to view the homogeneous symmetric function as an element in the direct product of vector spaces $$\prod _{\lambda \in P^{+}}\mathbb{Z}{\chi }^{\lambda }\text{.}$$ 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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