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

Representation theory

Fix a Cartan matrix $C=\left(⟨{\alpha }_i,{\alpha }_j⟩\right)$ and define $𝔘$ to be the associative algebra (over $ℂ\text{)}$ with $1$ generated by $x_i,$ $y_i,$ $h_i,$ $1\le i\le n$ with relations (Serre relations) $$\begin{array}{ccc}& [h_i,h_j]=0(1\le i,j\le n),& \text{(}S1\text{)}\\ & [x_i,y_i]=h_i,[x_i,y_j]=0\hspace{0.17em}\text{if}\hspace{0.17em}i\ne j,& \text{(}S2\text{)}\\ & [h_i,x_j]=⟨{\alpha }_j,{\alpha }_i⟩x_j,[h_i,y_j]=-⟨{\alpha }_j,{\alpha }_i⟩y_j,& \text{(}S3\text{)}\\ & {\left(\text{ad}\hspace{0.17em}{x}_i\right)}^{-⟨{\alpha }_j,{\alpha }_i⟩+1}\left(x_j\right)=0(i\ne j),& \text{(}{S}_{ij}^{+}\text{)}\\ & {\left(\text{ad}\hspace{0.17em}{y}_i\right)}^{-⟨{\alpha }_j,{\alpha }_i⟩+1}\left(y_j\right)=0(i\ne j)\text{.}& \text{(}{S}_{ij}^{-}\text{)}\end{array}$$ Here $[a,b]=ab-ba$ and ${\left(\text{ad}\hspace{0.17em}a\right)}^k\left(b\right)=[a,[a,[a,\cdots ,[a,b]]\cdots ]\text{.}$

Let ${𝔥}^{*}={\sum }_iℂ{\omega }_i\text{.}$ Let $V$ be a $𝔘$ module. A vector $v\in V$ is called a weight vector if, for each $i,$ $$h_{i}v={\lambda }_{i}v,$$ for some constant ${\lambda }_i\in ℂ\text{.}$ We associate to $v$ the weight $\lambda ={\sum }_i{\lambda }_i{\omega }_i\text{.}$

If $v$ is a weight vector of weight $\gamma$ then $x_{i}v$ is a weight vector of weight $\gamma +{\alpha }_i\text{.}$ This is the motivation for the definition of the raising operators $R_{\alpha }\text{.}$ This also gives the motivation for the definition of the dominance partial order as $\lambda \le \mu$ if and only if $\mu =R\lambda$ for some sequence of raising operators $R=R_{{\beta }_1}{R}_{{\beta }_2}\cdots R_{{\beta }_k}\text{.}$

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

(see [Hum1972]) There is a unique finite dimensional irreducible representation $V^{\lambda }$ of $𝔘$ corresponding to each dominant weight $\lambda \in P^{+}\text{.}$ 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 $\lambda \text{.}$

(Weyl) Every finite dimensional representation $V$ of $𝔘$ (corresponding to a simple Cartan matrix) is completely decomposable as a direct sum of irreducible representations $V^{\lambda },$ $\lambda \in P^{+}\text{.}$

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 $𝔘\text{-module.}$ 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 $B,$ $${\chi }_V=\sum _{b\in B}{e}^{wt\left(b\right)}\text{.}$$

The irreducibles $V^{\lambda },$ combinatorially.

In this section we shall define the vector spaces $V^{\lambda }$ 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 ${\left(\text{ad}\hspace{0.17em}a\right)}^k\left(b\right)=[a,[a,[a,\cdots ,[a,b]]\cdots ]\text{.}$ Let $y_1,y_2,\dots ,y_n$ be letters and suppose that they satisfy the relations $${\left(\text{ad}\hspace{0.17em}{y}_i\right)}^{-⟨{\alpha }_j,{\alpha }_i⟩+1}\left(y_j\right)=0,(i\ne j)\text{.}$$ (We should not be surprised by the appearance of the values $⟨{\alpha }_i,{\alpha }_j⟩$ 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 $y_i\text{.}$ Let $v^{+}$ be a dummy letter marking the end of a word. Fix a dominant integral weight $\lambda \in P^{+}\text{.}$ Define $V^{\lambda }$ to be the vector space which is a linear span of words in the $y_i$ ending with $v^{+},$ $$V^{\lambda }=\text{span}\left\{y_{i_1}{y}_{i_2}\cdots y_{i_k}{v}^{+}\right\},$$ with the additional relations $$y_j^{{\lambda }_j+1}{v}^{+}=0\text{.}$$

Given that the vectors of the form $y_{i_1}\cdots y_{i_k}{v}^{+}$ span $V^{\lambda }$ it is possible to choose a basis $B$ of $V^{\lambda }$ of vectors of this form. Define the weight of a word $v=y_{i_1}\cdots y_{i_k}{v}^{+}$ to be $$wt\left(v\right)=\lambda -\sum _{j=1}^k{\alpha }_{i_j},$$ where the ${\alpha }_{i_j}$ are simple roots. Define the Weyl character ${\chi }^{\lambda }$ to be the generating function of the weights of the basis $B,$ $$\begin{array}{cc}{\chi }^{\lambda }=\sum _{b\in B}{e}^{wt\left(b\right)}\text{.}& \text{(3.3)}\end{array}$$ (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 $\lambda \in P^{+},$ $${\chi }^{\lambda }=\frac{{\sum }_{w\in W}\epsilon \left(w\right)e^{w(\lambda +\rho )}}{{\sum }_{w\in W}\epsilon \left(w\right)e^{w\rho }},$$ where $\rho ={\sum }_i{\omega }_i\text{.}$

If we define $$\begin{array}{ccc}{h}_i{v}^{+}& =& {\lambda }_i{v}^{+},\\ x_i{v}^{+}& =& 0,\end{array}$$ for the generators $x_i,$ $h_i,$ $1\le i\le n$ of $𝔘$ then we can use the defining relations for $𝔘$ to rewrite the expressions of the form $x_i{y}_{i_1}{y}_{i_2}\cdots y_{i_k}{v}^{+}$ and $h_i{y}_{i_1}\cdots y_{i_k}{v}^{+}$ as elements of $V^{+}\text{.}$ In this way we get an action of $𝔘$ on $V^{\lambda }\text{.}$ $V^{\lambda }$ is an irreducible $𝔘\text{-module.}$

Weight multiplicities

Let $\mu \in P$ and define $${\left(V^{\lambda }\right)}_{\mu }=\text{span}\{v=y_{i_1}\cdots y_{i_k}{v}^{+}\hspace{0.17em}|\hspace{0.17em}wt\left(v\right)=\mu \}\text{.}$$ Define $$\begin{array}{cc}{K}_{\lambda \mu }=\text{dim}\left({\left(V^{\lambda }\right)}_{\mu }\right)\text{.}& \text{(3.5)}\end{array}$$ It is clear from the definitions 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 $\mu \nleqq \lambda \text{.}$

The key fact motivating the Weyl group is the following.

(see [Hum1972]) $$K_{\lambda \mu }=K_{\lambda ,w\mu },$$ for all $w\in W\text{.}$

To show this one uses the $𝔘$ action on $V^{\lambda }$ to define a map ${\stackrel{\sim }{s}}_i$ on $V^{\lambda }$ by $${\stackrel{\sim }{s}}_{i}v=\text{exp}\left(x_i\right)\text{exp}(-y_i)\text{exp}\left(x_i\right)v\text{.}$$ One can show that ${\stackrel{\sim }{s}}_i$ is a bijection from ${\left(V^{\lambda }\right)}_{\mu }$ to ${\left(V^{\lambda }\right)}_{s_i\mu }$ for each $\mu \in P\text{.}$ Then if $w=s_{i_1}\cdots s_{i_p}$ is an element of the Weyl group, ${\stackrel{\sim }{s}}_{i_1}\cdots {\stackrel{\sim }{s}}_{i_p}$ is a bijection between ${\left(V^{\lambda }\right)}_{\mu }$ and ${\left(V^{\lambda }\right)}_{w\mu }\text{.}$ This gives that $K_{\lambda \mu }=K_{\lambda ,w\mu }\text{.}$

From (3.3) and (2.3) we have that $$\begin{array}{cc}\begin{array}{ccc}{\chi }^{\lambda }& =& \sum _{\mu \in P}{K}_{\lambda \mu }{e}^{\mu }\\ & =& \sum _{\nu \in P^{+}}{K}_{\lambda \nu }\sum _{\mu \in W\nu }{e}^{\mu }\\ & =& \sum _{\nu \in P^{+}}{K}_{\lambda \nu }{m}_{\nu }\text{.}\end{array}& \text{(3.7)}\end{array}$$

Note that $$\text{dim}\hspace{0.17em}{V}^{\lambda }=\sum _{\mu \in P^{+}}\sum _{W\mu }{K}_{\lambda \mu }\text{.}$$ Since the orbits $W\mu$ are finite and the integers $K_{\lambda \mu }$ are finite we see that $V^{\lambda }$ is finite dimensional.

Tensor products

Let $\lambda ,\mu \in P^{+}$ and let $V^{\lambda }$ and $V^{\mu }$ be as given above so that $V^{\lambda }=\text{span}\left\{y_{i_1}\cdots y_{i_k}{v}^{+}\right\},$ and $V^{\mu }=\text{span}\left\{y_{i_1}\cdots y_{i_k}{\bar{v}}^{+}\right\}\text{.}$ Then the vector space $V^{\lambda }\otimes V^{\mu }$ is $$V^{\lambda }\otimes V^{\mu }=\text{span}\{y_{i_1}\cdots y_{i_r}{v}^{+}\otimes y_{j_1}\cdots y_{j_s}{\bar{v}}^{+}\}\text{.}$$ The weight of a composite word is given by $$wt(y_{i_1}\cdots y_{i_r}{v}^{+}\otimes y_{j_1}\cdots y_{j_s}{\bar{v}}^{+})=wt\left(y_{i_1}\cdots y_{i_r}{v}^{+}\right)+wt\left(y_{j_1}\cdots y_{j_s}{\bar{v}}^{+}\right)\text{.}$$ Now let $B$ be a basis of $V^{\lambda }$ of vectors of the form $y_1\cdots y_{i_r}{v}^{+}$ and let $\bar{B}$ be a basis of $V^{\mu }$ of vectors of the form $y_{j_1}\cdots y_{j_s}{\bar{v}}^{+}\text{.}$ Then the words $b\otimes \bar{b},$ $b\in B,$ $\bar{b}\in \bar{B}$ form a basis of $V^{\lambda }\otimes V^{\mu }\text{.}$ Then the character of $V^{\lambda }\otimes V^{\mu },$ the generating function of the basis $B\otimes \bar{B},$ is $$\sum _{b\otimes \bar{b}}{e}^{wt(b\otimes \bar{b})}=\sum _{b\in B}\sum _{\bar{b}\in \bar{B}}{e}^{wt\left(b\right)}{e}^{wt\left(\bar{b}\right)}={\chi }^{\lambda }{\chi }^{\mu }\text{.}$$ In this way the multiplication of elements of $A^W$ has a representation theoretical significance.

$V^{\lambda }\otimes V^{\mu }$ is also a $𝔘\text{-module.}$ If $g$ is one of the generators $x_i,$ $y_i$ or $h_i$ then we define $$g(y_{i_1}\cdots y_{i_r}{v}^{+}\otimes y_{j_1}\cdots y_{j_s}{\bar{v}}^{+})=g{y}_{i_1}\cdots y_{i_r}{v}^{+}\otimes y_{j_1}\cdots y_{j_s}{\bar{v}}^{+}+y_{i_1}\cdots y_{i_r}{v}^{+}\otimes g{y}_{j_1}\cdots y_{j_s}{\bar{v}}^{+}\text{.}$$ This defines a $𝔘$ action on $V^{\lambda }\otimes V^{\mu }\text{.}$

The elementary symmetric functions $e_{\lambda }$ are the characters of the tensor product ${\left(V^{{\omega }_1}\right)}^{\otimes {\lambda }_1}\otimes \cdots \otimes {\left(V^{{\omega }_n}\right)}^{\otimes {\lambda }_n}\text{.}$ 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 $\lambda ,$ $$\begin{array}{cc}{e}_{\lambda }={\chi }^{\lambda }+\sum _{\mu <\lambda }{\stackrel{\sim }{K}}_{\lambda \mu }{\chi }^{\mu },& \text{(3.8)}\end{array}$$ for nonnegative integers ${\stackrel{\sim }{K}}_{\lambda \mu }\text{.}$ This is the motivation for the definition of the elementary symmetric functions. Since the ${\omega }_r$ are the basic generators of the weight lattice $P,$ the $e_r={\chi }^{{\omega }_r}$ are the most fundamental Weyl characters. (3.8) shows that these do indeed generate $A^W\text{.}$

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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