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

Classical symmetric functions

This section gives a brief summary of the classical symmetric function theory. See [Mac] Chapter 1 for a complete treatment.

Fix a positive integer $n\text{.}$ A partition $\lambda =({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n)$ is a sequence ${\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_n\ge 0$ of nonnegative integers. Let $𝒫$ denote the set of partitions. We have the following sequence of inclusions $$\begin{array}{cc}𝒫\subset {\mathbb{Z}}^n\supset {ℂ}^n\text{.}& \text{(1.1)}\end{array}$$ There is a partial ordering, the dominance ordering, on elements of ${\mathbb{Z}}^n$ given by $$\begin{array}{cc}\gamma \ge \kappa \text{if}{\gamma }_1+{\gamma }_2+\cdots +{\gamma }_i\ge {\kappa }_1+{\kappa }_2+\cdots {\kappa }_i,\text{for all}\hspace{0.17em}i\text{.}& \text{(1.2)}\end{array}$$

Let $S_n$ denote the symmetric group. The sign $\epsilon \left(w\right)$ of a permutation $w\in S_n$ is the determinant of the corresponding permutation matrix. $S_n$ acts on elements of ${\mathbb{Z}}^n$ by permuting the positions.

For each $1\le i<j\le n$ the raising operator $R_{ij}$ is the operator which acts on elements of ${\mathbb{Z}}^n$ by $$\begin{array}{cc}{R}_{ij}({\gamma }_1,{\gamma }_2,\dots ,{\gamma }_n)=({\gamma }_1,{\gamma }_2,\dots ,{\gamma }_i+1,\dots ,{\gamma }_j-1,\dots ,{\gamma }_n)\text{.}& \text{(1.4)}\end{array}$$

Let $x_1,x_2,\dots ,x_n$ be commuting variables and for each $\gamma =({\gamma }_1,{\gamma }_2,\dots ,{\gamma }_n)\in {\mathbb{Z}}^n$ define $$\begin{array}{cc}{x}^{\gamma }=x_1^{{\gamma }_1}{x}_2^{{\gamma }_2}\cdots x_n^{{\gamma }_n}\text{.}& \text{(1.5)}\end{array}$$ Define an action of $S_n$ on monomials by $$\begin{array}{cc}w{x}^{\gamma }=x^{w\gamma }\text{.}& \text{(1.6)}\end{array}$$ The ring $${\Lambda }_n=\mathbb{Q}{[x_1,x_2,\dots ,x_n]}^{S_n}$$ is the ring of symmetric functions.

Bases of symmetric functions

For each partition $\lambda$ define the monomial symmetric function by $$\begin{array}{cc}{m}_{\lambda }=\sum _{\gamma \in S_n\lambda }{x}^{\gamma },& \text{(1.7)}\end{array}$$ where the sum runs over all $\gamma \in {\mathbb{Z}}^n$ in the $S_n$ orbit of $\lambda ,$ i.e., over all distinct permutations of $\lambda \text{.}$

The Schur functions are given by $$\begin{array}{cc}{s}_{\lambda }=\frac{{\sum }_{w\in S_n}\epsilon \left(w\right)x^{w(\lambda +\rho )}}{{\sum }_{w\in S_n}\epsilon \left(w\right)x^{w\rho }},& \text{(1.8)}\end{array}$$ where $\rho =(n-1,n-2,\dots ,1,0)\text{.}$

The elementary symmetric functions are given by defining $$\begin{array}{ccc}{e}_0& =& 1,\\ e_r& =& \sum _{1\le i_1<\cdots <i_r\le n}{x}_{i_1}{x}_{i_2}\cdots x_{i_r},\end{array}$$ for each positive integer $r,$ and $$\begin{array}{cc}{e}_{\lambda }=e_{{\lambda }_1}{e}_{{\lambda }_2}\cdots e_{{\lambda }_n},& \text{(1.9)}\end{array}$$ for all partitions $\lambda \text{.}$

The homogeneous symmetric functions are given by defining $$\begin{array}{ccc}{h}_0& =& 1,\\ h_{}& =& \sum _{1\le i_1\le \cdots \le i_r\le n}{x}_{i_1}{x}_{i_2}\cdots x_{i_r},\end{array}$$ for each positive integer $r,$ and $$h_{\gamma }=h_{{\gamma }_1}{h}_{{\gamma }_2}\cdots h_{{\gamma }_n},$$ for all sequences $\gamma \in {\mathbb{Z}}^n\text{.}$

Define integers $K_{\lambda \mu }$ by $$\begin{array}{cc}{s}_{\lambda }=\sum _{\mu \in 𝒫}{K}_{\lambda \mu }{m}_{\mu }\text{.}& \text{(1.10)}\end{array}$$ One has the following (nontrivial) facts:

(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 $\mu \nleqq \lambda \text{.}$

Each of the sets ${\left\{m_{\lambda }\right\}}_{\lambda \in 𝒫},$ ${\left\{s_{\lambda }\right\}}_{\lambda \in 𝒫},$ ${\left\{e_{\lambda }\right\}}_{\lambda \in 𝒫},$ ${\left\{h_{\lambda }\right\}}_{\lambda \in 𝒫},$ forms a $\mathbb{Z}\text{-basis}$ of ${\Lambda }_n\text{.}$

Inner product

There is an inner product on the ring of symmetric functions given by making the Schur functions orthonormal, $$\begin{array}{cc}⟨s_{\lambda },s_{\mu }⟩={\delta }_{\lambda \mu }\text{.}& \text{(1.11)}\end{array}$$

Further facts

The homogeneous symmetric functions are the dual basis to the basis of monomial symmetric functions, $$\begin{array}{cc}⟨h_{\lambda },m_{\mu }⟩={\delta }_{\lambda \mu }\text{.}& \text{(1.12)}\end{array}$$ A consequence of this is that $$\begin{array}{cc}{h}_{\mu }=\sum _{\lambda }{s}_{\lambda }{K}_{\lambda \mu }\text{.}& \text{(1.13)}\end{array}$$

One has the following "Jacobi-Trudi" formula for the Schur functions in terms of the homogeneous symmetric functions, $$\begin{array}{cc}{s}_{\lambda }=\sum _{w\in S_n}\epsilon \left(w\right)h_{\lambda +\rho -w\rho }\text{.}& \text{(1.14)}\end{array}$$ There is also a formula for the Schur function in terms of raising operators and the homogeneous symmetric function. $$\begin{array}{cc}{s}_{\lambda }=\prod _{i<j}(1-R_{ij})h_{\lambda }\text{.}& \text{(1.15)}\end{array}$$

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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