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

Orthogonality

In this section we introduce a formalism for which allows us to work in the universe of identities for Weyl group symmetric functions.

We return to the setup of section 2. Let $P^{+}$ be the set of dominant integral weights, ${\chi }^{\lambda },$ $\lambda \in P^{+},$ the Weyl characters and $m_{\mu },$ $\mu \in P^{+}$ the monomial symmetric functions (2.7). Let $\hat{M}$ be a subset of $P^{+}\text{.}$

Let $\hat{B}$ be a vector space of dimension $\text{Card}\left(\hat{M}\right)$ and let ${\xi }_{\lambda },$ $\lambda \in \hat{M}$ be linearly independent functions on $\hat{B}\text{.}$ Let $$\text{bichar}\hspace{0.17em}M=\sum _{\lambda \in \hat{M}}{\xi }_{\lambda }{\chi }^{\lambda },$$ and, for each $\mu \in P,$ define ${\eta }_{\mu }$ by $${\eta }_{\mu }={\left[\text{bichar}M\right]}_{e^{\mu }},$$ where ${\left[f\right]}_{e^{\mu }}$ denotes taking the coefficient of $e^{\mu }$ in $f\text{.}$ Let $P_M^{+}$ be the set of $\mu \in P^{+}$ such that ${\eta }_{\mu }$ is not identically $0\text{.}$ Define $$\begin{array}{ccc}{\Lambda }_{\hat{M}}& =& \text{span}\{{\chi }^{\lambda },\lambda \in \hat{M}\},\\ {\Lambda }_{\hat{M}}^{*}& =& \text{span}\{{\xi }_{\lambda },\lambda \in \hat{M}\},\end{array}$$ and define a pairing between ${\Lambda }_{\hat{M}}$ and ${\Lambda }_{\hat{M}}^{*}$ by defining $$\begin{array}{cc}⟨{\xi }_{\lambda },{\chi }^{\mu }⟩={\delta }_{\lambda \mu }\text{.}& \text{(5.1)}\end{array}$$

Suppose that $\hat{M}$ is finite and let $u_{\lambda }$ and $v_{\lambda }$ be bases of ${\Lambda }_{\hat{M}}$ and ${\Lambda }_{\hat{M}}^{*}$ indexed by the elements
of $\hat{M}\text{.}$ Then the following conditions are equivalent.

a)	$⟨u_{\lambda },v_{\mu }⟩={\delta }_{\lambda \mu }$ for all $\lambda ,\mu \in \hat{M}\text{.}$
b)	${\sum }_{\lambda \in \hat{M}}{u}_{\lambda }{v}_{\lambda }=\text{bichar}M\text{.}$

		Proof.
	The proof is exactly as in [Mac1979] Chapt. I (4.6) p.34. $\square$

1)	"Jacobi-Trudi" identity. For each $\lambda \in \hat{M},$ $${\xi }_{\lambda }=\sum _{w\in W}\epsilon \left(w\right){\eta }_{\lambda +\rho -w\rho }\text{.}$$
2)	Raising operator identity. Allow the raising operators (2.4) to act upon the ${\eta }_{\mu }$ by defining $R\left({\eta }_{\mu }\right)={\eta }_{R\left(\mu \right)}$ far each sequence $R=R_{{\beta }_1}{R}_{{\beta }_2}\cdots R_{{\beta }_k}\text{.}$ Then for all $\lambda \in \hat{M}$ $${\xi }^{\lambda }=\prod _{\alpha >0}(1-R_{\alpha }){\eta }_{\lambda }\text{.}$$
3)	For each $\mu \in P,$ $${\eta }_{\mu }=\sum _{\lambda \in \hat{M}}{\xi }_{\lambda }{K}_{\lambda \mu }\text{.}$$
4)	For all $\mu \in P$ and all $w\in W,$ $${\eta }_{\mu }={\eta }_{w\mu }\text{.}$$
5)	If $\hat{M}$ is finite and $P_M^{+}=\hat{M}$ then $m_{\mu }$ and ${\eta }_{\mu }$ are bases of ${\Lambda }_{\hat{M}}$ and ${\Lambda }_{\hat{M}}^{*}$ respectively and with respect to the pairing between ${\Lambda }_{\hat{M}}$ and ${\Lambda }_{\hat{M}}^{*}$ defined in (5.1) $$⟨{\eta }_{\lambda },m_{\mu }⟩={\delta }_{\lambda \mu }\text{.}$$

		Proof.
	1) follows exactly as in (4.6). 2) follows from 1) in the same fashion as in the proof of Theorem (2.15). By definition $$\begin{array}{ccc}{\eta }_{\mu }& =& {\left[\sum _{\lambda \in \hat{M}}{\xi }_{\lambda }{\chi }^{\lambda }\right]}_{e^{\mu }}\\ & =& \sum _{\lambda \in \hat{M}}{\xi }_{\lambda }{\left[{\chi }^{\lambda }\right]}_{e^{\mu }}\text{.}\end{array}$$ By (2.10), ${\left[{\chi }^{\lambda }\right]}_{e^{\mu }}=K_{\lambda \mu },$ giving 3). 4) follows from 3) and the definition of the monomial symmetric functions (using the fact (2.3) that the Weyl group orbits of elements in $P^{+}$ are all distinct). It follows from 1) that ${\xi }_{\lambda }$ is a finite linear combination of ${\eta }_{\mu }\text{.}$ Thus the ${\eta }_{\mu }$ span ${\Lambda }_{\hat{M}}^{*}\text{.}$ Since $\hat{M}=P_M^{+}$ the ${\eta }_{\mu }$ form a basis of ${\Lambda }_{\hat{M}}^{*}\text{.}$ That $⟨{\eta }_{\lambda },m_{\mu }⟩={\delta }_{\lambda \mu }$ now follows from (5.2), completing the proof of 5). $\square$

Examples

1. Let $x_1,\dots ,x_n$ and $y_1,\dots ,y_n$ be commuting variables. Let $𝒫$ denote the set of partitions of length $\le n\text{.}$ $\lambda \in 𝒫$ be a partition and let $s_{\lambda }\left(X_n\right)$ and $s_{\lambda }\left(Y_n\right)$ be the Schur functions associated to $\lambda$ in the $X$ and the $Y$ variables respectively. Consider the identity $$\prod _{1\le i,j\le n}\frac{1}{1-x_i{y}_j}=\sum _{\lambda \in 𝒫}{s}_{\lambda }\left(X_n\right)s_{\lambda }\left(Y_n\right)\text{.}$$ Then, for each $\mu =({\mu }_1,\dots ,{\mu }_n)\in 𝒫,$ define $$h_{\mu }\left(Y_n\right)={\left[\prod _{1\le i,j\le n}\frac{1}{1-x_i{y}_j}\right]}_{x^{\mu }}\text{.}$$ Then $h_{\mu }\left(Y_n\right)=h_{{\mu }_1}\left(Y_n\right)\cdots h_{{\mu }_n}\left(Y_n\right)$ where $$h_r\left(Y_n\right)={\left[\prod _j\frac{1}{1-x{y}_j}\right]}_{x^r},$$ for each integer $r\ge 0\text{.}$ We may define a pairing between symmetric functions in the $X$ variables and symmetric functions in the $Y$ variables by defining $$⟨s_{\lambda }\left(X_n\right),s_{\mu }\left(Y_n\right)⟩={\delta }_{\lambda \mu },$$ for all $\lambda ,\mu \in 𝒫\text{.}$ Then $$⟨m_{\lambda }\left(X_n\right),h_{\mu }\left(Y_n\right)⟩={\delta }_{\lambda \mu },$$ for all $\lambda ,\mu \in 𝒫,$ where $m_{\lambda }\left(X_n\right)$ denotes the monomial symmetric function. For each $\lambda \in 𝒫$ $$\begin{array}{ccc}{s}_{\lambda }\left(Y_n\right)& =& \sum _{w\in S_n}\epsilon \left(w\right)h_{\lambda +\delta -w\delta }\left(Y_n\right)\\ & =& \text{det}\left(h_{{\lambda }_i-i+j}\left(Y_n\right)\right),\end{array}$$ where $\delta =(n-1,n-2,\dots ,1,0)\text{.}$

2. Keeping the same notation as in example 1, let $\hat{M}$ be the set of partitions $\lambda$ such that $\lambda \in 𝒫$ and such that the conjugate partition $\lambda \prime \in 𝒫\text{.}$ Consider the identity $$\prod _{1\le i,j\le n}(1+x_i{y}_j)=\sum _{\lambda \in \hat{M}}{s}_{\lambda \prime }\left(X_n\right)s_{\lambda }\left(Y_n\right)\text{.}$$ Then, for each $\mu =({\mu }_1,\dots ,{\mu }_n)\in 𝒫,$ define $$e_{\mu }\left(Y_n\right)={\left[\prod _{1\le i,j\le n}(1+x_i{y}_j)\right]}_{x^{\mu }}\text{.}$$ Then $e_{\mu }\left(Y_n\right)=e_{{\mu }_1}\left(Y_n\right)\cdots e_{{\mu }_n}\left(Y_n\right)$ where $$e_r\left(Y_n\right)={\left[\prod _j(1+x{y}_j)\right]}_{x^r},$$ for each integer $r\ge 0\text{.}$ Define a pairing between the symmetric functions of degree $\le n$ in the $X$ variables and the symmetric functions of degree $\le n$ in the $Y$ variables by defining $$⟨s_{\lambda }\left(X_n\right),s_{\mu }\left(Y_n\right)⟩={\delta }_{\lambda \mu \prime },$$ for all $\lambda ,\mu \in \hat{M}\text{.}$ Then we have that $$⟨m_{\lambda }\left(X_n\right),e_{\mu }\left(Y_n\right)⟩{\delta }_{\lambda \mu },$$ for all $\lambda ,\mu \in \hat{M},$ and that $$\begin{array}{ccc}{s}_{\lambda }\left(Y_n\right)& =& \sum _{w\in S_n}\epsilon \left(w\right)e_{\lambda \prime +\delta -w\delta }\left(Y_n\right)\\ & =& \text{det}\hspace{0.17em}\left(e_{{\lambda }_i^{\prime }-i+j}\left(Y_n\right)\right)\text{.}\end{array}$$

3. Keeping the same notation as in example $1,$ set $k\le n$ and let ${\Lambda }_k$ be the set of partitions $\lambda$ such that $\lambda \in 𝒫$ and such that $\left|\lambda \right|={\sum }_i{\lambda }_i=k,$ i.e., $\lambda ⊢k\text{.}$ Consider the identity $$p_{\rho }\left(X_n\right)=\sum _{\lambda ⊢k}{\chi }^{\lambda }\left(\rho \right)s_{\lambda }\left(X_n\right),$$ where $\rho ⊢k,$ $p_{\mu }\left(X_n\right)$ denotes the power symmetric function corresponding to the partition $\mu ,$ and ${\chi }^{\lambda }$ is the irreducible character of the symmetric group $S_k\text{.}$ Then for $\mu ,p⊢k$ define $${\eta }_{\mu }\left(\rho \right)={\left[p_{\rho }\left(X_n\right)\right]}_{x^{\mu }}\text{.}$$ Then ${\eta }_{\mu }$ is the character of $S_k$ determined by inducing the trivial character of the Young subgroup $S_{{\mu }_1}\times \cdots \times S_{{\mu }_m}$ to $S_k\text{.}$ Define a pairing between the symmetric functions in the $X$ variables and characters of the symmetric group $S_k$ by defining $$⟨{\chi }^{\lambda },s_{\lambda }\left(X_n\right)⟩={\delta }_{\lambda \mu },$$ for all $\lambda ,\mu ⊢k\text{.}$ Then we have that $$⟨{\eta }_{\lambda }\left(X_n\right),m_{\mu }\left(X_n\right)⟩={\delta }_{\lambda \mu },$$ for all $\lambda ,\mu ⊢k,$ and that $${\chi }_{\lambda }\left(\rho \right)=\sum _{w\in S_n}\epsilon \left(w\right){\eta }_{\lambda +\delta -w\delta }\left(\rho \right),$$ for all $\lambda ,\rho ⊢k\text{.}$

4. Let $x_1,\dots ,x_n$ and $y_1,\dots ,y_n$ be commuting variables let $\rho =(n,n-1,\dots ,1)$ and let $W{C}_n$ be the hyperoctahedral group. For each $\lambda \in 𝒫$ define $$s{c}_{\lambda }\left(Y_n^{\pm 1}\right)=\frac{{\sum }_{w\in W{C}_n}\epsilon \left(w\right)w\left(y^{\lambda +\rho }\right)}{{\sum }_{w\in W{C}_n}\epsilon \left(w\right)w\left(y^{\rho }\right)}=\frac{\text{det}(y_i^{{\lambda }_i+n-j+1}-y_i^{-({\lambda }_i+n-j+1)})}{\text{det}(y_i^{n-j+1}-y_i^{-(n-j+1)})}\text{.}$$ Then consider the identity $$\frac{{\prod }_{1\le i<j\le n}(1-x_i{x}_j)}{{\prod }_{1\le i,j\le n}(1-x_i{y}_j)(1-x_i{y}_j^{-1})}=\sum _{\lambda \in 𝒫}s{c}_{\lambda }\left(Y_n^{\pm 1}\right)s_{\lambda }\left(X_n\right),$$ where $s_{\lambda }\left(X_n\right)$ denotes the Schur function in the $X$ variables. For each $\mu =({\mu }_1,\dots ,{\mu }_n)\in 𝒫$ define $$h{c}_{\mu }\left(Y_n^{\pm 1}\right)={\left[\frac{{\prod }_{1\le i<j\le n}(1-x_i{x}_j)}{{\prod }_{1\le i,j\le n}(1-x_i{y}_j)(1-x_i{y}_j^{-1})}\right]}_{x^{\mu }}\text{.}$$ Define a bilinear pairing between functions in the $y_i^{\pm 1}$ symmetric under the hyperoctahedral group and classical symmetric functions in the $X$ variables by $$⟨s_{\lambda }\left(X_n\right),s{c}_{\mu }\left(Y_n^{\pm 1}\right)⟩{\delta }_{\lambda \mu },$$ for all $\lambda ,\mu \in 𝒫\text{.}$ Then we have that $$⟨m_{\lambda }\left(X_n\right),h{c}_{\mu }\left(Y_n^{\pm 1}\right)⟩={\delta }_{\lambda \mu },$$ for all $\lambda ,\mu \in 𝒫,$ and that $$\begin{array}{ccc}s{c}_{\lambda }\left(Y_n\right)& =& \sum _{w\in S_n}\epsilon \left(w\right)h{c}_{\lambda +\delta -w\delta }\left(Y_n\right)\\ & =& \frac{1}{2}\text{det}\hspace{0.17em}(h_{{\lambda }_i-i-j+2}\left(Y_n^{\pm 1}\right)+h_{{\lambda }_i-i+j}\left(Y_n^{\pm 1}\right)),\end{array}$$ where $h_r\left(Y_n^{\pm 1}\right)$ is defined by $$\prod _{1\le j\le n}\frac{1}{(1-x{y}_j)(1-x{y}_j^{-1})}=\sum _{r\ge 0}{h}_r\left(Y_n^{\pm 1}\right)x^r\text{.}$$

5. In a similar fashion the identity $$\prod _{1\le i\le j\le n}(1-x_i{x}_j)\prod _{1\le i,j\le n}(1+x_i{y}_j)(1+x_i{y}_j^{-1})=\sum _{\lambda \in \hat{M}}s{c}_{\lambda }\left(Y_n^{\pm 1}\right)s_{\lambda \prime }\left(X_n\right),$$ where the set $\hat{M}$ is as in example 2, gives the formula $$s{c}_{\lambda }\left(Y_n^{\pm 1}\right)=\frac{1}{2}\text{det}\hspace{0.17em}((e_{{\lambda }_i^{\prime }-i-j+2}\left(Y_n^{\pm 1}\right)-e_{{\lambda }_i^{\prime }-i-j}\left(Y_n^{\pm 1}\right))+(e_{{\lambda }_i^{\prime }-i+j}\left(Y_n^{\pm 1}\right)-e_{{\lambda }_i^{\prime }-i+j-2}\left(Y_n^{\pm 1}\right))),$$ where $e_r\left(Y_n^{\pm 1}\right)$ is defined by $$\prod _{j=1}^N(1+x{y}_j)(1+x{y}_j^{-1})=\sum _{r\ge 0}{e}_r\left(Y_n^{\pm 1}\right)x^r\text{.}$$

6. Let $x_1,\dots ,x_n$ and $y_1,\dots ,y_N$ be two sets of variables and let $s{c}_{\lambda }\left(X_n^{\pm }\right)$ and $s{c}_{\lambda }\left(Y_N^{\pm }\right)$ be defined as in example 5. For a partition $\lambda ,$ we shall write $\lambda \subseteq \left(N^n\right)$ if $\lambda$ is of length $\le n$ and the conjugate partition $\lambda \prime$ is of length $\le N\text{.}$ Given a partition $\lambda \subseteq \left(N^n\right)$ and define the opposite partition $\stackrel{\sim }{\lambda }$ by ${\stackrel{\sim }{\lambda }}_{N-i+1}=n-{\lambda }_i^{\prime }\text{.}$ The Morris identity $$\prod _{i=1}^n\prod _{j=1}^N(x_i+x_i^{-1}+y_j+y_j^{-1})=\sum _{\lambda \subseteq \left(N^n\right)}s{c}_{\lambda }\left(X_n^{\pm 1}\right)s{c}_{\stackrel{\sim }{\lambda }}\left(Y_N^{\pm 1}\right)$$ gives the formula $$\begin{array}{ccc}s{c}_{\lambda }\left(Y_N^{\pm 1}\right)& =& \sum _{w\in W{C}_n}\epsilon \left(w\right)e_{\lambda \prime +\rho -w\rho }\left(Y_N^{\pm 1}\right)\\ & =& \text{det}\hspace{0.17em}(e_{{\lambda }_i^{\prime }-i+j}\left(Y_N^{\pm 1}\right)-e_{{\lambda }_i^{\prime }-i-j}\left(Y_N^{\pm 1}\right)),\end{array}$$ where $e_r\left(Y_N^{\pm 1}\right)$ is defined by $$\prod _{j=1}^N(1+x{y}_j)(1+x{y}_j^{-1})=\sum _{r\ge 0}{e}_r\left(Y_N^{\pm 1}\right)x^r,$$ and $\rho =(n,n-1,\dots ,1,0)\text{.}$

Remarks. Notice that the only case in the above examples where the "Jacobi-Trudi" formula is an alternating sum over a Weyl group other than the symmetric group is in example 6. The "Jacobi-Trudi" identity in example 6 can be obtained easily from the "Jacobi-Trudi" identity of example 5 and is thus given in [KTe1987]. The idea of deriving this identity from the Morris identity is, I believe, new.

The "Jacobi-Trudi" formulas given in examples 4 and 5 are analoguous to the classical ones given in examples 1 and 2. There are known analogues of the formulas in the above examples for the Weyl characters corresponding to all types A,B,C,D. The appropriate bicharacter identities and the corresponding "Jacobi-Trudi" formulas for these analogues as well as the above examples can be found with proofs in the forthcoming paper [RamSTLD]. Many of these results have been in the literature for a long time.

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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