## Weyl Group Symmetric Functions and the Representation Theory of Lie Algebras

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 $\stackrel{^}{M}$ be a subset of ${P}^{+}\text{.}$

Let $\stackrel{ˆ}{B}$ be a vector space of dimension $\text{Card}\left(\stackrel{^}{M}\right)$ and let ${\xi }_{\lambda },$ $\lambda \in \stackrel{^}{M}$ be linearly independent functions on $\stackrel{ˆ}{B}\text{.}$ Let $bichar M= ∑λ∈M^ ξλχλ,$ and, for each $\mu \in P,$ define ${\eta }_{\mu }$ by $ημ= [bicharM]eμ,$ 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 $ΛM^ = span{χλ,λ∈M^}, ΛM^* = span{ξλ,λ∈M^},$ and define a pairing between ${\Lambda }_{\stackrel{^}{M}}$ and ${\Lambda }_{\stackrel{^}{M}}^{*}$ by defining $⟨ξλ,χμ⟩ =δλμ. (5.1)$

Suppose that $\stackrel{^}{M}$ is finite and let ${u}_{\lambda }$ and ${v}_{\lambda }$ be bases of ${\Lambda }_{\stackrel{^}{M}}$ and ${\Lambda }_{\stackrel{^}{M}}^{*}$ indexed by the elementsâ€¨of $\stackrel{^}{M}\text{.}$ Then the following conditions are equivalent.

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

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

 1) "Jacobi-Trudi" identity. For each $\lambda \in \stackrel{^}{M},$ $ξλ=∑w∈W ε(w) ηλ+ρ-wρ.$ 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 $\lambda \in \stackrel{^}{M}$ $ξλ=∏α>0 (1-Rα)ηλ.$ 3) For each $\mu \in P,$ $ημ=∑λ∈M^ ξλKλμ.$ 4) For all $\mu \in P$ and all $w\in W,$ $ημ=ηwμ.$ 5) If $\stackrel{^}{M}$ is finite and ${P}_{M}^{+}=\stackrel{^}{M}$ then ${m}_{\mu }$ and ${\eta }_{\mu }$ are bases of ${\Lambda }_{\stackrel{^}{M}}$ and ${\Lambda }_{\stackrel{^}{M}}^{*}$ respectively and with respect to the pairing between ${\Lambda }_{\stackrel{^}{M}}$ and ${\Lambda }_{\stackrel{^}{M}}^{*}$ defined in (5.1) $⟨ηλ,mμ⟩ =δλμ.$

 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 $ημ = [∑λ∈M^ξλχλ]eμ = ∑λ∈M^ξλ [χλ]eμ.$ 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 ${\Lambda }_{\stackrel{^}{M}}^{*}\text{.}$ Since $\stackrel{^}{M}={P}_{M}^{+}$ the ${\eta }_{\mu }$ form a basis of ${\Lambda }_{\stackrel{^}{M}}^{*}\text{.}$ 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 $∏1≤i,j≤n 11-xiyj= ∑λ∈𝒫sλ (Xn)sλ (Yn).$ Then, for each $\mu =\left({\mu }_{1},\dots ,{\mu }_{n}\right)\in 𝒫,$ define $hμ(Yn)= [ ∏1≤i,j≤n 11-xiyj ] xμ .$ Then ${h}_{\mu }\left({Y}_{n}\right)={h}_{{\mu }_{1}}\left({Y}_{n}\right)\cdots {h}_{{\mu }_{n}}\left({Y}_{n}\right)$ where $hr(Yn)= [∏j11-xyj]xr,$ 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 $⟨sλ(Xn),sμ(Yn)⟩ =δλμ,$ for all $\lambda ,\mu \in 𝒫\text{.}$ Then $⟨mλ(Xn),hμ(Yn)⟩ =δλμ,$ for all $\lambda ,\mu \in 𝒫,$ where ${m}_{\lambda }\left({X}_{n}\right)$ denotes the monomial symmetric function. For each $\lambda \in 𝒫$ $sλ(Yn) = ∑w∈Sn ε(w) hλ+δ-wδ (Yn) = det(hλi-i+j(Yn)),$ where $\delta =\left(n-1,n-2,\dots ,1,0\right)\text{.}$

2. Keeping the same notation as in example 1, let $\stackrel{^}{M}$ 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 $∏1≤i,j≤n (1+xiyj) =∑λ∈M^ sλ′(Xn) sλ(Yn).$ Then, for each $\mu =\left({\mu }_{1},\dots ,{\mu }_{n}\right)\in 𝒫,$ define $eμ(Yn)= [∏1≤i,j≤n(1+xiyj)]xμ.$ Then ${e}_{\mu }\left({Y}_{n}\right)={e}_{{\mu }_{1}}\left({Y}_{n}\right)\cdots {e}_{{\mu }_{n}}\left({Y}_{n}\right)$ where $er(Yn)= [∏j(1+xyj)]xr,$ 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 $⟨sλ(Xn),sμ(Yn)⟩ =δλμ′,$ for all $\lambda ,\mu \in \stackrel{^}{M}\text{.}$ Then we have that $⟨mλ(Xn),eμ(Yn)⟩ δλμ,$ for all $\lambda ,\mu \in \stackrel{^}{M},$ and that $sλ(Yn) = ∑w∈Snε (w)eλ′+δ-wδ (Yn) = det (eλi′-i+j(Yn)).$

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 $|\lambda |={\sum }_{i}{\lambda }_{i}=k,$ i.e., $\lambda ⊢k\text{.}$ Consider the identity $pρ(Xn)= ∑λ⊢k χλ(ρ) sλ(Xn),$ 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 $ημ(ρ)= [pρ(Xn)]xμ.$ Then ${\eta }_{\mu }$ is the character of ${S}_{k}$ determined by inducing the trivial character of the Young subgroup ${S}_{{\mu }_{1}}×\cdots ×{S}_{{\mu }_{m}}$ 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 $⟨χλ,sλ(Xn)⟩ =δλμ,$ for all $\lambda ,\mu ⊢k\text{.}$ Then we have that $⟨ηλ(Xn),mμ(Xn)⟩ =δλμ,$ for all $\lambda ,\mu ⊢k,$ and that $χλ(ρ)= ∑w∈Snε (w)ηλ+δ-wδ (ρ),$ for all $\lambda ,\rho ⊢k\text{.}$

4. Let ${x}_{1},\dots ,{x}_{n}$ and ${y}_{1},\dots ,{y}_{n}$ be commuting variables let $\rho =\left(n,n-1,\dots ,1\right)$ and let $W{C}_{n}$ be the hyperoctahedral group. For each $\lambda \in 𝒫$ define $scλ(Yn±1)= ∑w∈WCn ε(w)w (yλ+ρ) ∑w∈WCn ε(w)w (yρ) = det ( yiλi+n-j+1- yi-(λi+n-j+1) ) det ( yin-j+1- yi-(n-j+1) ) .$ Then consider the identity $∏1≤i where ${s}_{\lambda }\left({X}_{n}\right)$ denotes the Schur function in the $X$ variables. For each $\mu =\left({\mu }_{1},\dots ,{\mu }_{n}\right)\in 𝒫$ define $hcμ(Yn±1)= [ ∏1≤i Define a bilinear pairing between functions in the ${y}_{i}^{±1}$ symmetric under the hyperoctahedral group and classical symmetric functions in the $X$ variables by $⟨sλ(Xn),scμ(Yn±1)⟩ δλμ,$ for all $\lambda ,\mu \in 𝒫\text{.}$ Then we have that $⟨mλ(Xn),hcμ(Yn±1)⟩ =δλμ,$ for all $\lambda ,\mu \in 𝒫,$ and that $scλ(Yn) = ∑w∈Snε (w)hcλ+δ-wδ (Yn) = 12det ( hλi-i-j+2 (Yn±1)+ hλi-i+j (Yn±1) ) ,$ where ${h}_{r}\left({Y}_{n}^{±1}\right)$ is defined by $∏1≤j≤n 1(1-xyj)(1-xyj-1) =∑r≥0hr (Yn±1)xr.$

5. In a similar fashion the identity $∏1≤i≤j≤n (1-xixj) ∏1≤i,j≤n (1+xiyj) (1+xiyj-1)= ∑λ∈M^s cλ(Yn±1) sλ′(Xn),$ where the set $\stackrel{^}{M}$ is as in example 2, gives the formula $scλ(Yn±1) =12det ( ( eλi′-i-j+2 (Yn±1)- eλi′-i-j (Yn±1) ) + ( eλi′-i+j (Yn±1)- eλi′-i+j-2 (Yn±1) ) ) ,$ where ${e}_{r}\left({Y}_{n}^{±1}\right)$ is defined by $∏j=1N (1+xyj) (1+xyj-1)= ∑r≥0er (Yn±1) xr.$

6. Let ${x}_{1},\dots ,{x}_{n}$ and ${y}_{1},\dots ,{y}_{N}$ be two sets of variables and let $s{c}_{\lambda }\left({X}_{n}^{±}\right)$ and $s{c}_{\lambda }\left({Y}_{N}^{±}\right)$ 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 $∏i=1n ∏j=1N ( xi+xi-1+ yj+yj-1 ) =∑λ⊆(Nn) scλ(Xn±1) scλ∼(YN±1)$ gives the formula $scλ(YN±1) = ∑w∈WCnε(w) eλ′+ρ-wρ (YN±1) = det ( eλi′-i+j (YN±1)- eλi′-i-j (YN±1) ) ,$ where ${e}_{r}\left({Y}_{N}^{±1}\right)$ is defined by $∏j=1N (1+xyj) (1+xyj-1)= ∑r≥0er (YN±1) xr,$ and $\rho =\left(n,n-1,\dots ,1,0\right)\text{.}$

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.