## Schur functions and multiplicities

Last update: 8 December 2012

## Schur functions

Use notations for the Weyl group $W$ and the lattice $P$ as in Section 2. The group algebra of $P$ is the ring $ℤ[P] with basis {Xλ | λ∈P} and product XλXμ = Xλ+μ,$ for $\lambda ,\mu \in P$. The group $W$ acts on $ℤ\left[P\right]$ by $wXλ =Xwλ, for w∈W,λ∈P .$

The ring of symmetric functions and Fock space are

 $ℤ[P] W={ f∈ ℤ[P] | wf=f for all w∈W} and ℤ[P] det={ f∈ ℤ[P] | wf=det(w)f for all w∈W},$ (5.1)
respectively. For $\lambda \in P$ define
 $mλ= ∑ γ∈Wλ Xγ and aλ =∑w∈W det(w) Xwλ.$ (5.2)
The straightening laws for these elements are
 $mwλ =mλ and awλ =det(w) aλ, for w∈W and λ∈P.$ (5.3)
The second relation implies that ${a}_{\lambda }=0$ if there exists $w\in {W}_{\lambda }$ with $\mathrm{det}\left(w\right)\ne 1$, and it follows from the straightening laws that
 $ℤ[P] W has basis {mλ | λ∈P+ } and ℤ[P] det has basis {aλ+ρ | λ∈P+}$ (5.4)
where ${P}^{+}$ and $\rho$ are as in (2.14) and (2.16), respectively.

The Weyl characters or Schur functions are defined by

 $sλ= aλ+ρ aρ, for λ∈P.$ (5.5)
The following theorem shows that the ${s}_{\lambda }$ are the elements of $ℤ\left[P\right]$ and that
 $ℤ[P]W has basis {sλ | λ∈P+}.$ (5.6)

Fock space ${ℤ\left[P\right]}^{\mathrm{det}}$ is a free ${ℤ\left[P\right]}^{W}$ module with generator

 $aρ=xρ ∏ α∈R+ (1-x-α ) and the map ℤ[P]W → ℤ[P]det f ↦ aρf sλ ↦ aλ+ρ$
is a ${ℤ\left[P\right]}^{W}$ module isomorphism.

Proof.
Let $f\in {ℤ\left[P\right]}^{\mathrm{det}}$ and let $\alpha \in {R}^{+}$. If ${f}_{\gamma }$ is the coefficient of ${x}^{\gamma }$ in $f$ then $∑γ∈P fγxγ =f=-sαf =∑γ∈P -fγ xsαγ , and so f= ∑ γ∈P ⟨γ, α∨⟩≥0 fγ (xγ- xsαγ ),$ since ${f}_{{s}_{\alpha }\gamma }=-{f}_{\gamma }$. Since each term ${x}^{\gamma }-{x}^{{s}_{\alpha }\gamma }$ is divisible by $1-{x}^{-\alpha }$, $f$ is divisible by $1-{x}^{-\alpha }$, and thus
 $each f∈ ℤ[P]det is divisible by xρ ∏ α∈R+ (1-x-α )$ (5.7)
since the polynomials $1-{x}^{-\alpha }$, $\alpha \in {R}^{+}$, are coprime in $ℤ\left[P\right]$ and ${x}^{\rho }$ is a unit in $ℤ\left[P\right]$. Comparing coefficients of the maximal terms in ${a}_{\rho }$ and ${x}^{\rho }\prod _{\alpha \in {R}^{+}}\left(1-{x}^{-\alpha }\right)$ shows that $aρ= xρ ∏ α∈R+ (1-x-α ).$

Thus each $f\in {ℤ\left[P\right]}^{\mathrm{det}}$ is divisible by ${a}_{\rho }$ and so the inverse of multiplication by ${a}_{\rho }$ is well defined. $\square$

The dot action of $W$on $P$ is given by

 $w∘μ =w(μ+ρ) -ρ, for w∈W, μ∈P.$ (5.8)
The straightening law
 $sw∘μ =det(w)sμ ,for μ∈P,w∈W.$ (5.9)
for the Schur functions follows from the straightening law for the ${a}_{\mu }$ in (5.3).

Let $f\in {ℤ\left[P\right]}^{W}$ and write $f=\sum _{\gamma }{f}_{\gamma }{x}^{\gamma }$ so that ${f}_{\gamma }$ is the coefficient of ${x}^{\gamma }$ in $f$. Then $f=∑ μ∈P+ fμmμ =∑ λ∈P+ ηλsλ, where ηλ= ∑w∈W det(w) fλ+ρ -wρ.$

 Proof. The first equality is immediate from the definition of ${m}_{\mu }$. Since $f\in {ℤ\left[P\right]}^{W}$ and the ${s}_{\lambda }$, $\lambda \in {P}^{+}$, are a basis of ${ℤ\left[P\right]}^{W}$, the element $f$ can be written as a linear combination of ${s}_{\lambda }$. Then, since ${e}^{\lambda +\rho }$ is the unique dominant term in ${a}_{\lambda +\rho }$, $ηλ =(coefficient of sλ in f) =(coefficient of aλ+ρ in faρ) = (coefficient of eλ+ρ in ∑μ∈P ∑w∈W det(w)fμ eμ+wρ) .□$

If $\nu \in {𝔥}_{ℝ}^{*}$ and $f=\sum _{\mu \in P}{f}_{\mu }{e}^{\mu }\in ℤ\left[P\right]$ define $f\left({e}^{\nu }\right)=\sum _{\mu \in P}{f}_{\mu }{e}^{⟨\mu ,\nu ⟩}$. Let $\lambda \in {P}^{+},t\in {ℝ}_{>0},q={e}^{t}$ and ${\rho }^{\vee }=\frac{1}{2}\sum _{\alpha \in {R}^{+}}{\alpha }^{\vee }$. Then $sλ( qρ∨) =∏ α∈R+ [⟨λ+ρ, α∨⟩] [⟨ρ, α∨⟩] and sλ(1) =∏ α∈R+ ⟨λ+ρ, α∨⟩ ⟨ρ, α∨⟩$ where $\left[k\right]=\left({q}^{k}-1\right)/\left(q-1\right)$ for an integer $k\ne 0$.

 Proof. $aλ+ρ ( etρ∨ ) =∑w∈W det(w) e⟨w( λ+ρ), tρ∨⟩ =∑w∈W det(w) e ⟨wρ∨, t(λ+ρ)⟩ =aρ∨ (e t(λ+ρ) ) =e⟨ρ∨ , t(λ+ρ) ⟩ ∏ α∈R+ (1-e ⟨-α∨ ,t(λ+ρ) ⟩ ).$ Thus $sλ(e tρ∨) = aλ+ρ (etρ∨ ) aρ( etρ∨ ) = e ⟨ρ∨, t(λ+ρ) ⟩ e⟨ρ∨, tρ⟩ ∏ α∈R+ 1- e⟨ -α∨, t(λ+ρ) ⟩ 1- e⟨ -α∨, tρ⟩ = q- ⟨λ,ρ∨ ⟩ ∏ α∈R+ q⟨λ+ρ, α∨⟩ -1 q ⟨ρ, α∨⟩ -1$ and $sλ(1) =limq→1 sλ( qρ∨) =∏ α∈R+ ⟨λ+ρ, α∨⟩ ⟨ρ,α∨ ⟩ .□$

## Multiplicities

The weight multiplicities are the integers ${K}_{\lambda \gamma }$, $\lambda \in {P}^{+}$, $\gamma \in P$, defined by the equations
 $sλ =∑γ∈P Kλγ xγ =∑ μ∈P+ Kλμ mμ.$ (5.10)
The tensor product multiplicities are the integers ${c}_{\mu \nu }^{\lambda }$, $\mu ,\nu ,\lambda \in {P}^{+}$, defined by the equations
 $sμsν =∑ λ∈P+ cμνλ sλ.$ (5.11)
The partition function is the function $p:P\to {ℤ}_{\ge 0}$ defined by the equation
 $∏ α∈R+ 1 1-x-α = ∑γ∈P p(γ) x-γ.$ (5.12)

Let $\lambda ,\mu ,\nu \in {P}^{+}$.

1. (a) ${K}_{\lambda \lambda }=1$,   ${K}_{\lambda ,w\mu }={K}_{\lambda \mu }$, for $w\in W$,   and   ${K}_{\lambda \mu }=0$ unless $\mu \le \lambda$.
2. (b) ${K}_{\lambda \mu }=\sum _{w\in W}\mathrm{det}\left(w\right)p\left(w\left(\lambda +\rho \right)-\left(\mu +\rho \right)\right)$.
3. (c) ${c}_{\mu \nu }^{\lambda }=\sum _{v,w\in W}\mathrm{det}\left(vw\right)p\left(v\left(\mu +\rho \right)+w\left(\nu +\rho \right)-\left(\lambda +\rho \right)-\rho \right)$.

 Proof (a). The equality ${K}_{\lambda ,w\mu }={K}_{\lambda \mu }$ follows from the definition and the fact that ${s}_{\lambda }\in {ℤ\left[P\right]}^{W}$. If $w\ne 1$ then $w\left(\lambda +\rho \right)<\lambda +\rho$ so that $w\left(\lambda +\rho \right)-\rho <\lambda$ and $sλ = ( ∑w∈W det(w) xw(λ+ρ) -ρ ) ⋅ ∏ α∈R+ 1 1-x-α = xλ+ (lower terms in dominance order).$ Thus ${K}_{\lambda \lambda =1}$and ${K}_{\lambda \mu }=0$ unless $\mu \le \lambda$. $\square$

 Proof (b). The coefficient of ${x}^{\mu }$ in $sλ= ( ∑w∈W det(w) xw(λ+ρ) -ρ) ∏ α∈R+ 1 1-x-α =∑ w∈W γ∈Q+ det(w) p(γ) xw(λ+ρ) -ρ-γ,$ has a contribution $\mathrm{det}\left(w\right)p\left(\gamma \right)$ when $w\left(\lambda +\rho \right)-\rho -\gamma =\mu$ so that $\gamma =w\left(\lambda +\rho \right)-\left(\mu +\rho \right)$. $\square$

 Proof (c). Let $\epsilon =\sum _{w\in W}\mathrm{det}\left(w\right)w$. Since ${c}_{\lambda }^{\mu \nu }$ is the coefficient of ${x}^{\nu +\rho }$ in $sμsν aρ = ε( xμ+ρ) ε( xν+ρ) aρ =( ∑v,w∈W det(vw) xv(μ+ρ) +w(ν+ρ) -ρ ) ( ∏ α∈R+ 1 1-x-α ) = ∑ v,w∈W γ∈Q+ det(vw) p(γ) xv(μ+ρ) +w(ν+ρ) -γ-ρ ,$ there is a contribution $\mathrm{det}\left(vw\right)p\left(\gamma \right)$ to the coefficient ${c}_{\mu \nu }^{\lambda }$ when $\lambda +\rho =v\left(\mu +\rho \right)+w\left(\nu +\rho \right)-\gamma -\rho$ so that $\gamma =v\left(\mu +\rho \right)+w\left(\mu +\rho \right)-\left(\lambda +\rho \right)-\rho$. $\square$

Fix $J\subseteq \left\{1,2,\dots ,n\right\}$. The subgroup of $W$ generated by the reflections in the hyperplanes ${H}_{{\alpha }_{j}}$, $j\in J$,

 (5.13)
as a fundamental chamber. The group ${W}_{J}$ acts on $P$ and
 $ℤ[P] WJ ={f∈ℤ[P] | wf=f for w∈WJ}$ (5.14)
is a subalgebra of $ℤ\left[P\right]$ which contains ${ℤ\left[P\right]}^{W}$. If
 $CJ‾ = {μ∈𝔥ℝ* | ⟨μ, αj∨⟩ ≥0 for j∈J}, P+J =P∩ CJ‾, ρJ =∑j∈J ωj,$ (5.15)
 $aJμ =∑ w∈WJ det(w)wXμ ,for μ∈P, and sλJ = aλ+ρJJ aρJJ , for λ∈P,$ (5.16)
then ${sλJ | λ∈PJ+} is a basis of ℤ[P] WJ.$ The restriction multiplicities are the integers ${c}_{J,\nu }^{\lambda }$ given by
 $sλ =∑ ν∈PJ+ cJ,νλ sνJ.$ (5.17)

## Notes and References

These notes are intended to supplement various lecture series given by Arun Ram. This is a typed version of section 5.1 of the paper [Ram2006].

## References

[Ram2006] Alcove walks, Hecke algebras, Spherical functions, crystals and column strict tableaux, Pure and Applied Mathematics Quarterly 2 no. 4 (Special Issue: In honor of Robert MacPherson, Part 2 of 3) (2006) 963-1013. MR2282411