Schur functions and multiplicities

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

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 [P] with basis {Xλ | λP} and product XλXμ = Xλ+μ, for λ,μP. The group W acts on [P] by wXλ =Xwλ, for wW,λP .

The ring of symmetric functions and Fock space are

[P] W={ f [P] | wf=f for all wW} and [P] det={ f [P] | wf=det(w)f for all wW}, (5.1)
respectively. For λP define
mλ= γWλ Xγ and aλ =wW det(w) Xwλ. (5.2)
The straightening laws for these elements are
mwλ =mλ and awλ =det(w) aλ, for wW and λP. (5.3)
The second relation implies that aλ=0 if there exists wWλ with det(w)1, and it follows from the straightening laws that
[P] W has basis {mλ | λP+ } and [P] det has basis {aλ+ρ | λP+} (5.4)
where P+ and ρ are as in (2.14) and (2.16), respectively.

The Weyl characters or Schur functions are defined by

sλ= aλ+ρ aρ, for λP. (5.5)
The following theorem shows that the sλ are the elements of [P] and that
[P]W has basis {sλ | λP+}. (5.6)

Fock space [P]det is a free [P]W module with generator

aρ=xρ αR+ (1-x-α ) and the map [P]W [P]det f aρf sλ aλ+ρ
is a [P]W module isomorphism.

Let f [P]det and let αR+. If fγ is the coefficient of xγ in f then γP fγxγ =f=-sαf =γP -fγ xsαγ , and so f= γP γ, α0 fγ (xγ- xsαγ ), since fsα γ =-fγ. Since each term xγ- xsαγ is divisible by 1-x-α, f is divisible by 1-x-α, and thus
each f [P]det is divisible by xρ αR+ (1-x-α ) (5.7)
since the polynomials 1-x-α, αR+, are coprime in [P] and xρ is a unit in [P]. Comparing coefficients of the maximal terms in aρ and xρ αR+ (1-x-α ) shows that aρ= xρ αR+ (1-x-α ).

Thus each f [P]det is divisible by aρ and so the inverse of multiplication by aρ is well defined.

The dot action of Won P is given by

wμ =w(μ+ρ) -ρ, for wW, μP. (5.8)
The straightening law
swμ =det(w)sμ ,for μP,wW. (5.9)
for the Schur functions follows from the straightening law for the aμ in (5.3).

Let f [P]W and write f= γ fγxγ so that fγ is the coefficient of xγ in f. Then f= μP+ fμmμ = λP+ ηλsλ, where ηλ= wW det(w) fλ+ρ -wρ.

The first equality is immediate from the definition of mμ. Since f [P]W and the sλ, λP+, are a basis of [P]W, the element f can be written as a linear combination of sλ. Then, since eλ+ρ is the unique dominant term in aλ+ρ, ηλ =(coefficient of sλ in f) =(coefficient of aλ+ρ in faρ) = (coefficient of eλ+ρ in μP wW det(w)fμ eμ+wρ) .

If ν 𝔥* and f= μP fμeμ [P] define f(eν) =μP fμe μ,ν . Let λP+, t>0 ,q=et and ρ =12 αR+ α. Then sλ( qρ) = αR+ [λ+ρ, α] [ρ, α] and sλ(1) = αR+ λ+ρ, α ρ, α where [k] =(qk-1) /(q-1) for an integer k0.


aλ+ρ ( etρ ) =wW det(w) ew( λ+ρ), tρ =wW det(w) e wρ, t(λ+ρ) =aρ (e t(λ+ρ) ) =eρ , t(λ+ρ) αR+ (1-e -α ,t(λ+ρ) ). Thus sλ(e tρ) = aλ+ρ (etρ ) aρ( etρ ) = e ρ, t(λ+ρ) eρ, tρ αR+ 1- e -α, t(λ+ρ) 1- e -α, tρ = q- λ,ρ αR+ qλ+ρ, α -1 q ρ, α -1 and sλ(1) =limq1 sλ( qρ) = αR+ λ+ρ, α ρ,α .


The weight multiplicities are the integers Kλγ, λ P+, γ P, defined by the equations
sλ =γP Kλγ xγ = μP+ Kλμ mμ. (5.10)
The tensor product multiplicities are the integers cμνλ , μ,ν,λ P+, defined by the equations
sμsν = λP+ cμνλ sλ. (5.11)
The partition function is the function p:P 0 defined by the equation
αR+ 1 1-x-α = γP p(γ) x-γ. (5.12)

Let λ,μ,ν P+.

  1. (a) Kλλ =1,   Kλ,wμ =Kλμ, for w W,   and   Kλμ =0 unless μλ.
  2. (b) Kλμ =wW det(w) p(w(λ+ρ) -(μ+ρ)).
  3. (c) cμνλ =v,w W det(vw) p(v(μ+ρ) +w(ν+ρ) -(λ+ρ) -ρ).

Proof (a).
The equality Kλ,wμ =Kλμ follows from the definition and the fact that sλ [P]W. If w 1 then w(λ+ρ) < λ+ρ so that w(λ+ρ) -ρ<λ and sλ = ( wW det(w) xw(λ+ρ) -ρ ) αR+ 1 1-x-α = xλ+ (lower terms in dominance order). Thus Kλλ=1 and Kλμ =0 unless μλ.

Proof (b).
The coefficient of xμ in sλ= ( wW det(w) xw(λ+ρ) -ρ) αR+ 1 1-x-α = wW γQ+ det(w) p(γ) xw(λ+ρ) -ρ-γ, has a contribution det(w) p(γ) when w(λ+ρ) -ρ-γ=μ so that γ= w(λ+ρ) -(μ+ρ).

Proof (c).
Let ε= wW det(w)w. Since cλμν is the coefficient of xν+ρ in sμsν aρ = ε( xμ+ρ) ε( xν+ρ) aρ =( v,wW det(vw) xv(μ+ρ) +w(ν+ρ) -ρ ) ( αR+ 1 1-x-α ) = v,wW γQ+ det(vw) p(γ) xv(μ+ρ) +w(ν+ρ) -γ-ρ , there is a contribution det(vw) p(γ) to the coefficient cμνλ when λ+ρ =v(μ+ρ) +w(ν+ρ) -γ-ρ so that γ= v(μ+ρ) +w(μ+ρ) -(λ+ρ) -ρ.

Fix J{1,2, ,n}. The subgroup of W generated by the reflections in the hyperplanes Hαj, jJ,

WJ =sj | jJ acts on 𝔥*   with   CJ ={μ 𝔥* | μ, αj >0 for jJ} (5.13)
as a fundamental chamber. The group WJ acts on P and
[P] WJ ={f[P] | wf=f for wWJ} (5.14)
is a subalgebra of [P] which contains [P] W. If
CJ = {μ𝔥* | μ, αj 0 for jJ}, P+J =P CJ, ρJ =jJ ωj, (5.15)
aJμ = wWJ det(w)wXμ ,for μP, and sλJ = aλ+ρJJ aρJJ , for λP, (5.16)
then {sλJ | λPJ+} is a basis of [P] WJ. The restriction multiplicities are the integers cJ,νλ given by
sλ = νPJ+ 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].


[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

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