The Pieri-Chevalley formula

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

Last update: 21 February 2013

The Pieri-Chevalley formula

In this section we shall inductively apply formula (3.1d) to obtain an expansion of the product eλ[𝒪Xw] in K(G/B) in terms of the basis { [𝒪Xv] vW } . We use the path model of P. Littelmann to keep track of the combinatorics involved in iterating formula (3.1d).

The path model

Let Λ=iωi be the weight lattice and let 𝔥*=iωi. A path in 𝔥* is a piecewise linear map π:[0,1]𝔥* such that π(0)=0. Let π be a path, let α be a simple root and let hα:[0,1] be the function given by

hα: [0,1] t π(t),α.

At t this function gives the position of π(t) in the α-direction. Let mα be the minimal value of hα and define functions l:[0,1][0,1] and r:[0,1][0,1] by

l(t)=min { 1,hα(s)- mα ts1 } ,r(t)=1-min { 1,hα(s)- mα 0st } .

The root operators (see [Lit1997] Definitions 2.1 and 2.2) are operators on the paths given by

eαπ = { tπ(t) +r(t)α, ifr(0)=0, 0, otherwise, and fαπ = { tπ(t) +1(t)α, ifl(1)=1, 0, otherwise,

where we use 0 to denote the “null path”.

Fix a dominant weight λΛ. Let πλ be the path given by πλ(t)=tλ, 0t1, and let

𝒯λ= { fi1 fi2 fil πλ }

be the set of all paths obtained by applying sequences of root operators fi=fαi, 1in to πλ. This is the set of Lakshmibai-Seshadri paths of shape λ. P. Littelmann [Lit1994] has shown that this set of paths is finite and can be characterized in terms of an integrality condition. We shall not need this alternative characterization.

Let Wλ be the stabilizer of λ. The cosets in W/Wλ are partially ordered by the Bruhat-Chevalley order. Use a pair of sequences

τ = ( τ1> τ2> > τ ) , τiW/Wλ, and a = ( 0=a0<a1< a2<<a =1 ) , ai,

to encode the path π:[0,1]𝔥* given by

π(t)= (t-aj-1) τjλ+ i=1j-1 (ai-ai-1) τiλ,for aj-1taj.

We shall write π=(τ,a). Every path π𝒯λ is of this form. Littelmann introduced this set of paths 𝒯λ as a model for the Weyl character formula. He proved that

η𝒯λ eη(1)= wWε(w) ew(λ+ρ) wWε(w) ewρ ,

where ρ=12α>0α is the half-sum of the positive roots.

Application of the path model

Fix a dominant weight λΛ and let π=(τ,a)= ( (τ1>>τr), (a0<<ar) ) 𝒯λ. The initial direction of π is ι(π)=τ1. Fix wW, and let w=wWλ W/Wλ and define

𝒯wλ= { π𝒯λ ι(π)w } .

Let π=(τ,a)𝒯wλ. A maximal lift of τ with respect to w is a choice of representatives tiW of the cosets τi such that wt1tr and each ti is maximal in Bruhat order such that ti-1>ti. The final direction of π with respect to w is


where wt1tr is a maximal lift of τ1>>τr with respect to w.

For each π𝒯λ such that eα(π)=0 the α-string of π is the set of paths

Sα(π)= { π,fαπ,,fαmπ } ,

where m is maximal such that fαmπ0. We have:

  1. If fαjπ0 then (fαjπ) (1)=π(1) -jα.
  2. ι(fαjπ)= sαι(π) for all 1jm.
  3. If Sα(π)𝒯wλ then v(fαmπ,w)= sαv(π,w) and v(fαjπ,w)= v(π,w) for 1j<m.

Statement (a) is [Lit1995] Lemma 2.1a, statement (b) is [Lit1994] Lemma 5.3b, and statement (c) follows from [Lit1994] Lemma 5.3c and [Lit1995] Lemma 2.1e. All of these facts are really coming from the explicit form of the action of the root operators on the Lakshmibai-Seshadri paths which is given in [Lit1994] Proposition 4.2. The consequence of (a) and (c) is that

ηSα(π) Tv(η,w)-1 eη(1)= Tv(π,w)-1 ( Tαesαπ(1)+ eπ(1)- esαπ(1) 1-e-α ) =Tv(π,w)-1 eπ(1)Tα.

Let w=sαw where (w)=(w)+1. Let π𝒯wλ be such that eα(π)=0. It follows from (a) that

Sα(π)𝒯wλ ,andeither Sα(π) 𝒯wλ= {π}orSα (π)𝒯wλ.

Suppose that wt1>>tr and wt1>>tr are maximal lifts of π with respect to w and w respectively.

If m>0 then t1 is not divisible by sα. It follows that t1=t1 and thus that tr=tr.
If m=0 then all the ti are divisible by sα and it follows that tr=sαtr.

Thus v(π,w)=v(π,w) if m>0 and v(π,w)=v (π,w)sα if m=0. We conclude that

Tv(π,w)-1 eπ(1)Tα = Tv(π,w)-1 ( Tαesαπ(1)+ eπ(1)- esαπ(1) 1-e-α ) = ηSα(π) Tv(η,w)-1 eη(1). (4.1)

Theorem 4.2. Let λ be a dominant integral weight and let wW. Then

eλTw-1= η𝒯wλ Tv(η,w)-1 eη(1)

as operators on R(T).


The proof is by induction on (w). The base case (w)=1 is formula (3.1d). Let w=sαw with (w)=(w)+1. Then

eλTw-1 = eλT(w)-1 Tα = ( η𝒯wλ Tv(η,w1)-1 eη(1) ) Tα(by induction) = π𝒯wλ eα(π)=0 ( Sα(π)𝒯wλ Tv(π,w)-1 eπ(1)Tα+ Sα(π) 𝒯wλ= {π} Tv(π,w)-1 eη(1) ) Tα = π𝒯wλ eα(π)=0 ( Sα(π)𝒯wλ Tv(π,w)-1 eπ(1)Tα+ Sα(π) 𝒯wλ= {π} Tv(π,w)-1 eη(1)Tα ) = η𝒯wλ Tv(η,w)-1 eη(1)(by (4.1)).

Theorem 4.3. Let λ be a dominant integral weight. In K(G/B)

eλ[𝒪Xw]= η𝒯wλ [𝒪Xv(η,w)]


Since the sheaf 𝒪X1 is supported on the single point X1G/B, any product [][𝒪X1], where is a vector bundle on G/B is the class of a bundle supported on the single point X1. More precisely, [][𝒪X1]= rk()[𝒪X1]. Thus, since eλ is the class of a line bundle we have eλ[𝒪X1]= [𝒪X1]. By Corollary 3.6, [𝒪Xw]= Tw-1 [𝒪X1] and so the result follows from Theorem 4.2.

The following example illustrates how one computes the product eω2 [ 𝒪 X s1s2 s1s2 ] in K(G/B) for the group G of type G2. In this case λ=ω2, w-1=s2s1s2s1 and the starting path πλ is the straight line path from the origin to the point ω2. The paths in the set 𝒯 s2s1s2s1 ω2 are the paths in the following diagrams.

ω2 α2 α1 ω2 α2 α1 ω2 α2 α1

These paths yield the following data:

endpoint maximal liftt ι(η)=τ1 v(w,η)-1 ω2 (s1) 1 s1 s2ω2 (s2s1) s2 s1s2 s2ω2-α1 ( s1s2s1 >s2s1 ) s1s2 s1s2 s2ω2-2α1 ( s1s2s1 >s2s1 ) s1s2 s1s2 s1s2ω2 ( s1s2s1 ) s1s2 s1s2s1 s2s1s2ω2 (s2s1s2) s2s1s2 s2s1s2 s2s1s2ω2-α1 ( s1s2s1s2 >s2s1s2 ) s1s2s1s2 s2s1s2 s2s1s2ω2-2α1 ( s1s2s1s2 >s2s1s2 ) s1s2s1s2 s2s1s2 s1s2s1s2ω2 (s1s2s1s2) s1s2s1s2 s2s1s2s1 α1 ( s2s1s2> s1s2>s2 ) s2s1s2 s2 -α1 ( s1s2s1s2> s2s1s2>s1 s2 ) s1s2s1s2 s2s1 0 (s2s1s2>s1s2) s2s1s2 s2s1 0 ( s1s2s1s2> s2s1s2>s1 s2>s2 ) s1s2s1s2 s2

and thus we get

eω2 Ts2s1s2s1 =Ts1eω2 + Ts1s2 es2ω2+ Ts1s2 es2ω2-α1+ Ts1s2 es2ω2-2α1 +Ts1s2s1 es1s2ω2 + Ts2s1s2 es1s2s1ω2+ Ts1s1s2 es2s1s2ω2-α1+ Ts2s1s2 es2s1s2ω2-2α1 + Ts2s1s2s1 es1s2s1s2ω2+ Ts2eα1+ Ts2s1e-α1 +Ts2s1e0+ Ts2e0


eω2 [ 𝒪Xs1s2s1s2 ] = [ 𝒪Xs1s2s1s2 ] + [ 𝒪Xs1s2s1 ] +3 [ 𝒪Xs2s1s2 ] +3 [ 𝒪Xs2s1 ] +2 [ 𝒪Xs1s2 ] +2 [ 𝒪Xs2 ] + [ 𝒪Xs1 ] .

Notes and References

This is an excerpt of the paper entitled A Pieri-Chevalley formula for K(G/B) authored by H. Pittie and Arun Ram (preprint May 19, 1998).

Research supported in part by National Science Foundation grant DMS-9622985.

page history