Pieri-Chevalley formulas

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

Last update: 22 February 2013

Pieri-Chevalley formulas

Recall that both

{ XλTw-1 λP, wW } and { Tz-1Xμ μP, zW } are bases ofH.

If cw,λμ,z are the entries of the transition matrix between these two bases,

XλTw-1= zW,μP cw,λμ,z Tz-1Xμ, (3.1)

then applying each side of (3.1) to [OX1] gives that

[Xλ] [𝒪Xw]= zW,μP cw,λμ,z eμ[𝒪Xz], inKT(G/B).

This is the most general form of “Pieri–Chevalley rule”. The problem is to determine the coefficients cw,λμ,z.

3.1. The path model

A path in 𝔥* is a piecewise linear map p:[0,1]𝔥* such that p(0)=0. For each 1in there are root operators ei and fi (see [Lit1997] Definitions 2.1 and 2.2) which act on the paths. If λP+ the path model for λ is

𝒯λ= { fi1 fi2 filpλ } ,

the set of all paths obtained by applying the root operators to pλ, where pλ is the straight path from 0 to λ, that is, pλ(t)=tλ, 0t1. Each path p in 𝒯λ is a concatenation of segments

p= pw1λα1 pw2λα2 pwrλαr withw1w2 wrand a1+a2++ar=1, (3.2)

where, for vW and a(0,1], pvλa is a piece of length a from the straight line path pvλ=vpλ. If Wλ=Stab(λ) then the wj should be viewed as cosets in W/Wλ and denotes the order on W/Wλ inherited from the Bruhat–Chevalley order on W. The total length of p is the same as the total length of pλ which is assumed (or normalized) to be 1. For p𝒯λ let

p(1) = i=1rai wiλ be the endpoint ofp, ι(p) = w1, the initial direction ofp,and ϕ(p) = wr, the final direction ofp.

If h𝒯λ is such that ei(h)=0 then h is the head of its i-string

Siλ(h)= { h,fih,,fimh } ,

where m is the smallest positive integer such that fimh0 and fim+1h=0. The full path model 𝒯λ is the union of its i-strings. The endpoints and the inital and final directions of the paths in the i-string Siλ(h) have the following properties:

(fikh)(1) =h(1)-kαi ,for0km, eitherι(h)=ι (fih)==ι (fimh)<si ι(h) orι(h)<ι (fih)== ι(fimh)= siι(h), and eithersiϕ (fimh)<ϕ (h)==ϕ (fim-1h)= ϕ(fimh) orsiϕ (fimh)= ϕ(h)==ϕ (fim-1h) <ϕ(fimh). (3.3)

The first property is [Lit1995, Lemma 2.1a], the second is [Lit1994, Lemma 5.3], and the last is a result of applying [Lit1995, Lemma 2.1e] to [Lit1994, Lemma 5.3]. All of these facts are really coming from the explicit form of the action of the root operators on the paths in 𝒯λ which is given in [Lit1994, Proposition 4.2].

Let λP+, wW and zW/Wλ, and let p𝒯λ be such that ι(p)wWλ and ϕ(p)z. Write p in the form (3.2) and let w1,,wr,z be the maximal (in Bruhat order) coset representatives of the cosets w1,,wr,z such that

ww1 w2 wrz. (3.4)

Theorem 3.5. Recall the notation εv from (1.11). Let λP+ and let Wλ=Stab(λ). Let wW. Then, in the affine nil-Hecke algebra H,

XλTw-1 = p𝒯λι(p)wWλ Tϕ(p)-1 Xp(1)and Xλεw-1 = p𝒯λι(p)=w zW/Wλzϕ(p) (-1)(w)+(z) εz-1 Xp(1),

where, if Wλ{1} then Tϕ(p)-1= Twr-1 and εz-1= εz-1 with wr and z as in (3.4).


(a) The proof is by induction on (w). Let w=siv where siv>v. Define

𝒯wλ= { p𝒯λ ι(p)wWλ } .

Assume w=siv>v. Then the facts in (3.3) imply that

  1. 𝒯wλ is a union of strings Si(h) such that h𝒯vλ, and
  2. If h𝒯vλ then either si(h)𝒯vλ or Si(h)𝒯vλ={h}.

Using the facts in (3.3), a direct computation with the relation (1.3) establishes that, if h𝒯vλ then

pSi(h) Tϕ(p)-1 Xη(1) = Tϕ(h)-1 Xh(1)Ti, and pSi(h) Tϕ(p)-1 Xη(1) = { Tϕ(h)-1 Xh(1)Ti, ifSi(h) 𝒯vλ, Tϕ(h)-1 Xh(1)Ti, ifSi(h) 𝒯vλ= {h}.


XλTw-1 = XλTv-1 Ti= ( p𝒯vλ Tϕ(p)-1 Xp(1) ) Ti(by induction) = h𝒯vλ ei(h)=0 ( Si(h)𝒯vλ pSi(h) Tϕ(p)-1 Xp(1)+ Si(h)𝒯vλ={h} Tϕ(h)-1 Xh(1) ) Ti = h𝒯wλ ei(h)=0 ( Si(h)𝒯vλ Tϕ(h)-1 Xh(1)Ti+ Si(h)𝒯vλ={h} Tϕ(h)-1 Xh(1) ) Ti = h𝒯wλ ei(h)=0 ( Si(h)𝒯vλ Tϕ(h)-1 Xh(1)Ti+ Si(h)𝒯vλ={h} pSi(h) Tϕ(p)-1 Xp(1) ) = p𝒯wλ Tϕ(p)-1 Xp(1).

(b) The proof is similar to case (a). For wW let

𝒯=wλ= { p𝒯λ ι(p)=wWλ } .

Assume w=siv>v. Then the facts in (3.3) imply that

  1. 𝒯=wλ is a union of the strings Si(h) such that h𝒯-hλ, and
  2. If h𝒯=vλ then either Si(h)𝒯=vλ or Si(h)𝒯=vλ={h}.


Εϕ(p)= zW/Wλ zϕ(p) (-1)(z) εz-1. (3.6)

Using (3.3), a direct computation with the relation (1.3) establishes that, if h𝒯=vλ with eih=0 then

pSi(h) Εϕ(p) Xp(1)Ti=0 ,andΕϕ(0h) Xh(1)Ti=- pSi(h)-{] h} Εϕ(p) Xp(1).


Xλεw-1 = Xλεv-1 εi= (-1)(v) ( p𝒯=vλ Εϕ(p) Xp(1) ) Ti = (-1)(v) ( Si(h)𝒯=vλ pSi(h) Εϕ(p) Xp(1)+ Si(h)𝒯=vλ={h} Εϕ(h) Xh(1) ) Ti = (-1)(v) ( 0- Si(h)𝒯=vλ={h} pSi(h)-{h} Εϕ(p) Xp(1) ) = (-1)(v) ( p𝒯=wλ Εϕ(p) Xp(1) ) .

Corollary 3.7. Let λ,μP+ and let wW. Then, in the affine nil-Hecke algebra H,

X-λTw-1 = p𝒯-w0λ ϕ(p)=ww0 zW/W-w0λ zw0ι(p) (-1)(w)+(z) Tz-1 Xp(1)and Xw0μTw-1 = p𝒯μ ϕ(p)=ww0 zW/Wμ zw0ϕ(p) (-1)(w)+(z) Tz-1 Xp(1).


The second identity is a restatement of the first with a change of variable μ=-w0λ. The first identity is obtained by applying the algebra involution

HH Twεw XλX-λ and the bijection 𝒯λ𝒯-w0λ pp*

where p* is the same path as p except translated so that its endpoint is at the origin. Representation theoretically, this bijection corresponds to the fact that L(λ)*L (-w0λ), if L(λ) is the simple G-module of highest weight λ. Note that p*(1)=-p(1), ι(p*)=ϕ(p)w0, and ϕ(p*)=ι(p)w0.

Applying the identities from Theorem 3.5 and Corollary 3.7 to [OX1] yields the following product formulas in KT(G/B). In particular, this gives a combinatorial proof of the (T-equivariant extension) of the duality theorem of Brion [Bri2002, Theorem 4]. For λP and wW let [Xλ] =Xλ[𝒪Xw0] =XλTw0[OX1] and let cλ,wz be given by

[Xλ] [𝒪Xw]= zW cλ,wz [𝒪Xz]. (3.8)

Corollary 3.9. Let λP+, wW and Wλ=Stab(λ). Then, with notation as in (3.8),

cλ,wz = p𝒯λ wWλι(p) ϕ(p)=zWλ ep(1), cw0λ,wz = (-1)(w)+(z) cλ,zw0ww0, and c-λ,wz= (-1)(w)+(z) c -w0λ,zw0 ww0 .

Proposition 3.10. For 1in, [𝒪Xw0si] =1-ew0ωi [X-ωi].


We shall show that

X-ωi [𝒪Xw0]= e-w0ωi ( [𝒪Xw0]- [𝒪Xw0si] ) , (3.11)

and the result will follow by solving for [𝒪Xsiw0]. Let ωj=-w0ωi. By Corollary 3.9,

c-ωi,w0z= (-1)(w0)+(z) cωj,zw01= (-1)(w0)+(z) p𝒯ωj zw0ι(p) ϕ(p)=1 ep(1).

The straight line path to ωj, pωj, has ιzw0 (pωj)= ϕzw0 (ωj) and is the unique path in 𝒯ωj which may have final direction 1. Suppose ϕzw0(pωj)=1. Then, since sj is the only simple reflection which is not in Stab(ωj), it must be that zw0sk for all kj. Thus zw0=1 or zw0=sj and so c-ωi,w0z0 only if z=w0 or z=sjw0=w0si. Now (3.11) follows since pωj has endpoint ωj=-w0ωi.

Corollary 3.12. Let cwvz be as in (3.8). Then, for

cw0si,wλ=- ( e-(wωi-w0ωi) -1 )


cw0si,wz= (-1)(w)+(z)+1 p𝒯-w0ωi zw0ι(p) ϕ(p)=ww0 ew0ωi+p(1), forzw.


This follows from Proposition 3.10 and Corollary 3.9 and the fact that, in the case when z=w, there is a unique path p with ww-=ι(p)=ϕ(p)=ww0 and endpoint p(1)=ww0 (-w0ωi)= -wωi.

Notes and References

This is an excerpt of the paper entitled Affine Hecke algebras and the Schubert calculus authored by Stephen Griffeth and Arun Ram. It was dedicated to Alain Lascoux.

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