Pieri-Chevalley formulas

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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).

Proof.

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).

Proof.

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].

Proof.

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

cw0si,wλ=- ( e-(wωi-w0ωi) -1 )

and

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

Proof.

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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