## The Pieri-Chevalley formula

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 ${e}^{\lambda }\left[{𝒪}_{{X}_{w}}\right]$ in $K\left(G/B\right)$ in terms of the basis $\left\{\left[{𝒪}_{{X}_{v}}\right] \mid v\in W\right\}\text{.}$ We use the path model of P. Littelmann to keep track of the combinatorics involved in iterating formula (3.1d).

The path model

Let $\Lambda ={\sum }_{i}ℤ{\omega }_{i}$ be the weight lattice and let ${𝔥}^{*}={\sum }_{i}ℝ{\omega }_{i}\text{.}$ A path in ${𝔥}^{*}$ is a piecewise linear map $\pi :\left[0,1\right]\to {𝔥}^{*}$ such that $\pi \left(0\right)=0\text{.}$ Let $\pi$ be a path, let $\alpha$ be a simple root and let ${h}_{\alpha }:\left[0,1\right]\to ℝ$ be the function given by

$hα: [0,1] ⟶ ℝ t ⟼ ⟨π(t),α∨⟩.$

At $t$ this function gives the position of $\pi \left(t\right)$ in the $\alpha \text{-direction.}$ Let ${m}_{\alpha }$ be the minimal value of ${h}_{\alpha }$ and define functions $l:\left[0,1\right]\to \left[0,1\right]$ and $r:\left[0,1\right]\to \left[0,1\right]$ by

$l(t)=min { 1,hα(s)- mα ∣ t≤s≤1 } ,r(t)=1-min { 1,hα(s)- mα ∣ 0≤s≤t } .$

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

$eαπ = { t↦π(t) +r(t)α, if r(0)=0, 0, otherwise, and fαπ = { t↦π(t) +1(t)α, if l(1)=1, 0, otherwise,$

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

Fix a dominant weight $\lambda \in \Lambda \text{.}$ Let ${\pi }_{\lambda }$ be the path given by ${\pi }_{\lambda }\left(t\right)=t\lambda ,$ $0\le t\le 1,$ and let

$𝒯λ= { fi1 fi2… fil πλ }$

be the set of all paths obtained by applying sequences of root operators ${f}_{i}={f}_{{\alpha }_{i}},$ $1\le i\le n$ to ${\pi }_{\lambda }\text{.}$ This is the set of Lakshmibai-Seshadri paths of shape $\lambda \text{.}$ 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 ${W}_{\lambda }$ be the stabilizer of $\lambda \text{.}$ The cosets in $W/{W}_{\lambda }$ are partially ordered by the Bruhat-Chevalley order. Use a pair of sequences

$τ→ = ( τ1> τ2> …> τℓ ) , τi∈W/Wλ, and a→ = ( 0=a0

to encode the path $\pi :\left[0,1\right]\to {𝔥}^{*}$ given by

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

We shall write $\pi =\left(\stackrel{\to }{\tau },\stackrel{\to }{a}\right)\text{.}$ Every path $\pi \in {𝒯}^{\lambda }$ is of this form. Littelmann introduced this set of paths ${𝒯}^{\lambda }$ as a model for the Weyl character formula. He proved that

$∑η∈𝒯λ eη(1)= ∑w∈Wε(w) ew(λ+ρ) ∑w∈Wε(w) ewρ ,$

where $\rho =\frac{1}{2}{\sum }_{\alpha >0}\alpha$ is the half-sum of the positive roots.

Application of the path model

Fix a dominant weight $\lambda \in \Lambda$ and let $\pi =\left(\stackrel{\to }{\tau },\stackrel{\to }{a}\right)=\left(\left({\tau }_{1}>\dots >{\tau }_{r}\right),\left({a}_{0}<\dots <{a}_{r}\right)\right)\in {𝒯}^{\lambda }\text{.}$ The initial direction of $\pi$ is $\iota \left(\pi \right)={\tau }_{1}\text{.}$ Fix $w\in W,$ and let $\stackrel{‾}{w}=w{W}_{\lambda }\in W/{W}_{\lambda }$ and define

$𝒯wλ= { π∈𝒯λ ∣ ι(π)≤w‾ } .$

Let $\pi =\left(\stackrel{\to }{\tau },\stackrel{\to }{a}\right)\in {𝒯}_{w}^{\lambda }\text{.}$ A maximal lift of $\stackrel{\to }{\tau }$ with respect to $w$ is a choice of representatives ${t}_{i}\in W$ of the cosets ${\tau }_{i}$ such that $w\ge {t}_{1}\ge \dots \ge {t}_{r}$ and each ${t}_{i}$ is maximal in Bruhat order such that ${t}_{i-1}>{t}_{i}\text{.}$ The final direction of $\pi$ with respect to $w$ is

$v(π,w)=tr,$

where $w\ge {t}_{1}\ge \dots \ge {t}_{r}$ is a maximal lift of ${\tau }_{1}>\dots >{\tau }_{r}$ with respect to $w\text{.}$

For each $\pi \in {𝒯}^{\lambda }$ such that ${e}_{\alpha }\left(\pi \right)=0$ the $\alpha \text{-string}$ of $\pi$ is the set of paths

$Sα(π)= { π,fαπ,…,fαmπ } ,$

where $m$ is maximal such that ${f}_{\alpha }^{m}\pi \ne 0\text{.}$ We have:

1. If ${f}_{\alpha }^{j}\pi \ne 0$ then $\left({f}_{\alpha }^{j}\pi \right)\left(1\right)=\pi \left(1\right)-j\alpha \text{.}$
2. $\iota \left({f}_{\alpha }^{j}\pi \right)={s}_{\alpha }\iota \left(\pi \right)$ for all $1\le j\le m\text{.}$
3. If ${S}_{\alpha }\left(\pi \right)\subseteq {𝒯}_{w}^{\lambda }$ then $v\left({f}_{\alpha }^{m}\pi ,w\right)={s}_{\alpha }v\left(\pi ,w\right)$ and $v\left({f}_{\alpha }^{j}\pi ,w\right)=v\left(\pi ,w\right)$ for $1\le j

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

$∑η∈Sα(π) Tv(η,w)-1 eη(1)= Tv(π,w)-1 ( Tαesαπ(1)+ eπ(1)- esαπ(1) 1-e-α ) =Tv(π,w)-1 eπ(1)Tα.$

Let $w={s}_{\alpha }w\prime$ where $\ell \left(w\right)=\ell \left(w\prime \right)+1\text{.}$ Let $\pi \in {𝒯}_{w}^{\lambda }$ be such that ${e}_{\alpha }\left(\pi \right)=0\text{.}$ It follows from (a) that

$Sα(π)⊆𝒯wλ ,andeither Sα(π)∩ 𝒯w′λ= {π} or Sα (π)⊆𝒯w′λ.$

Suppose that $w\ge {t}_{1}>\dots >{t}_{r}$ and $w\prime \ge {t}_{1}^{\prime }>\dots >{t}_{r}^{\prime }$ are maximal lifts of $\pi$ with respect to $w$ and $w\prime$ respectively.

If $m>0$ then ${t}_{1}$ is not divisible by ${s}_{\alpha }\text{.}$ It follows that ${t}_{1}^{\prime }={t}_{1}$ and thus that ${t}_{r}={t}_{r}^{\prime }\text{.}$
If $m=0$ then all the ${t}_{i}$ are divisible by ${s}_{\alpha }$ and it follows that ${t}_{r}={s}_{\alpha }{t}_{r}^{\prime }\text{.}$

Thus $v\left(\pi ,w\right)=v\left(\pi ,w\prime \right)$ if $m>0$ and $v\left(\pi ,w\right)=v\left(\pi ,w\prime \right){s}_{\alpha }$ if $m=0\text{.}$ We conclude that

$Tv(π,w′)-1 eπ(1)Tα = Tv(π,w′)-1 ( Tαesαπ(1)+ eπ(1)- esαπ(1) 1-e-α ) = ∑η∈Sα(π) Tv(η,w)-1 eη(1). (4.1)$

Theorem 4.2. Let $\lambda$ be a dominant integral weight and let $w\in W\text{.}$ Then

$eλTw-1= ∑η∈𝒯wλ Tv(η,w)-1 eη(1)$

as operators on $R\left(T\right)\text{.}$

 Proof. The proof is by induction on $\ell \left(w\right)\text{.}$ The base case $\ell \left(w\right)=1$ is formula (3.1d). Let $w={s}_{\alpha }{w}^{\prime }$ with $\ell \left(w\right)=\ell \left(w\prime \right)+1\text{.}$ Then $eλTw-1 = eλT(w′)-1 Tα = ( ∑η∈𝒯w′λ Tv(η,w1)-1 eη(1) ) Tα(by induction) = ∑ π∈𝒯wλ eα(π)=0 ( ∑Sα(π)⊆𝒯w′λ Tv(π,w′)-1 eπ(1)Tα+ ∑ Sα(π)∩ 𝒯w′λ= {π} Tv(π,w′)-1 eη(1) ) Tα = ∑ π∈𝒯wλ eα(π)=0 ( ∑Sα(π)⊆𝒯w′λ Tv(π,w′)-1 eπ(1)Tα+ ∑ Sα(π)∩ 𝒯w′λ= {π} Tv(π,w′)-1 eη(1)Tα ) = ∑η∈𝒯wλ Tv(η,w)-1 eη(1)(by (4.1)).$ $\square$

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

$eλ[𝒪Xw]= ∑η∈𝒯wλ [𝒪Xv(η,w)]$

 Proof. Since the sheaf ${𝒪}_{{X}_{1}}$ is supported on the single point ${X}_{1}\in G/B,$ any product $\left[ℱ\right]\left[{𝒪}_{{X}_{1}}\right],$ where $ℱ$ is a vector bundle on $G/B$ is the class of a bundle supported on the single point ${X}_{1}\text{.}$ More precisely, $\left[ℱ\right]\left[{𝒪}_{{X}_{1}}\right]=\text{rk}\left(ℱ\right)\left[{𝒪}_{{X}_{1}}\right]\text{.}$ Thus, since ${e}^{\lambda }$ is the class of a line bundle we have ${e}^{\lambda }\left[{𝒪}_{{X}_{1}}\right]=\left[{𝒪}_{{X}_{1}}\right]\text{.}$ By Corollary 3.6, $\left[{𝒪}_{{X}_{w}}\right]={T}_{{w}^{-1}}\left[{𝒪}_{{X}_{1}}\right]$ and so the result follows from Theorem 4.2. $\square$

The following example illustrates how one computes the product ${e}^{{\omega }_{2}}\left[{𝒪}_{{X}_{{s}_{1}{s}_{2}{s}_{1}{s}_{2}}}\right]$ in $K\left(G/B\right)$ for the group $G$ of type ${G}_{2}\text{.}$ In this case $\lambda ={\omega }_{2},$ ${w}^{-1}={s}_{2}{s}_{1}{s}_{2}{s}_{1}$ and the starting path ${\pi }_{\lambda }$ is the straight line path from the origin to the point ${\omega }_{2}\text{.}$ The paths in the set ${𝒯}_{{s}_{2}{s}_{1}{s}_{2}{s}_{1}}^{{\omega }_{2}}$ are the paths in the following diagrams.

$ω2 α2 α1 ω2 α2 α1$ $ω2 α2 α1$

These paths yield the following data:

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

and thus we get

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

and

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

## 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.