The Pieri-Chevalley formula

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

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 $K(G/B)$ in terms of the basis $\{\left[{𝒪}_{X_v}\right]\hspace{0.17em}\mid \hspace{0.17em}v\in W\}\text{.}$ We use the path model of P. Littelmann to keep track of the combinatorics involved in iterating formula (3.1d).

The path model

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

$$\begin{array}{cccc}{h}_{\alpha }:& [0,1]& ⟶& ℝ\\ & t& ⟼& ⟨\pi \left(t\right),{\alpha }^{\vee }⟩\text{.}\end{array}$$

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 $l:[0,1]\to [0,1]$ and $r:[0,1]\to [0,1]$ by

$$l\left(t\right)=\text{min}\{1,h_{\alpha }\left(s\right)-m_{\alpha }\hspace{0.17em}\mid \hspace{0.17em}t\le s\le 1\},r\left(t\right)=1-\text{min}\{1,h_{\alpha }\left(s\right)-m_{\alpha }\hspace{0.17em}\mid \hspace{0.17em}0\le s\le t\}\text{.}$$

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

$$\begin{array}{ccc}{e}_{\alpha }\pi & =& \{\begin{array}{cc}t\mapsto \pi \left(t\right)+r\left(t\right)\alpha ,& \text{if}\hspace{0.17em}r\left(0\right)=0,\\ 0,& \text{otherwise,}\end{array}\text{and}\\ f_{\alpha }\pi & =& \{\begin{array}{cc}t\mapsto \pi \left(t\right)+1\left(t\right)\alpha ,& \text{if}\hspace{0.17em}l\left(1\right)=1,\\ 0,& \text{otherwise,}\end{array}\end{array}$$

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

$${𝒯}^{\lambda }=\left\{f_{i_1}{f}_{i_2}\dots f_{i_l}{\pi }_{\lambda }\right\}$$

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

$$\begin{array}{ccccc}\overrightarrow{\tau }& =& ({\tau }_1>{\tau }_2>\dots >{\tau }_{\ell }),& {\tau }_i\in W/W_{\lambda },& \text{and}\\ \overrightarrow{a}& =& (0=a_0<a_1<a_2<\dots <a_{\ell }=1),& a_i\in \mathbb{Q},\end{array}$$

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

$$\pi \left(t\right)=(t-a_{j-1}){\tau }_j\lambda +\sum _{i=1}^{j-1}(a_i-a_{i-1}){\tau }_i\lambda ,\text{for}\hspace{0.17em}{a}_{j-1}\le t\le a_j\text{.}$$

We shall write $\pi =(\overrightarrow{\tau },\overrightarrow{a})\text{.}$ 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

$$\sum _{\eta \in {𝒯}^{\lambda }}{e}^{\eta \left(1\right)}=\frac{{\sum }_{w\in W}\epsilon \left(w\right)e^{w(\lambda +\rho )}}{{\sum }_{w\in W}\epsilon \left(w\right)e^{w\rho }},$$

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 $\pi =(\overrightarrow{\tau },\overrightarrow{a})=(({\tau }_1>\dots >{\tau }_r),(a_0<\dots <a_r))\in {𝒯}^{\lambda }\text{.}$ The initial direction of $\pi$ is $\iota \left(\pi \right)={\tau }_1\text{.}$ Fix $w\in W,$ and let $\bar{w}=w{W}_{\lambda }\in W/W_{\lambda }$ and define

$${𝒯}_w^{\lambda }=\{\pi \in {𝒯}^{\lambda }\hspace{0.17em}\mid \hspace{0.17em}\iota \left(\pi \right)\le \bar{w}\}\text{.}$$

Let $\pi =(\overrightarrow{\tau },\overrightarrow{a})\in {𝒯}_w^{\lambda }\text{.}$ A maximal lift of $\overrightarrow{\tau }$ 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

$$v(\pi ,w)=t_r,$$

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

$$S_{\alpha }\left(\pi \right)=\{\pi ,f_{\alpha }\pi ,\dots ,f_{\alpha }^m\pi \},$$

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 $v(f_{\alpha }^m\pi ,w)=s_{\alpha }v(\pi ,w)$ and $v(f_{\alpha }^j\pi ,w)=v(\pi ,w)$ for $1\le j<m\text{.}$

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

$$\sum _{\eta \in S_{\alpha }\left(\pi \right)}{T}_{v{(\eta ,w)}^{-1}}{e}^{\eta \left(1\right)}=T_{v{(\pi ,w)}^{-1}}(T_{\alpha }{e}^{s_{\alpha }\pi \left(1\right)}+\frac{e^{\pi \left(1\right)}-e^{s_{\alpha }\pi \left(1\right)}}{1-e^{-\alpha }})=T_{v{(\pi ,w)}^{-1}}{e}^{\pi \left(1\right)}{T}_{\alpha }\text{.}$$

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

$$S_{\alpha }\left(\pi \right)\subseteq {𝒯}_w^{\lambda },\text{and}\text{either}\hspace{0.17em}{S}_{\alpha }\left(\pi \right)\cap {𝒯}_{w\prime }^{\lambda }=\left\{\pi \right\}\hspace{0.17em}\text{or}\hspace{0.17em}{S}_{\alpha }\left(\pi \right)\subseteq {𝒯}_{w\prime }^{\lambda }\text{.}$$

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 $v(\pi ,w)=v(\pi ,w\prime )$ if $m>0$ and $v(\pi ,w)=v(\pi ,w\prime )s_{\alpha }$ if $m=0\text{.}$ We conclude that

$$\begin{array}{cc}\begin{array}{ccc}{T}_{v{(\pi ,w\prime )}^{-1}}{e}^{\pi \left(1\right)}{T}_{\alpha }& =& T_{v{(\pi ,w\prime )}^{-1}}(T_{\alpha }{e}^{s_{\alpha }\pi \left(1\right)}+\frac{e^{\pi \left(1\right)}-e^{s_{\alpha }\pi \left(1\right)}}{1-e^{-\alpha }})\\ & =& \sum _{\eta \in S_{\alpha }\left(\pi \right)}{T}_{v{(\eta ,w)}^{-1}}{e}^{\eta \left(1\right)}\text{.}\end{array}& \text{(4.1)}\end{array}$$

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

$$e^{\lambda }{T}_{w^{-1}}=\sum _{\eta \in {𝒯}_w^{\lambda }}{T}_{v{(\eta ,w)}^{-1}}{e}^{\eta \left(1\right)}$$

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 $$\begin{array}{ccc}{e}^{\lambda }{T}_{w^{-1}}& =& e^{\lambda }{T}_{{\left(w\prime \right)}^{-1}}{T}_{\alpha }\\ & =& \left(\sum _{\eta \in {𝒯}_{w\prime }^{\lambda }}{T}_{v{(\eta ,w1)}^{-1}}{e}^{\eta \left(1\right)}\right)T_{\alpha }\text{(by induction)}\\ & =& \sum _{\underset{e_{\alpha }\left(\pi \right)=0}{\pi \in {𝒯}_w^{\lambda }}}(\sum _{S_{\alpha }\left(\pi \right)\subseteq {𝒯}_{w\prime }^{\lambda }}{T}_{v{(\pi ,w\prime )}^{-1}}{e}^{\pi \left(1\right)}{T}_{\alpha }+\sum _{S_{\alpha }\left(\pi \right)\cap {𝒯}_{w\prime }^{\lambda }=\left\{\pi \right\}}{T}_{v{(\pi ,w\prime )}^{-1}}{e}^{\eta \left(1\right)})T_{\alpha }\\ & =& \sum _{\underset{e_{\alpha }\left(\pi \right)=0}{\pi \in {𝒯}_w^{\lambda }}}(\sum _{S_{\alpha }\left(\pi \right)\subseteq {𝒯}_{w\prime }^{\lambda }}{T}_{v{(\pi ,w\prime )}^{-1}}{e}^{\pi \left(1\right)}{T}_{\alpha }+\sum _{S_{\alpha }\left(\pi \right)\cap {𝒯}_{w\prime }^{\lambda }=\left\{\pi \right\}}{T}_{v{(\pi ,w\prime )}^{-1}}{e}^{\eta \left(1\right)}{T}_{\alpha })\\ & =& \sum _{\eta \in {𝒯}_w^{\lambda }}{T}_{v{(\eta ,w)}^{-1}}{e}^{\eta \left(1\right)}\text{(by (4.1)).}\end{array}$$ $\square$

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

$$e^{\lambda }\left[{𝒪}_{X_w}\right]=\sum _{\eta \in {𝒯}_w^{\lambda }}\left[{𝒪}_{X_{v(\eta ,w)}}\right]$$

		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 $K(G/B)$ 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.

$$\begin{array}{cc} ω2 α2 α1 & ω2 α2 α1 \end{array}$$ $$ω2 α2 α1$$

These paths yield the following data:

$$\begin{array}{cccc}\text{endpoint}& \text{maximal lift}\hspace{0.17em}\overrightarrow{t}& \iota \left(\eta \right)={\tau }_1& v{(w,\eta )}^{-1}\\ {\omega }_2& \left(s_1\right)& \bar{1}& s_1\\ s_2{\omega }_2& \left(s_2{s}_1\right)& \overline{s_2}& s_1{s}_2\\ s_2{\omega }_2-{\alpha }_1& (s_1{s}_2{s}_1>s_2{s}_1)& \overline{s_1{s}_2}& s_1{s}_2\\ s_2{\omega }_2-2{\alpha }_1& (s_1{s}_2{s}_1>s_2{s}_1)& \overline{s_1{s}_2}& s_1{s}_2\\ s_1{s}_2{\omega }_2& \left(s_1{s}_2{s}_1\right)& \overline{s_1{s}_2}& s_1{s}_2{s}_1\\ s_2{s}_1{s}_2{\omega }_2& \left(s_2{s}_1{s}_2\right)& \overline{s_2{s}_1{s}_2}& s_2{s}_1{s}_2\\ s_2{s}_1{s}_2{\omega }_2-{\alpha }_1& (s_1{s}_2{s}_1{s}_2>s_2{s}_1{s}_2)& \overline{s_1{s}_2{s}_1{s}_2}& s_2{s}_1{s}_2\\ s_2{s}_1{s}_2{\omega }_2-2{\alpha }_1& (s_1{s}_2{s}_1{s}_2>s_2{s}_1{s}_2)& \overline{s_1{s}_2{s}_1{s}_2}& s_2{s}_1{s}_2\\ s_1{s}_2{s}_1{s}_2{\omega }_2& \left(s_1{s}_2{s}_1{s}_2\right)& \overline{s_1{s}_2{s}_1{s}_2}& s_2{s}_1{s}_2{s}_1\\ {\alpha }_1& (s_2{s}_1{s}_2>s_1{s}_2>s_2)& \overline{s_2{s}_1{s}_2}& s_2\\ -{\alpha }_1& (s_1{s}_2{s}_1{s}_2>s_2{s}_1{s}_2>s_1{s}_2)& \overline{s_1{s}_2{s}_1{s}_2}& s_2{s}_1\\ 0& (s_2{s}_1{s}_2>s_1{s}_2)& \overline{s_2{s}_1{s}_2}& s_2{s}_1\\ 0& (s_1{s}_2{s}_1{s}_2>s_2{s}_1{s}_2>s_1{s}_2>s_2)& \overline{s_1{s}_2{s}_1{s}_2}& s_2\end{array}$$

and thus we get

$$\begin{array}{ccc}{e}^{{\omega }_2}{T}_{s_2{s}_1{s}_2{s}_1}=T_{s_1}{e}^{{\omega }_2}& +& T_{s_1{s}_2}{e}^{s_2{\omega }_2}+T_{s_1{s}_2}{e}^{s_2{\omega }_2-{\alpha }_1}+T_{s_1{s}_2}{e}^{s_2{\omega }_2-2{\alpha }_1}+T_{s_1{s}_2{s}_1}{e}^{s_1{s}_2{\omega }_2}\\ & +& T_{s_2{s}_1{s}_2}{e}^{s_1{s}_2{s}_1{\omega }_2}+T_{s_1{s}_1{s}_2}{e}^{s_2{s}_1{s}_2{\omega }_2-{\alpha }_1}+T_{s_2{s}_1{s}_2}{e}^{s_2{s}_1{s}_2{\omega }_2-2{\alpha }_1}\\ & +& T_{s_2{s}_1{s}_2{s}_1}{e}^{s_1{s}_2{s}_1{s}_2{\omega }_2}+T_{s_2}{e}^{{\alpha }_1}+T_{s_2{s}_1}{e}^{-{\alpha }_1}+T_{s_2{s}_1}{e}^0+T_{s_2}{e}^0\end{array}$$

and

$$e^{{\omega }_2}\left[{𝒪}_{X_{s_1{s}_2{s}_1{s}_2}}\right]=\left[{𝒪}_{X_{s_1{s}_2{s}_1{s}_2}}\right]+\left[{𝒪}_{X_{s_1{s}_2{s}_1}}\right]+3\left[{𝒪}_{X_{s_2{s}_1{s}_2}}\right]+3\left[{𝒪}_{X_{s_2{s}_1}}\right]+2\left[{𝒪}_{X_{s_1{s}_2}}\right]+2\left[{𝒪}_{X_{s_2}}\right]+\left[{𝒪}_{X_{s_1}}\right]\text{.}$$

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.

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