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

$$\{X^{\lambda }{T}_{w^{-1}}\hspace{0.17em}\mid \hspace{0.17em}\lambda \in P,w\in W\}\text{and}\{T_{z^{-1}}{X}^{\mu }\hspace{0.17em}\mid \hspace{0.17em}\mu \in P,z\in W\}\text{are bases of}\hspace{0.17em}\stackrel{\sim }{H}\text{.}$$

If $c_{w,\lambda }^{\mu ,z}\in \mathbb{Z}$ are the entries of the transition matrix between these two bases,

$$\begin{array}{cc}{X}^{\lambda }{T}_{w^{-1}}=\sum _{z\in W,\mu \in P}{c}_{w,\lambda }^{\mu ,z}{T}_{z^{-1}}{X}^{\mu },& \text{(3.1)}\end{array}$$

then applying each side of (3.1) to $\left[O_{X_1}\right]$ gives that

$$\left[X^{\lambda }\right]\left[{𝒪}_{X_w}\right]=\sum _{z\in W,\mu \in P}{c}_{w,\lambda }^{\mu ,z}{e}^{\mu }\left[{𝒪}_{X_z}\right],\text{in}\hspace{0.17em}{K}_T(G/B)\text{.}$$

This is the most general form of “Pieri–Chevalley rule”. The problem is to determine the coefficients $c_{w,\lambda }^{\mu ,z}\text{.}$

3.1. The path model

A path in ${𝔥}^{*}$ is a piecewise linear map $p:[0,1]\to {𝔥}^{*}$ such that $p\left(0\right)=0\text{.}$ For each $1\le i\le n$ there are root operators $e_i$ and $f_i$ (see [Lit1997] Definitions 2.1 and 2.2) which act on the paths. If $\lambda \in P^{+}$ the path model for $\lambda$ is

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

the set of all paths obtained by applying the root operators to $p_{\lambda },$ where $p_{\lambda }$ is the straight path from 0 to $\lambda ,$ that is, $p_{\lambda }\left(t\right)=t\lambda ,$ $0\le t\le 1\text{.}$ Each path $p$ in ${𝒯}^{\lambda }$ is a concatenation of segments

$$\begin{array}{cc}p=p_{w_1\lambda }^{{\alpha }_1}\otimes p_{w_2\lambda }^{{\alpha }_2}\otimes \dots \otimes p_{w_r\lambda }^{{\alpha }_r}\text{with}\hspace{0.17em}{w}_1\ge w_2\ge \dots \ge w_r\text{and}{a}_1+a_2+\dots +a_r=1,& \text{(3.2)}\end{array}$$

where, for $v\in W$ and $a\in (0,1],$ $p_{v\lambda }^a$ is a piece of length $a$ from the straight line path $p_{v\lambda }=v{p}_{\lambda }\text{.}$ If $W_{\lambda }=\text{Stab}\left(\lambda \right)$ then the $w_j$ should be viewed as cosets in $W/W_{\lambda }$ and $\ge$ denotes the order on $W/W_{\lambda }$ inherited from the Bruhat–Chevalley order on $W\text{.}$ The total length of $p$ is the same as the total length of $p_{\lambda }$ which is assumed (or normalized) to be 1. For $p\in {𝒯}^{\lambda }$ let

$$\begin{array}{cccc}p\left(1\right)& =& \sum _{i=1}^r{a}_i{w}_i\lambda & \text{be the endpoint of}\hspace{0.17em}p,\\ \iota \left(p\right)& =& w_1,& \text{the initial direction of}\hspace{0.17em}p,\text{and}\\ \varphi \left(p\right)& =& w_r,& \text{the final direction of}\hspace{0.17em}p\text{.}\end{array}$$

If $h\in {𝒯}^{\lambda }$ is such that $e_i\left(h\right)=0$ then $h$ is the head of its i-string

$$S_i^{\lambda }\left(h\right)=\{h,f_{i}h,\dots ,f_i^{m}h\},$$

where $m$ is the smallest positive integer such that $f_i^{m}h\ne 0$ and $f_i^{m+1}h=0\text{.}$ The full path model ${𝒯}^{\lambda }$ is the union of its $i\text{-strings.}$ The endpoints and the inital and final directions of the paths in the $i\text{-string}$ $S_i^{\lambda }\left(h\right)$ have the following properties:

$$\begin{array}{cc}\begin{array}{c}\left(f_i^{k}h\right)\left(1\right)=h\left(1\right)-k{\alpha }_i,\text{for}\hspace{0.17em}0\le k\le m,\\ \text{either}\iota \left(h\right)=\iota \left(f_{i}h\right)=\dots =\iota \left(f_i^{m}h\right)<s_i\iota \left(h\right)\\ \text{or}\iota \left(h\right)<\iota \left(f_{i}h\right)=\dots =\iota \left(f_i^{m}h\right)=s_i\iota \left(h\right),\text{and}\\ \text{either}{s}_i\varphi \left(f_i^{m}h\right)<\varphi \left(h\right)=\dots =\varphi \left(f_i^{m-1}h\right)=\varphi \left(f_i^{m}h\right)\\ \text{or}{s}_i\varphi \left(f_i^{m}h\right)=\varphi \left(h\right)=\dots =\varphi \left(f_i^{m-1}h\right)<\varphi \left(f_i^{m}h\right)\text{.}\end{array}& \text{(3.3)}\end{array}$$

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 ${𝒯}^{\lambda }$ which is given in [Lit1994, Proposition 4.2].

Let $\lambda \in P^{+},$ $w\in W$ and $z\in W/W_{\lambda },$ and let $p\in {𝒯}^{\lambda }$ be such that $\iota \left(p\right)\le w{W}_{\lambda }$ and $\varphi \left(p\right)\ge z\text{.}$ Write $p$ in the form (3.2) and let ${\stackrel{\sim }{w}}_1,\dots ,{\stackrel{\sim }{w}}_r,\stackrel{\sim }{z}$ be the maximal (in Bruhat order) coset representatives of the cosets $w_1,\dots ,w_r,z$ such that

$$\begin{array}{cc}w\ge {\stackrel{\sim }{w}}_1\ge {\stackrel{\sim }{w}}_2\ge \dots \ge {\stackrel{\sim }{w}}_r\ge \stackrel{\sim }{z}\text{.}& \text{(3.4)}\end{array}$$

Theorem 3.5. Recall the notation ${\epsilon }_v$ from (1.11). Let $\lambda \in P^{+}$ and let $W_{\lambda }=\text{Stab}\left(\lambda \right)\text{.}$ Let $w\in W\text{.}$ Then, in the affine nil-Hecke algebra $\stackrel{\sim }{H},$

$$\begin{array}{ccc}{X}^{\lambda }{T}_{w^{-1}}& =& \sum _{\underset{\iota \left(p\right)\le w{W}_{\lambda }}{p\in {𝒯}^{\lambda }}}{T}_{\varphi {\left(p\right)}^{-1}}{X}^{p\left(1\right)}\text{and}\\ X^{\lambda }{\epsilon }_{w^{-1}}& =& \sum _{\underset{\iota \left(p\right)=w}{p\in {𝒯}^{\lambda }}}\sum _{\underset{z\le \varphi \left(p\right)}{z\in W/W_{\lambda }}}{(-1)}^{\ell \left(w\right)+\ell \left(z\right)}{\epsilon }_{{\stackrel{\sim }{z}}^{-1}}{X}^{p\left(1\right)},\end{array}$$

where, if $W_{\lambda }\ne \left\{1\right\}$ then $T_{\varphi {\left(p\right)}^{-1}}=T_{{\stackrel{\sim }{w}}_r^{-1}}$ and ${\epsilon }_{z^{-1}}={\epsilon }_{{\stackrel{\sim }{z}}^{-1}}$ with ${\stackrel{\sim }{w}}_r$ and $\stackrel{\sim }{z}$ as in (3.4).

		Proof.
	(a) The proof is by induction on $\ell \left(w\right)\text{.}$ Let $w=s_{i}v$ where $s_{i}v>v\text{.}$ Define $${𝒯}_{\le w}^{\lambda }=\{p\in {𝒯}^{\lambda }\hspace{0.17em}\mid \hspace{0.17em}\iota \left(p\right)\le w{W}_{\lambda }\}\text{.}$$ Assume $w=s_{i}v>v\text{.}$ Then the facts in (3.3) imply that ${𝒯}_{\le w}^{\lambda }$ is a union of strings $S_i\left(h\right)$ such that $h\in {𝒯}_{\le v}^{\lambda },$ and If $h\in {𝒯}_{\le v}^{\lambda }$ then either $s_i\left(h\right)\subseteq {𝒯}_{\le v}^{\lambda }$ or $S_i\left(h\right)\cap {𝒯}_{\le v}^{\lambda }=\left\{h\right\}\text{.}$ Using the facts in (3.3), a direct computation with the relation (1.3) establishes that, if $h\in {𝒯}_{\le v}^{\lambda }$ then $$\begin{array}{ccc}\sum _{p\in S_i\left(h\right)}{T}_{\varphi {\left(p\right)}^{-1}}{X}^{\eta \left(1\right)}& =& T_{\varphi {\left(h\right)}^{-1}}{X}^{h\left(1\right)}{T}_i,\text{and}\\ \sum _{p\in S_i\left(h\right)}{T}_{\varphi {\left(p\right)}^{-1}}{X}^{\eta \left(1\right)}& =& \{\begin{array}{cc}{T}_{\varphi {\left(h\right)}^{-1}}{X}^{h\left(1\right)}{T}_i,& \text{if}\hspace{0.17em}{S}_i\left(h\right)\subseteq {𝒯}_{\le v}^{\lambda },\\ T_{\varphi {\left(h\right)}^{-1}}{X}^{h\left(1\right)}{T}_i,& \text{if}\hspace{0.17em}{S}_i\left(h\right)\cap {𝒯}_{\le v}^{\lambda }=\left\{h\right\}\text{.}\end{array}\end{array}$$ Thus $$\begin{array}{ccc}{X}^{\lambda }{T}_{w^{-1}}& =& X^{\lambda }{T}_{v^{-1}}{T}_i=\left(\sum _{p\in {𝒯}_{\le v}^{\lambda }}{T}_{\varphi {\left(p\right)}^{-1}}{X}^{p\left(1\right)}\right)T_i\text{(by induction)}\\ & =& \sum _{\underset{e_i\left(h\right)=0}{h\in {𝒯}_{\le v}^{\lambda }}}(\sum _{S_i\left(h\right)\subseteq {𝒯}_{\le v}^{\lambda }}\sum _{p\in S_i\left(h\right)}{T}_{\varphi {\left(p\right)}^{-1}}{X}^{p\left(1\right)}+\sum _{S_i\left(h\right){𝒯}_{\le v}^{\lambda }=\left\{h\right\}}{T}_{\varphi {\left(h\right)}^{-1}}{X}^{h\left(1\right)})T_i\\ & =& \sum _{\underset{e_i\left(h\right)=0}{h\in {𝒯}_{\le w}^{\lambda }}}(\sum _{S_i\left(h\right)\subseteq {𝒯}_{\le v}^{\lambda }}{T}_{\varphi {\left(h\right)}^{-1}}{X}^{h\left(1\right)}{T}_i+\sum _{S_i\left(h\right)\cap {𝒯}_{\le v}^{\lambda }=\left\{h\right\}}{T}_{\varphi {\left(h\right)}^{-1}}{X}^{h\left(1\right)})T_i\\ & =& \sum _{\underset{e_i\left(h\right)=0}{h\in {𝒯}_{\le w}^{\lambda }}}(\sum _{S_i\left(h\right)\subseteq {𝒯}_{\le v}^{\lambda }}{T}_{\varphi {\left(h\right)}^{-1}}{X}^{h\left(1\right)}{T}_i+\sum _{S_i\left(h\right)\cap {𝒯}_{\le v}^{\lambda }=\left\{h\right\}}\sum _{p\in S_i\left(h\right)}{T}_{\varphi {\left(p\right)}^{-1}}{X}^{p\left(1\right)})\\ & =& \sum _{p\in {𝒯}_{\le w}^{\lambda }}{T}_{\varphi {\left(p\right)}^{-1}}{X}^{p\left(1\right)}\text{.}\end{array}$$ (b) The proof is similar to case (a). For $w\in W$ let $${𝒯}_{=w}^{\lambda }=\{p\in {𝒯}^{\lambda }\hspace{0.17em}\mid \hspace{0.17em}\iota \left(p\right)=w{W}_{\lambda }\}\text{.}$$ Assume $w=s_{i}v>v\text{.}$ Then the facts in (3.3) imply that ${𝒯}_{=w}^{\lambda }$ is a union of the strings $S_i\left(h\right)$ such that $h\in {𝒯}_{-h}^{\lambda },$ and If $h\in {𝒯}_{=v}^{\lambda }$ then either $S_i\left(h\right)\subseteq {𝒯}_{=v}^{\lambda }$ or $S_i\left(h\right)\cap {𝒯}_{=v}^{\lambda }=\left\{h\right\}\text{.}$ Let $$\begin{array}{cc}{{\rm E}}_{\varphi \left(p\right)}=\sum _{\underset{z\le \varphi \left(p\right)}{z\in W/W_{\lambda }}}{(-1)}^{\ell \left(z\right)}{\epsilon }_{{\stackrel{\sim }{z}}^{-1}}\text{.}& \text{(3.6)}\end{array}$$ Using (3.3), a direct computation with the relation (1.3) establishes that, if $h\in {𝒯}_{=v}^{\lambda }$ with $e_{i}h=0$ then $$\sum _{p\in S_i\left(h\right)}{{\rm E}}_{\varphi \left(p\right)}{X}^{p\left(1\right)}{T}_i=0,\text{and}{{\rm E}}_{\varphi (0h)}{X}^{h\left(1\right)}{T}_i=-\sum _{p\in S_i\left(h\right)-\{]h\}}{{\rm E}}_{\varphi \left(p\right)}{X}^{p\left(1\right)}\text{.}$$ Thus $$\begin{array}{ccc}{X}^{\lambda }{\epsilon }_{w^{-1}}& =& X^{\lambda }{\epsilon }_{v^{-1}}{\epsilon }_i={(-1)}^{\ell \left(v\right)}\left(\sum _{p\in {𝒯}_{=v}^{\lambda }}{{\rm E}}_{\varphi \left(p\right)}{X}^{p\left(1\right)}\right)T_i\\ & =& {(-1)}^{\ell \left(v\right)}(\sum _{S_i\left(h\right)\subseteq {𝒯}_{=v}^{\lambda }}\sum _{p\in S_i\left(h\right)}{{\rm E}}_{\varphi \left(p\right)}{X}^{p\left(1\right)}+\sum _{S_i\left(h\right)\cap {𝒯}_{=v}^{\lambda }=\left\{h\right\}}{{\rm E}}_{\varphi \left(h\right)}{X}^{h\left(1\right)})T_i\\ & =& {(-1)}^{\ell \left(v\right)}(0-\sum _{S_i\left(h\right)\cap {𝒯}_{=v}^{\lambda }=\left\{h\right\}}\sum _{p\in S_i\left(h\right)-\left\{h\right\}}{{\rm E}}_{\varphi \left(p\right)}{X}^{p\left(1\right)})\\ & =& {(-1)}^{\ell \left(v\right)}\left(\sum _{p\in {𝒯}_{=w}^{\lambda }}{{\rm E}}_{\varphi \left(p\right)}{X}^{p\left(1\right)}\right)\text{.}\end{array}$$ $\square$

Corollary 3.7. Let $\lambda ,\mu \in P^{+}$ and let $w\in W\text{.}$ Then, in the affine nil-Hecke algebra $\stackrel{\sim }{H},$

$$\begin{array}{ccc}{X}^{-\lambda }{T}_{w^{-1}}& =& \sum _{\underset{\varphi \left(p\right)=w{w}_0}{p\in {𝒯}^{-w_0\lambda }}}\sum _{\underset{z{w}_0\ge \iota \left(p\right)}{z\in W/W_{-w_0\lambda }}}{(-1)}^{\ell \left(w\right)+\ell \left(z\right)}{T}_{{\stackrel{\sim }{z}}^{-1}}{X}^{p\left(1\right)}\text{and}\\ X^{w_0\mu }{T}_{w^{-1}}& =& \sum _{\underset{\varphi \left(p\right)=w{w}_0}{p\in {𝒯}^{\mu }}}\sum _{\underset{z{w}_0\ge \varphi \left(p\right)}{z\in W/W_{\mu }}}{(-1)}^{\ell \left(w\right)+\ell \left(z\right)}{T}_{{\stackrel{\sim }{z}}^{-1}}{X}^{p\left(1\right)}\text{.}\end{array}$$

		Proof.
	The second identity is a restatement of the first with a change of variable $\mu =-w_0\lambda \text{.}$ The first identity is obtained by applying the algebra involution $$\begin{array}{ccc}\stackrel{\sim }{H}& ⟶& \stackrel{\sim }{H}\\ T_w& ⟼& {\epsilon }_w\\ X^{\lambda }& ⟼& X^{-\lambda }\end{array}\text{and the bijection}\begin{array}{ccc}{𝒯}^{\lambda }& ⟶& {𝒯}^{-w_0\lambda }\\ p& ⟶& p^{*}\end{array}$$ 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 $L{\left(\lambda \right)}^{*}\cong L(-w_0\lambda ),$ if $L\left(\lambda \right)$ is the simple $G\text{-module}$ of highest weight $\lambda \text{.}$ Note that $p^{*}\left(1\right)=-p\left(1\right),$ $\iota \left(p^{*}\right)=\varphi \left(p\right)w_0,$ and $\varphi \left(p^{*}\right)=\iota \left(p\right)w_0\text{.}$ $\square$

Applying the identities from Theorem 3.5 and Corollary 3.7 to $\left[O_{X_1}\right]$ yields the following product formulas in $K_T(G/B)\text{.}$ In particular, this gives a combinatorial proof of the $\text{(}T\text{-equivariant}$ extension) of the duality theorem of Brion [Bri2002, Theorem 4]. For $\lambda \in P$ and $w\in W$ let $\left[X^{\lambda }\right]=X^{\lambda }\left[{𝒪}_{X_{w_0}}\right]=X^{\lambda }{T}_{w_0}\left[O_{X_1}\right]$ and let $c_{\lambda ,w}^z$ be given by

$$\begin{array}{cc}\left[X^{\lambda }\right]\left[{𝒪}_{X_w}\right]=\sum _{z\in W}{c}_{\lambda ,w}^z\left[{𝒪}_{X_z}\right]\text{.}& \text{(3.8)}\end{array}$$

Corollary 3.9. Let $\lambda \in P^{+},$ $w\in W$ and $W_{\lambda }=\text{Stab}\left(\lambda \right)\text{.}$ Then, with notation as in (3.8),

$$\begin{array}{ccc}{c}_{\lambda ,w}^z& =& \sum _{\underset{w{W}_{\lambda }\ge \iota \left(p\right)\ge \varphi \left(p\right)=z{W}_{\lambda }}{p\in {𝒯}^{\lambda }}}{e}^{p\left(1\right)},\\ c_{w_0\lambda ,w}^z& =& {(-1)}^{\ell \left(w\right)+\ell \left(z\right)}{c}_{\lambda ,z{w}_0}^{w{w}_0},\text{and}{c}_{-\lambda ,w}^z={(-1)}^{\ell \left(w\right)+\ell \left(z\right)}{c}_{-w_0\lambda ,z{w}_0}^{w{w}_0}\text{.}\end{array}$$

Proposition 3.10. For $1\le i\le n,$ $\left[{𝒪}_{X_{w_0{s}_i}}\right]=1-e^{w_0{\omega }_i}\left[X^{-{\omega }_i}\right]\text{.}$

		Proof.
	We shall show that $$\begin{array}{cc}{X}^{-{\omega }_i}\left[{𝒪}_{X_{w_0}}\right]=e^{-w_0{\omega }_i}(\left[{𝒪}_{X_{w_0}}\right]-\left[{𝒪}_{X_{w_0{s}_i}}\right]),& \text{(3.11)}\end{array}$$ and the result will follow by solving for $\left[{𝒪}_{X_{s_i{w}_0}}\right]\text{.}$ Let ${\omega }_j=-w_0{\omega }_i\text{.}$ By Corollary 3.9, $$c_{-{\omega }_i,w_0}^z={(-1)}^{\ell \left(w_0\right)+\ell \left(z\right)}{c}_{{\omega }_j,z{w}_0}^1={(-1)}^{\ell \left(w_0\right)+\ell \left(z\right)}\sum _{\underset{z{w}_0\ge \iota \left(p\right)\ge \varphi \left(p\right)=1}{p\in {𝒯}^{\omega }j}}{e}^{p\left(1\right)}\text{.}$$ The straight line path to ${\omega }_j,$ $p_{{\omega }_j},$ has ${\iota }_{z{w}_0}\left(p_{{\omega }_j}\right)={\varphi }_{z{w}_0}\left({\omega }_j\right)$ and is the unique path in ${𝒯}^{{\omega }_j}$ which may have final direction 1. Suppose ${\varphi }_{z{w}_0}\left(p_{{\omega }_j}\right)=1\text{.}$ Then, since $s_j$ is the only simple reflection which is not in $\text{Stab}\left({\omega }_j\right),$ it must be that $z{w}_0\ngeqq s_k$ for all $k\ne j\text{.}$ Thus $z{w}_0=1$ or $z{w}_0=s_j$ and so $c_{-{\omega }_i,w_0}^z\ne 0$ only if $z=w_0$ or $z=s_j{w}_0=w_0{s}_i\text{.}$ Now (3.11) follows since $p_{{\omega }_j}$ has endpoint ${\omega }_j=-w_0{\omega }_i\text{.}$ $\square$

Corollary 3.12. Let $c_{wv}^z$ be as in (3.8). Then, for

$$c_{w_0{s}_i,w}^{\lambda }=-(e^{-(w{\omega }_i-w_0{\omega }_i)}-1)$$

and

$$c_{w_0{s}_i,w}^z={(-1)}^{\ell \left(w\right)+\ell \left(z\right)+1}\sum _{\underset{z{w}_0\ge \iota \left(p\right)\ge \varphi \left(p\right)=w{w}_0}{p\in {𝒯}^{-w_0{\omega }_i}}}{e}^{w_0{\omega }_i+p\left(1\right)},\text{for}\hspace{0.17em}z\ne w\text{.}$$

		Proof.
	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 $ww-=\iota \left(p\right)=\varphi \left(p\right)=w{w}_0$ and endpoint $p\left(1\right)=w{w}_0(-w_0{\omega }_i)=-w{\omega }_i\text{.}$ $\square$

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.

page history