The moment graph model

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

Last update: 17 February 2013

The moment graph model

$T\text{-fixed}$ points and the map $\Phi$

Following Goresky-Kottwitz-MacPherson [GKM1489894, Theorem 1.2.2] a powerful way to think about this theory is via the moment graph model. This means that for a $T\text{-variety}$ $X$ where the imbeddings of the $T\text{-fixed}$ points of $X$ into $X$ are

$$\begin{array}{cc}\begin{array}{cccc}{\iota }_w:& \text{pt}& \to & X\\ & *& \mapsto & w\end{array}\text{consider}{\iota }^{*}={\oplus }_{w\in W}{\iota }_w^{*}:{\Omega }_T^{*}\left(X\right)⟶\underset{w\in W}{⨁}{\Omega }_T\left(\text{pt}\right),& \text{(3.1)}\end{array}$$

where the sums are over an index set $W$ for the $T\text{-fixed}$ points in $X\text{.}$ When $X$ is a “GKM-space” (see [GKM1489894, Theorem 14] for several equivalent characterization of a GKM space for equivariant ordinary cohomology and [?, HHH0409305] or equivariant generalized cohomology theories) the ring homomorphism ${\iota }^{*}$ is injective with image

$$\text{im}\hspace{0.17em}{\iota }^{*}=\{{\left(g_w\right)}_{w\in W_0}\in \underset{w\in W_0}{⨁}{\Omega }_T\left(\text{pt}\right),\hspace{0.17em}\mid \hspace{0.17em}\begin{array}{c}{g}_w-g_{w\prime }\in y_{\alpha }{\Omega }_T\left(\text{pt}\right)\hspace{0.17em}\text{if there is a}\\ \text{1-dimensional}\hspace{0.17em}T\text{-orbit containing}\hspace{0.17em}w\hspace{0.17em}\text{and}\hspace{0.17em}w\prime \end{array}\},$$

where $y_{\alpha }$ is the $T\text{-equivariant}$ Chern class of the tangent along the 1-dimensional orbit connecting $w$ and $w\prime \text{.}$

Computations are facilitated by encoding the information of $\text{im}\hspace{0.17em}{\iota }^{*}$ with a moment graph, which has vertices corresponding to the $T\text{-fixed}$ points of $X$ and labeled edges $w\stackrel{\alpha }{⟶}w\prime$ corresponding to 1-dimensional $T\text{-orbits}$ in $X\text{.}$ For example, for $G/B$ for type $G{L}_3$ the graph is

$$\begin{array}{cc} 1 s1 s2 s1s2 s2s1 s1s2s1=s2s1s2 y-α2 y-(α1+α2) y-(α1+α2) y-α1 y-α1 y-(α1+α2) y-α2 y-α2 y-α1 & \text{(3.2)}\end{array}$$

A moment graph section is a tuple ${\left(g_2\right)}_{w\in W}$ of elements of ${\Omega }_T\left(\text{pt}\right)$ which is an element of $\text{im}\hspace{0.17em}{\iota }^{*}\text{.}$

A morphism of GKM-spaces is a morphism of $T\text{-spaces}$

$$f:X\to Y\text{which provides, by restriction,}f:W\to V$$

from the set W of $T\text{-fixed}$ points of $X$ to the set $V$ of $T\text{-fixed}$ points of $Y\text{.}$ Viewing elements of $H_T\left(X\right)$ and $H_T\left(Y\right)$ as moment graph sections the maps

$$f^{*}:H_T\left(Y\right)\to H_T\left(X\right)\text{and}{f}_{!}:H_T\left(X\right)\to H_T\left(Y\right)$$

are given by

$$\begin{array}{cc}{\left(f^{*}\left(c\right)\right)}_w=c_{f\left(w\right)},\text{and}{\left(f_{!}\left(\gamma \right)\right)}_v=\sum _{w\in f^{-1}\left(v\right)}{\gamma }_w\frac{1}{e{\left(f\right)}_{wv}},& \text{(3.3)}\end{array}$$

where the Euler class of $f$ from $v$ to $w$ is

$$e{\left(f\right)}_{wv}=\left(\prod _{\genfrac{}{}{0ex}{}{\text{edges of}\hspace{0.17em}W}{\text{adjacent to}\hspace{0.17em}w}}{y}_{\beta }\right){\left(\prod _{\genfrac{}{}{0ex}{}{\text{edges of}\hspace{0.17em}V}{\text{adjacent to}\hspace{0.17em}v}}{y}_{\beta }\right)}^{-1}\text{.}$$

The second formula in (3.3) is a form of the familiar formula for push forwards by “localization at the $T\text{-fixed}$ points” as found, for example, in [ABo0721448, (3.8)]. The Euler class of $f$ from $v$ to $w$ is the contribution measured by the difference between the tangent space at the $T\text{-fixed}$ point $w$ in $X$ to the tangent space to the $T\text{-fixed}$ point $v=f\left(w\right)$ in $Y\text{.}$

The Borel model and the moment graph model for $G/B$ for equivariant algebraic cobordism ${\Omega }_T(G/B)$ are summarized in the following Theorem, which is a combination of [KKr1104.1089, Theorem 4.7] and [HHH0409305, Theorem 3.1]. The ring $S$ which takes the role of ${\Omega }_T\left(\text{pt}\right)$ is as in [CPZ0905.1341, §2.4]. For comparison to the $K\text{-theory}$ case see [KKu0895705, Theorem 3.13] and [LSS2660675, Theorem 3.1].

Theorem 3.1. ([HHH0409305, Theorem 3.1], [KKr1104.1089, Theorem 4.7] and [CPZ0905.1341, §2.4] combined) Let $G\supseteq B\supseteq T$ be a reductive group as in (2.1) and let $W_0$ and ${𝔥}_{\mathbb{Z}}^{*}$ be the Weyl group and the weight lattice ${𝔥}_{\mathbb{Z}}^{*}$ as in (2.2). Let $𝕃$ be the Lazard ring generated by $a_{ij}$ as in (2.13) and let $S$ be the $𝕃\text{-algebra}$

$$\begin{array}{cc}S=𝕃\left[[y_{\lambda }\hspace{0.17em}\mid \hspace{0.17em}\lambda \in {𝔥}_{\mathbb{Z}}^{*}]\right],\text{with}{y}_{\lambda +\mu }=y_{\lambda }+y_{\mu }+a_{11}{y}_{\lambda }{y}_{\mu }+a_{12}{y}_{\lambda }{y}_{\mu }^2+a_{21}{y}_{\lambda }^2{y}_{\mu }+\dots \text{.}& \text{(3.4)}\end{array}$$

The Weyl group

$$W_0\text{acts}\hspace{0.17em}𝕃\text{-linaerly on}\hspace{0.17em}S\hspace{0.17em}\text{by}w{y}_{\lambda }=y_{w\lambda },$$

for $w\in W_0,$ $\lambda \in {𝔥}_{\mathbb{Z}}^{*}\text{.}$ Define a product on ${\oplus }_{w\in W_0}S$ pointwise,

$$\begin{array}{cc}{\left(f_w\right)}_{w\in W_0}·{\left(g_w\right)}_{w\in W_0}={\left(f_w{g}_w\right)}_{w\in W_0},& \text{(3.5)}\end{array}$$

and let $S{\otimes }_{S^{W_0}}S$ be the coinvariant ring as defined in (2.15). The $S\text{-algebra}$ homomorphism

$$\begin{array}{cc}\begin{array}{cccc}\Phi :& S{\otimes }_{S^{W_0}}S& \stackrel{\sim }{⟶}{\Omega }_T(G/B)\stackrel{\sim }{⟶}\text{im}\hspace{0.17em}\Phi \hookrightarrow & {\oplus }_{w\in W_0}S\\ & f\otimes g& ⟼& {(f·\left(w^{-1}g\right))}_{w\in W_0}\end{array}& \text{(3.6)}\end{array}$$

is well defined and injective with

$$\text{im}\hspace{0.17em}\Phi =\{{\left(g_w\right)}_{w\in W_0}\in \underset{w\in W_0}{⨁}S\hspace{0.17em}\mid \hspace{0.17em}{g}_w-g_{w{s}_{\alpha }}\in y_{-\alpha }S\hspace{0.17em}\text{for}\hspace{0.17em}\alpha \in R^{+}\hspace{0.17em}\text{and}\hspace{0.17em}w\in W_0\},$$

where $R^{+}$ is the set of positive roots corresponding to $B$ and $s_{\alpha }\in W_0$ denotes the reflection corresponding to $\alpha \text{.}$

To provide a feel for the ring $S$ of (3.4), let us provide some formulas which will be useful for computations later. To recapitulate and summarize previous definitions,

$$\begin{array}{cc}S=𝕃\left[[y_{\lambda }\hspace{0.17em}\mid \hspace{0.17em}\lambda \in {𝔥}_{\mathbb{Z}}^{*}]\right]\text{with}{y}_{\lambda +\mu }=y_{\lambda }+y_{\mu }-p(y_{\lambda },y_{\mu })y_{\lambda }{y}_{\mu },& \text{(3.7)}\end{array}$$

where $p(y_{\lambda },y_{\mu })\in 𝕃\left[[y_{\lambda },y_{\mu }]\right]$ is a power series

$$\begin{array}{cc}p(y_{\lambda },y_{\mu })=-a_{11}-a_{12}{y}_{\mu }-a_{21}{y}_{\lambda }-a_{31}{y}_{\lambda }^2-a_{22}{y}_{\lambda }{y}_{\mu }-a_{13}{y}_{\mu }{y}_{\lambda }-\dots ,& \text{(3.8)}\end{array}$$

with $a_{ij}\in 𝕃$ satisfying relations such that

$$\begin{array}{cc}{y}_{-\lambda +\lambda }=y_0=0,y_{\lambda +\mu }=y_{\mu +\lambda },y_{(\lambda +\mu )+\nu }=y_{\lambda +(\mu +\nu )}\text{.}& \text{(3.9)}\end{array}$$

Then

$$\begin{array}{cc}{y}_{\alpha }=\frac{-y_{-\alpha }}{1-p(y_{\alpha },y_{-\alpha })y_{-\alpha }},\frac{1}{y_{-\alpha }}+\frac{1}{y_{\alpha }}=p(y_{\alpha },y_{-\alpha }),& \text{(3.10)}\end{array}$$

and the formula

$$\begin{array}{cc}\frac{y_{-\ell \alpha }}{y_{-\alpha }}=\ell -\sum _{j=1}^{\ell -1}p\left(y_{-\alpha }{y}_{-j\alpha }\right)y_{-j\alpha }=1+\sum _{j=1}^{\ell -1}(1-p\left(y_{-\alpha }{y}_{-j\alpha }\right)y_{-j\alpha }),\text{for}\hspace{0.17em}\ell \in {\mathbb{Z}}_{>0},& \text{(3.11)}\end{array}$$

is proved by induction on $\ell \text{.}$ Using (3.11) and the formula $s_i\lambda =\lambda -⟨\lambda {\alpha }_i^{\vee }⟩{\alpha }_i$ for the action of a simple reflection on ${𝔥}^{*}$ produces

$$\begin{array}{cc}\frac{y_{s_i\lambda }-y_{\lambda }}{y_{-{\alpha }_i}}=(1-p(y_{\lambda },y_{-⟨\lambda ,{\alpha }_i^{\vee }⟩{\alpha }_i})y_{\lambda })(1+\sum _{j=1}^{⟨\lambda ,{\alpha }^{\vee }⟩-1}(1-p(y_{-{\alpha }_i},y_{-j{\alpha }_i})y_{-j{\alpha }_i})),& \text{(3.12)}\end{array}$$

for $⟨\lambda ,{\alpha }_i^{\vee }⟩\in {\mathbb{Z}}_{\ge 0}\text{.}$ Formula (3.12) generalizes one of the favorite formulas for the action of a Demazure operator (see [Kum1923198, Lemma 8.2.8]). This cobordism case specializes to $H_T$ and $K_T$ by setting

$$\begin{array}{cc}p(y_{\lambda },y_{\mu })=\{\begin{array}{cc}0,& \text{in}\hspace{0.17em}{H}_T,\\ 1,& \text{in}\hspace{0.17em}{K}_T,\end{array}\text{and}{y}_{\lambda }=\{\begin{array}{cc}{y}_{\lambda },& \text{in}\hspace{0.17em}{H}_T,\\ 1-e^{\lambda },& \text{in}\hspace{0.17em}{K}_T\text{.}\end{array}& \text{(3.13)}\end{array}$$

The nil affine Hecke algebra

Let S be as in (3.4) and (3.7). The point of view of [GRa0405333] is that the homomorphism $\Phi$ of (3.6) arises naturally from the nil affine Hecke algebra.

The nil affine Hecke algebra $H$ is

$$\begin{array}{ccc}H& =& \left(S{\otimes }_{𝕃}S\right)⋉𝕃\left[W_0\right]\\ & =& S\text{-span}\{g{t}_w\hspace{0.17em}\mid \hspace{0.17em}g\in S,w\in W_0\}=𝕃\text{-span}\{(f\otimes g)t_w\hspace{0.17em}\mid \hspace{0.17em}f,g\in S,w\in W_0\}\end{array}$$

with

$$\begin{array}{cc}{t}_u{t}_v=t_{uv}\text{and}{t}_w(f\otimes g)=(f\otimes \left(wg\right))t_w,& \text{(3.14)}\end{array}$$

for $u,v,w\in W_0$ and $f,g\in S\text{.}$ The nil affine Hecke algebra $H$ acts on $S{\otimes }_{𝕃}S$ and on $S{\otimes }_{S^{W_0}}S$ by

$$\begin{array}{cc}{t}_w(f\otimes g)=f\otimes wg\text{and}(h\otimes p)(f\otimes g)=hf\otimes pg,& \text{(3.15)}\end{array}$$

for $h,p,f,g\in S$ and $w\in W_0\text{.}$ These actions arise from the realization of $S{\otimes }_{S^{W_0}}S$ as an induced up $H\text{-module}$ in (3.16) below.

Let $b_1$ be a symbol and let $S{b}_1$ be the $S{\otimes }_{𝕃}S$ module (a rank 1 free $S\text{-module}$ with basis $\left\{b_1\right\}\text{)}$ corresponding to the ring homomorphism

$$\begin{array}{cccc}\epsilon :& S{\otimes }_{𝕃}S& ⟶& S\\ & f\otimes g& ⟼& fg\end{array}\text{so that the}\hspace{0.17em}S{\otimes }_{𝕃}S\hspace{0.17em}\text{action on}\hspace{0.17em}S{b}_1\hspace{0.17em}\text{is given by}\hspace{0.17em}(f\otimes g)b_1=fg{b}_1,$$

for $f,g\in S\text{.}$ The induced module

$$H{b}_1={\text{Ind}}_{S{\otimes }_{𝕃}S}^H\left(S{b}_1\right)\text{has}\hspace{0.17em}S\text{-basis}\{b_w\hspace{0.17em}\mid \hspace{0.17em}w\in W_0\},\text{where}\hspace{0.17em}{b}_w=t_w{b}_1\text{.}$$

Let $1_0={\sum }_{w\in W_0}{t}_w\text{.}$ With the definition of the $H$ action on $S{\otimes }_{𝕃}S$ as in (3.15), the sequence of maps (see [GRa0405333, Theorem 2.12])

$$\begin{array}{cc}\begin{array}{cccccccc}S{\otimes }_{𝕃}S& ⟶& H{1}_0& \hookrightarrow & H& ⟶& H{b}_1& \cong {⨁}_{w\in W_0}S\\ (f\otimes g)& ⟼& (f\otimes g)1_0\\ & & & & h& ⟼& h{b}_1\end{array}& \text{(3.16)}\end{array}$$

is a homomorphism of $H\text{-modules}$ (with kernel generated by $\{f\otimes 1-1\otimes f\hspace{0.17em}\mid \hspace{0.17em}f\in S^{W_0}\}\text{).}$ The maps in (3.16) allow for the expansion of any element of $S{\otimes }_{𝕃}S$ in terms of the basis $\{b_w\hspace{0.17em}\mid \hspace{0.17em}w\in W_0\}$ of $H{b}_1,$ giving

$$\begin{array}{ccccc}(f\otimes g)1_0{b}_1& =& (f\otimes g)\left(\sum _{w\in W_0}{t}_w\right)b_1& =& \sum _{w\in W_0}{t}_w(f\otimes \left(w^{-1}g\right))b_1\\ & =& \sum _{w\in W_0}{t}_w(f·\left(w^{-1}g\right))b_1& =& \sum _{w\in W_0}(f·\left(w^{-1}g\right))b_w\text{.}\end{array}$$

This formula illustrates that computing $\Phi (f\otimes g)$ in (3.6) is equivalent to expanding $(f\otimes g)b_1$ in terms of the $b_w\text{.}$ Because of this we use (3.6) and (3.16) to

$$\text{identify}{\Omega }_T(G/B)=H{b}_1=S\text{-span}\{b_w\hspace{0.17em}\mid \hspace{0.17em}w\in W_0\}\cong {⨁}_{w\in W_0}S$$

and write elements

$$\begin{array}{cc}f\in {\Omega }_T(G/B)\text{as}f=\sum _{w\in W_0}{f}_w{b}_w\text{.}& \text{(3.17)}\end{array}$$

The product in ${\Omega }_T(G/B)$ is then given by (3.5). To more easily keep track of the left and right factors in $S{\otimes }_{𝕃}S$ use the notation

$$\begin{array}{cc}{x}_{\mu }=1\otimes y_{\mu }\text{and}{y}_{\mu }=y_{\mu }\otimes 1\text{.}& \text{(3.18)}\end{array}$$

Then the formulas

$$\begin{array}{cc}{x}_{\lambda }·1=x_{\lambda }\sum _{w\in W_0}{t}_w{b}_1=\sum _{w\in W_0}{t}_w{x}_{w^{-1}\lambda }{b}_1=\sum _{w\in W_0}{y}_{w^{-1}\lambda }{b}_w,\text{and}& \text{(3.19)}\\ t_v\sum _{w\in W_0}{f}_w{b}_w=\sum _{w\in W_0}{f}_w{t}_v{b}_w=\sum _{w\in W_0}{f}_w{b}_{vw}=\sum _{z\in W_0}{f}_{v^{-1}z}{b}_z,& \text{(3.20)}\end{array}$$

provide the formulas for action of the nil affine Hecke algebra in terms of moment graph sections (see (3.15)). We often view the values $f_w$ as labels on the vertices of the moment graph so that, for exmaple, in type $G{L}_3$ where the moment graph is as in (3.2), (3.19) can be written

$$\begin{array}{cccc}& & & y_{\lambda }\\ x^{\lambda }& =& \begin{array}{c}{y}_{s_1\lambda }\\ y_{s_2{s}_1\lambda }\end{array}& & \begin{array}{c}{y}_{s_2\lambda }\\ y_{s_1{s}_2\lambda }\end{array}\\ & & & y_{s_1{s}_2{s}_1\lambda }\end{array}$$

Notes and References

This is an excerpt from a paper entitled Generalized Schubert Calculus authored by Nora Ganter and Arun Ram. It was dedicated to C.S. Seshadri on the occasion of his 90th birthday.

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