## The moment graph model

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

$ιw: pt → X * ↦ w considerι*= ⊕w∈Wιw*: ΩT*(X) ⟶⨁w∈WΩT (pt), (3.1)$

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

$im ι*= { (gw) w∈W0 ∈⨁w∈W0 ΩT(pt) , ∣ gw-gw′∈ yαΩT (pt) if there is a 1-dimensional T-orbit containing w and w′ } ,$

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 $\text{im} {\iota }^{*}$ 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

$1 {s}_{1} {s}_{2} {s}_{1}{s}_{2} {s}_{2}{s}_{1} {s}_{1}{s}_{2}{s}_{1}={s}_{2}{s}_{1}{s}_{2} {y}_{-{\alpha }_{2}} {y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)} {y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)} {y}_{-{\alpha }_{1}} {y}_{-{\alpha }_{1}} {y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)} {y}_{-{\alpha }_{2}} {y}_{-{\alpha }_{2}} {y}_{-{\alpha }_{1}} (3.2)$

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 $\text{im} {\iota }^{*}\text{.}$

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

$f:X→Ywhich provides, by restriction, f:W→V$

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

$f*:HT(Y)→ HT(X)and f!:HT(X)→ HT(Y)$

are given by

$(f*(c))w= cf(w),and (f!(γ))v =∑w∈f-1(v) γw 1e(f)wv, (3.3)$

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

$e(f)wv= ( ∏ edges of W adjacent to w yβ ) ( ∏ edges of V adjacent to v yβ ) -1 .$

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 ${\Omega }_{T}\left(G/B\right)$ 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 ${𝔥}_{ℤ}^{*}$ be the Weyl group and the weight lattice ${𝔥}_{ℤ}^{*}$ as in (2.2). Let $𝕃$ be the Lazard ring generated by ${a}_{ij}$ as in (2.13) and let $S$ be the $𝕃\text{-algebra}$

$S=𝕃 [ [ yλ ∣ λ∈𝔥ℤ* ] ] ,with yλ+μ=yλ+ yμ+a11yλ yμ+a12yλ yμ2+a21 yλ2yμ+…. (3.4)$

The Weyl group

$W0acts 𝕃-linaerly on S bywyλ= ywλ,$

for $w\in {W}_{0},$ $\lambda \in {𝔥}_{ℤ}^{*}\text{.}$ Define a product on ${\oplus }_{w\in {W}_{0}}S$ pointwise,

$(fw) w∈W0 · (gw) w∈W0 = (fwgw) w∈W0 , (3.5)$

and let $S{\otimes }_{{S}^{{W}_{0}}}S$ be the coinvariant ring as defined in (2.15). The $S\text{-algebra}$ homomorphism

$Φ: S⊗SW0S ⟶∼ΩT (G/B)⟶∼im Φ ↪ ⊕w∈W0S f⊗g ⟼ (f·(w-1g)) w∈W0 (3.6)$

is well defined and injective with

$im Φ= { (gw)w∈W0 ∈⨁w∈W0S ∣ gw- gwsα∈ y-αS for α∈R+ and w∈W0 } ,$

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,

$S=𝕃 [ [ yλ ∣ λ∈𝔥ℤ* ] ] withyλ+μ =yλ+yμ- p(yλ,yμ) yλyμ, (3.7)$

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

$p(yλ,yμ)= -a11-a12yμ- a21yλ-a31 yλ2-a22yλ yμ-a13yμ yλ-…, (3.8)$

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

$y-λ+λ=y0 =0,yλ+μ= yμ+λ, y(λ+μ)+ν= yλ+(μ+ν). (3.9)$

Then

$yα= -y-α 1-p (yα,y-α) y-α ,1y-α+ 1yα=p (yα,y-α), (3.10)$

and the formula

$y-ℓα y-α =ℓ-∑j=1ℓ-1 p(y-αy-jα) y-jα=1+ ∑j=1ℓ-1 ( 1-p (y-αy-jα) y-jα ) ,for ℓ∈ℤ>0, (3.11)$

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

$ysiλ-yλ y-αi = ( 1-p ( yλ, y -⟨λ,αi∨⟩ αi ) yλ ) ( 1+ ∑ j=1 ⟨λ,α∨⟩-1 ( 1-p ( y-αi, y-jαi ) y-jαi ) ) , (3.12)$

for $⟨\lambda ,{\alpha }_{i}^{\vee }⟩\in {ℤ}_{\ge 0}\text{.}$ 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

$p(yλ,yμ)= { 0, in HT, 1, in KT, andyλ= { yλ, in HT, 1-eλ, in KT. (3.13)$

### 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

$H = (S⊗𝕃S)⋉𝕃 [W0] = S-span { gtw ∣ g∈S,w∈W0 } =𝕃-span { (f⊗g)tw ∣ f,g∈S,w ∈W0 }$

with

$tutv=tuv andtw (f⊗g)= (f⊗(wg))tw, (3.14)$

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

$tw(f⊗g)=f ⊗wgand (h⊗p) (f⊗g)=hf ⊗pg, (3.15)$

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

$ε: S⊗𝕃S ⟶ S f⊗g ⟼ fg so that the S⊗𝕃S action on Sb1 is given by (f⊗g)b1=fgb1,$

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

$Hb1= IndS⊗𝕃SH (Sb1)has S-basis {bw ∣ w∈W0} ,where bw=tw b1.$

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])

$S⊗𝕃S ⟶ H10 ↪ H ⟶ Hb1 ≅⨁w∈W0S (f⊗g) ⟼ (f⊗g)10 h ⟼ hb1 (3.16)$

is a homomorphism of $H\text{-modules}$ (with kernel generated by $\left\{f\otimes 1-1\otimes f \mid f\in {S}^{{W}_{0}}\right\}\text{).}$ The maps in (3.16) allow for the expansion of any element of $S{\otimes }_{𝕃}S$ in terms of the basis $\left\{{b}_{w} \mid w\in {W}_{0}\right\}$ of $H{b}_{1},$ giving

$(f⊗g)10b1 = (f⊗g) ( ∑w∈W0 tw ) b1 = ∑w∈W0tw ( f⊗(w-1g) ) b1 = ∑w∈W0tw ( f·(w-1g) ) b1 = ∑w∈W0 ( f· (w-1g) ) bw.$

This formula illustrates that computing $\Phi \left(f\otimes g\right)$ in (3.6) is equivalent to expanding $\left(f\otimes g\right){b}_{1}$ in terms of the ${b}_{w}\text{.}$ Because of this we use (3.6) and (3.16) to

$identifyΩT (G/B)=Hb1=S -span { bw ∣ w ∈W0 } ≅⨁w∈W0S$

and write elements

$f∈ΩT (G/B)as f=∑w∈W0 fwbw. (3.17)$

The product in ${\Omega }_{T}\left(G/B\right)$ 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

$xμ=1⊗yμ and yμ=yμ⊗1. (3.18)$

Then the formulas

$xλ·1=xλ ∑w∈W0tw b1=∑w∈W0 twxw-1λ b1=∑w∈W0 yw-1λbw, and (3.19) tv∑w∈W0 fwbw= ∑w∈W0fw tvbw= ∑w∈W0fw bvw= ∑z∈W0 fv-1zbz, (3.20)$

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

$yλ xλ = ys1λ ys2s1λ ys2λ ys1s2λ ys1s2s1λ$

## 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.