The Schubert calculus framework

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

Last update: 17 February 2013

The Schubert calculus framework

Flag an Schubert varieties

The basic data is

$$\begin{array}{cc}\begin{array}{cc}G& \text{a connected complex reductive algebraic group}\\ \cup \mid \\ B& \text{a Borel subgroup}\\ \cup \mid \\ T& a\; maximal\; torus.\end{array}& \text{(2.1)}\end{array}$$

The Weyl group, the character lattice and cocharacter lattice are, respectively,

$$\begin{array}{cc}{W}_0=N\left(T\right)/T,{𝔥}_{\mathbb{Z}}^{*}=\text{Hom}(T,{ℂ}^{\times })\text{and}{𝔥}_{\mathbb{Z}}=\text{Hom}({ℂ}^{\times },T),& \text{(2.2)}\end{array}$$

where $\text{Hom}(H,K)$ is the abelian group of algebraic group homomorphisms from $H$ to $K$ with product given by pointwise multiplication, $\left(\varphi \phi \right)\left(h\right)=\varphi \left(h\right)\phi \left(h\right)\text{.}$ Since the Weyl group acts on $T,$ it also acts on ${𝔥}_{\mathbb{Z}}^{*}$ and on ${𝔥}_{\mathbb{Z}}\text{.}$

A standard parabolic subgroup of $G$ is a subgroup $P_J\supseteq B$ such that $G/P_J$ is a projective variety. A parabolic subgroup of $G$ is a conjugate of a standard parabolic subgroup.

$$\begin{array}{cc}\text{The}\hspace{0.17em}\text{flag variety}\hspace{0.17em}\text{is}\hspace{0.17em}G/B\text{and}G/P_J\hspace{0.17em}\text{are the}\hspace{0.17em}\text{partial flag varieties.}& \text{(2.3)}\end{array}$$

These are studied via the Bruhat decomposition

$$\begin{array}{cc}G=\underset{w\in W_0}{⨆}BwB\text{and}G=\underset{u\in W^J}{⨆}Bu{P}_J& \text{(2.4)}\end{array}$$

where $W_J=\{v\in W_0\hspace{0.17em}\mid \hspace{0.17em}vT\subseteq P_J\}$ and

$$\begin{array}{cc}{W}^J=\{\text{coset representatives}\hspace{0.17em}u\hspace{0.17em}\text{of cosets in}\hspace{0.17em}{W}_0/W_J\}\text{.}& \text{(2.5)}\end{array}$$

The Schubert varieties are

$$\begin{array}{cc}{X}_w=\overline{BwB}\text{in}\hspace{0.17em}G/B\text{and}{X}_u^J=\overline{Bu{P}_J}\hspace{0.17em}\text{in}\hspace{0.17em}G/P_J,& \text{(2.6)}\end{array}$$

and the Bruhat orders are the partial orders on $W_0$ and $W_J$ given by

$$\begin{array}{cc}{X}_w=\overline{BwB}=\underset{v\le w}{⨆}BvB\text{and}{X}_u^J=\overline{Bu{P}_J}=\underset{z\le u}{⨆}Bz{P}_J\text{.}& \text{(2.7)}\end{array}$$

The $T\text{-fixed}$ points

$$\begin{array}{cc}\text{in}\hspace{0.17em}G/B\hspace{0.17em}\text{are}\hspace{0.17em}\{wB\hspace{0.17em}\mid \hspace{0.17em}w\in W_0\}\text{and}\text{in}\hspace{0.17em}G/P_J\hspace{0.17em}\text{are}\hspace{0.17em}\{u{P}_J\hspace{0.17em}\mid \hspace{0.17em}u\in W^J\}\text{.}& \text{(2.8)}\end{array}$$

Let $P_1,\dots ,P_n$ be the minimal parabolic subgroups $P_i\ne B\text{.}$ Then

$$\begin{array}{cc}{W}_i=W_{\left\{i\right\}}=\{1,s_i\}\text{and}{s}_1,\dots ,s_n\hspace{0.17em}\text{are the}\hspace{0.17em}\text{simple reflections}\hspace{0.17em}\text{in}\hspace{0.17em}{W}_0\text{.}& \text{(2.9)}\end{array}$$

With respect to the action $W_0$ on ${𝔥}_{ℝ}^{*}=ℝ{\otimes }_{\mathbb{Z}}{𝔥}_{\mathbb{Z}}^{*},$ the $s_i$ are reflections in the hyperplanes ${\left({𝔥}^{*}\right)}^{s_i}=\{\mu \in {𝔥}_{ℝ}^{*}\hspace{0.17em}\mid \hspace{0.17em}{s}_i\mu =\mu \}\text{.}$ An alternative description of the standard parabolic subgroups is to let $J\subseteq \{1,2,\dots ,n\}$ and let

$$\begin{array}{cc}{W}_J=⟨s_j\hspace{0.17em}\mid \hspace{0.17em}j\in J⟩\text{.}\text{Then}{P}_J=\underset{v\in W_J}{⨆}BvB\text{.}& \text{(2.10)}\end{array}$$

In particular, $P_i=P_{\left\{i\right\}}=B\bigsqcup B{s}_{i}B,$ for $i=1,2,\dots ,n\text{.}$

Theorem 2.1. (Coxeter) The group $W_0$ is generated by $s_1,\dots ,s_n$ with relations

$$s_i^2=1\text{and}\underset{\underset{m_{ij}\hspace{0.17em}\text{factors}}{⏟}}{s_i{s}_j{s}_i\dots }=\underset{\underset{m_{ij}\hspace{0.17em}\text{factors}}{⏟}}{s_j{s}_i{s}_j\dots }$$

where $\pi /m_{ij}={\left({𝔥}^{*}\right)}^{s_i}\angle {\left({𝔥}^{*}\right)}^{s_j}$ is the angle between ${\left({𝔥}^{*}\right)}^{s_i}$ and ${\left({𝔥}^{*}\right)}^{s_j}\text{.}$

The definitions in (2.3), (2.8) and (2.6) provide $T\text{-equivariant}$ maps

$$\begin{array}{cc}\begin{array}{cccc}{p}_J:& G/B& ⟶& G/P_J\\ & gB& ⟼& g{P}_J\end{array}\begin{array}{cccc}{\iota }_w:& \text{pt}& \hookrightarrow & G/B\\ & \text{pt}& ⟼& wB\end{array}\begin{array}{cccc}{\sigma }_w:& X_w& \hookrightarrow & G/B\\ & gB& ⟼& gB\end{array}& \text{(2.11)}\end{array}$$

and

$$\begin{array}{cc}\begin{array}{cccc}{\iota }_u^J:& \text{pt}& \hookrightarrow & G/P_J\\ & \text{pt}& ⟼& u{P}_J\end{array}\begin{array}{cccc}{\sigma }_u^J:& X_u^J& \hookrightarrow & G/P_J\\ & g{P}_J& ⟼& g{P}_J\end{array}& \text{(2.12)}\end{array}$$

for $J\subseteq \{1,2,\dots ,\ell \},$ $w\in W_0,$ and $u\in W^J\text{.}$

For example, in type $G=G{L}_3,$ with $T$ and $B$ the subgroups given by

$$T=\left\{\left(\begin{array}{ccc}*& 0& 0\\ 0& *& 0\\ 0& 0& *\end{array}\right)\right\}\text{and}B=\left\{\left(\begin{array}{ccc}*& *& *\\ 0& *& *\\ 0& 0& *\end{array}\right)\right\},$$

then $W_0=⟨s_1,s_2\hspace{0.17em}\mid \hspace{0.17em}{s}_i^2=1,s_1{s}_2{s}_1=s_2{s}_1{s}_2⟩,$ where

$$s_1=\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right)\text{and}{s}_2=\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right)\text{.}$$

Then

$$\begin{array}{ccc} Xw0=G/B Xs1s2 Xs2s1 Xs1 Xs2 pt=X1 & \text{and}& G/B G/P1G/P2 G/G=pt \end{array}$$

where $P_1$ and $P_2$ are the subgroups of $G=G{L}_3\left(ℂ\right)$ given by

$$P_1=\left\{\left(\begin{array}{ccc}*& *& *\\ *& *& *\\ 0& 0& *\end{array}\right)\right\}=B\bigsqcup B{s}_{1}B\text{and}{P}_2=\left\{\left(\begin{array}{ccc}*& *& *\\ 0& *& *\\ 0& *& *\end{array}\right)\right\}=B\bigsqcup B{s}_{2}B\text{.}$$

Generalized cohomology theories

Schubert calculus is the study of the cohomology of flag and Schubert varieties. Although the home for our computations is the particular ring $S=𝕃\left[\left[y_{\lambda }\right]\right]$ of (3.4) the motivation comes from the formalism of generalized cohomology theories $h\text{.}$ Model examples are: ordinary cohomology $H,$ $K\text{-theory}$ $K,$ elliptic cohomology (see [MRa2330502, GKV9505012, Gro1994, And1637129, Lur2597740]) and complex and algebraic cobordism $\Omega$ (see [LMo2286826]). Key to our point of view is that if $f:X\to Y$ is a morphism of spaces, the contravariance of the cohomology theory provides

$$\text{a pullback}{f}^{*}:h\left(Y\right)\to h\left(X\right),\text{and}\text{a pushforward}{f}_{!}:h\left(X\right)\to h\left(Y\right)$$

exists if the morphism $f$ is nice enough. Our true interest is in the morphisms in (2.11) and (2.12) and (4.5). Sometimes we will try to consider, by combinatorial gadgetry, pushforwards across these morphisms even in cases where we are not sure that, for any given cohomology theory, the pushforward properly exists.

As in [CPZ0905.1341, §8.2], the important property for the analysis of Schubert calculus is that an oriented cohomology theory $h$ comes with a formal group law $F$ over the coefficient ring $h\left(pt\right)$ such that

$$F(c_1^h\left({ℒ}_1\right),c_1^h\left({ℒ}_2\right))=c_1^h({ℒ}_1\otimes {ℒ}_2),$$

where ${ℒ}_1$ and ${ℒ}_2$ are line bundles on $X$ and $c_1^h$ denotes the first Chern class in the cohomology theory $h$ (see [LMo2286286, Cor. 4.1.8]). The Lazard ring $𝕃$ is generated by symbols $a_{ij},$ for $i,j\in {\mathbb{Z}}_{>0},$ which satisfy the relations given by the equations

$$\begin{array}{cc}F(x,F(y,z))=F(F(x,y),z),F(x,y)=F(y,x),F(x,0)=x,& \text{(2.13)}\end{array}$$

where

$$F(x,y)=x+y+a_{11}xy+a_{12}x{y}^2+a_{21}{x}^{2}y+\dots$$

The ring $𝕃$ is the universal coefficient ring for a formal group law $F\text{.}$ This ring is one of the ingredients for the construction of the ring $S$ where we do our computations.

A equivariant cohomology theory $h_T$ is a functor from $T\text{-spaces}$ (some appropriate class of topological or geometric objects with $T\text{-action)}$ to some class of algebraic objects (in most of our model examples, $h_T\text{(pt)-algebras).}$ Important features and properties of the theory include:

0. Normalization: specification of $h_T\text{(pt),}$
1. nice behaviour under products, smashes, suspensions: such as axioms for computing $h_{G\times K}(M\times N),$
2. functoriality/pullbacks: if $f:X\to Y$ then we have $f^{*}:h_T\left(Y\right)\to h_T\left(X\right)$
3. Thom isomorphism/orientability/pushforwards: For certain classes of maps $f:X\to Y$ there exists a pushforward $f_{!}:h_T\left(X\right)\to h_T\left(Y\right),$
4. Change of groups: For certain classes of groups $G$ and $K$ and group homomorphisms $\phi :G\to K$ there exist ${\chi }_{\phi }:h_G\to h_K$ and ${\chi }^{\phi }:h_K\to h_G\text{.}$

The art of choosing appropriate categories of input $\text{"}T\text{-spaces",}$ of output “algebraic objects” and widening the classes of maps on which pushforwards and/or change of groups homomorphisms are defined is a beautiful chapter in algebraic topology and geometry. The challenge of extending a nonequivariant generalized cohomology theory to the equivariant case can be considerable. For such a genuinely equivariant theory the formal groups above will be replaced by actual groups but we do not emphasize this point of view here. For a small selection of references we refer the reader to [Ada1324104, p. 37-29] for a discussion of the connection to formal group laws and spectra, [May1413302, Chapt. XIII] and [Oko0645496] for a discussion of equivariant orientable theories as Mackey functors and [GKV9505012, (1.5)] for discussion of axioms for equivariant elliptic cohomology.

In order to specify a home for our computations in Schubert calculus in equivariant cohomology theories we follow [HHH0409305]. They restrict their class of spaces to GKM spaces: stratified $T\text{-spaces}$

$$X=\bigcup _{i\in {\mathbb{Z}}_{>0}}{X}_i,X_1\subseteq X_2\subseteq X_3\subseteq \dots ,$$

where the successive quotients $X_i/X_{i-1}$ are homeomorphic to the Thom spaces $Th\left(V_i\right)$ of some $h\text{-orientable}$ $T\text{-vector}$ bundles $V_i\to F_i$ (see [HHH0409305, (2.1)]). As pointed out in [HHH0409305, Remark 3.3], for the case of flag and Schubert varieties that are the focus of this paper, the $F_i$ are points and the $V_i$ are one dimensional representations of $T\text{.}$ In particular, the assumptions of [HHH0409305, §3] hold for these cases.

The Borel model for $h_T(G/B)$

The general combinatorial Schubert calculus uses ${𝔥}_{\mathbb{Z}}^{*}$ and ${𝔥}_{\mathbb{Z}}$ to build a $C\text{-algebra}$ $R$ with an action of $W_0$ on $R$ by $C\text{-algebra}$ automorphisms (in favorite examples $C$ may be $\mathbb{Z},$ or the ring ${\stackrel{\sim }{Th}}_0$ of holomorphic functions on the upper half plane, or the Lazard ring $𝕃,$ see the examples below). If

$$R^{W_0}=\{f\in R\hspace{0.17em}\mid \hspace{0.17em}wf=f\hspace{0.17em}\text{for}\hspace{0.17em}w\in W_0\}\text{is the}\hspace{0.17em}\text{invariant ring},$$

then, conceptually,

$$\begin{array}{cc}R=h_T\left(\text{pt}\right)\text{and}{R}^{W_0}=h_G\left(\text{pt}\right),& \text{(2.14)}\end{array}$$

for the equivariant cohomology theory $h_T$ under analysis. By definition, the coinvariant ring is

$$\begin{array}{cc}R{\otimes }_{R^{W_0}}R=\frac{R{\otimes }_{C}R}{⟨f\otimes 1-1\otimes f\hspace{0.17em}\mid \hspace{0.17em}f\in R^{W_0}⟩},& \text{(2.15)}\end{array}$$

where the terminology is chosen to be representative of the classical terminology in the study of the cohomology of $G/B,$ not to reflect a notion of coinvariants with respect to a group action. Then (see [Bor0051508, Proposition 26.1], [KLu0862716, Proposition 1.6], [KKr1104.1089, Theorem 4.7]) the ring

$$\begin{array}{cc}R{\otimes }_{R^{W_0}}R\text{is a good combinatorial model for}{h}_T(G/B),& \text{(2.16)}\end{array}$$

where the product on $R{\otimes }_{R^{W_0}}R$ is given by $(f_1\otimes g_1)(f_2\otimes g_2)=f_1{f}_2\otimes g_1{g}_2\text{.}$

There are four favorite examples:

Cohomology: $h_T=H_T\text{.}$ Here

$$H_T\left(\text{pt}\right)=S\left({𝔥}_{\mathbb{Z}}^{*}\right)=ℂ[x_1,\dots ,x_n]\text{and}{H}_G\left(\text{pt}\right)=H_T{\left(\text{pt}\right)}^{W_0}=ℂ{[x_1,\dots ,x_n]}^{W_0},$$

where $x_i=x_{{\omega }_i},$ where ${\omega }_1,\dots ,{\omega }_n$ is a $\mathbb{Z}\text{-basis}$ of ${𝔥}_{\mathbb{Z}}^{*}\text{.}$ Alternatively, $H_T\left(\text{pt}\right)$ is the ring

$$ℂ[x_{\lambda }\hspace{0.17em}\mid \hspace{0.17em}\lambda \in {𝔥}_{\mathbb{Z}}^{*}]\text{with}{x}_{\lambda +\mu }=x_{\lambda }+x_{\mu },$$

for $\lambda ,\mu \in {𝔥}_{\mathbb{Z}}^{*}$ and with $w{x}_{\lambda }=x_{w\lambda }$ for $w\in W_0$ and $\lambda \in {𝔥}_{\mathbb{Z}}^{*}\text{.}$ Then

$$H_T(G/B)=H_T\left(\text{pt}\right){\otimes }_{H_G\left(\text{pt}\right)}{H}_T\left(\text{pt}\right)=\frac{ℂ[y_1,\dots ,y_n,x_1,\dots ,x_n]}{⟨f(x_1,\dots ,x_n)-f(y_1,\dots ,y_n)\hspace{0.17em}\mid \hspace{0.17em}f\in ℂ{[x_1,\dots ,x_n]}^{W_0}⟩}\text{.}$$

K-theory: $h_T=K_T\text{.}$ Here

$$K_T\left(\text{pt}\right)=ℂ\left[{𝔥}_{\mathbb{Z}}^{*}\right]=ℂ[X_1^{\pm 1},\dots ,X_n^{\pm 1}]\text{and}{K}_G\left(\text{pt}\right)=K_T{\left(\text{pt}\right)}^{W_0}=ℂ{[X_1^{\pm 1},\dots ,X_n^{\pm 1}]}^{W_0},$$

where $X_i=e^{{\omega }_i},$ where ${\omega }_1,\dots ,{\omega }_n$ is a $\mathbb{Z}\text{-basis}$ of ${𝔥}_{\mathbb{Z}}^{*}\text{.}$ Alternatively, $K_T\left(\text{pt}\right)$ is the ring

$$ℂ[e^{\lambda }\hspace{0.17em}\mid \hspace{0.17em}\lambda \in {𝔥}_{\mathbb{Z}}^{*}]\text{with}{e}^{\lambda +\mu }=e^{\lambda }{e}^{\mu },$$

for $\lambda ,\mu \in {𝔥}_{\mathbb{Z}}^{*}$ and with $w{e}^{\lambda }=e^{w\lambda }$ for $w\in W_0$ and $\lambda \in {𝔥}_{\mathbb{Z}}^{*}\text{.}$ Then

$$K_T(G/B)=K_T\left(\text{pt}\right){\otimes }_{K_G\left(\text{pt}\right)}{K}_T\left(\text{pt}\right)=\frac{ℂ[Y_1^{\pm 1},\dots ,Y_n^{\pm 1},X_1^{\pm 1},\dots ,X_n^{\pm 1}]}{⟨f(X_1,\dots ,X_n)-f(Y_1,\dots ,Y_n)\hspace{0.17em}\mid \hspace{0.17em}f\in ℂ{[X_1^{\pm 1},\dots ,X_n^{\pm 1}]}^{W_0}⟩}\text{.}$$

Elliptic cohomology: $h_T={Ell}_T\text{.}$ Here ${Ell}_T\left(\text{pt}\right)$ is the structure sheaf of the abelian variety $A_{\tau }={𝔥}_{ℂ}^{*}/({𝔥}_{\mathbb{Z}}^{*}+\tau {𝔥}_{\mathbb{Z}}^{*})\text{.}$ The homogeneous coordinate ring

$$\text{for}\hspace{0.17em}{A}_{\tau }\hspace{0.17em}\text{is}\stackrel{\sim }{Th}=\underset{m\in {\mathbb{Z}}_{\ge 0}}{⨁}{\stackrel{\sim }{Th}}_m,\text{and}\text{for}\hspace{0.17em}{A}_{\tau }/W_0\hspace{0.17em}\text{is}{\stackrel{\sim }{Th}}^{W_0}\text{.}$$

Then the graded $\stackrel{\sim }{Th}\text{-module}$ corresponding to

$$\text{the sheaf}\hspace{0.17em}{Ell}_T(G/B)\hspace{0.17em}\text{on}\hspace{0.17em}{A}_{\tau }\hspace{0.17em}\text{is}\stackrel{\sim }{Th}{\otimes }_{{\stackrel{\sim }{Th}}^{W_0}}\stackrel{\sim }{Th}\text{.}$$

Complex or algebraic cobordism: $h_T={\Omega }_T\text{.}$ Algebraic cobordism is treated in the book of Levine-Morel [LMo2286826] and $T\text{-equivariant}$ algebraic cobordism ${\Omega }_T$ is treated in [Kri1006.3176] and [KKr1104.1089]. The following summary of our setting is made precise by Theorem 3.1 below.

The Lazard ring $𝕃$ is the coefficient ring for the universal formal group law $F$ so that $𝕃$ is given by generators $a_{ij}$ with relations given by setting

$$F(x,y)=x+y+\sum _{i,j\in {\mathbb{Z}}_{>0}}{a}_{ij}{x}^i{y}^i\text{in}\hspace{0.17em}𝕃\left[[x,y]\right],$$

and requiring

$$F(x,0)=F(0,x)=x,F(x,y)=F(y,x),F(x,F(y,z))=F(F(x,y),z)\text{.}$$

Then

$${\Omega }_T\left(\text{pt}\right)=𝕃\left[[x_{\lambda }\hspace{0.17em}\mid \hspace{0.17em}\lambda \in {𝔥}_{\mathbb{Z}}^{*}]\right]\text{with}{x}_{\lambda +\mu }=x_{\lambda }{+}_F{x}_{\mu }=F(x_{\lambda },x_{\mu }),$$

for $\lambda ,\mu \in {𝔥}_{\mathbb{Z}}^{*}\text{.}$ Then

$${\Omega }_G\left(\text{pt}\right)={\Omega }_T{\left(\text{pt}\right)}^{W_0}=𝕃{\left[[x_{\lambda }\hspace{0.17em}\mid \hspace{0.17em}\lambda \in {𝔥}_{\mathbb{Z}}^{*}]\right]}^{W_0},\text{where}w{x}_{\lambda }=x_{w\lambda },$$

for $w\in W_0$ and $\lambda \in {𝔥}_{\mathbb{Z}}^{*},$ and

$${\Omega }_T(G/B)={\Omega }_T\left(\text{pt}\right){\otimes }_{{\Omega }_G\left(\text{pt}\right)}{\Omega }_T\left(\text{pt}\right)=\frac{𝕃\left[[y_{\lambda },x_{\mu }\hspace{0.17em}\mid \hspace{0.17em}\lambda \in {𝔥}_{\mathbb{Z}}^{*}]\right]}{⟨f\left(x\right)-f\left(y\right)\hspace{0.17em}\mid \hspace{0.17em}f\in 𝕃{\left[[x_{\lambda }\hspace{0.17em}\mid \hspace{0.17em}\lambda \in {𝔥}_{\mathbb{Z}}^{*}]\right]}^{W_0}⟩}\text{.}$$

Sample references for such identities are [KKu0866159] for the case of $H_T(G/B),$ [KKu0895705], [KLu0862716], [CGi1433132] for $K_T(G/B),$ [KPe0750341], [Gro1994], [GKV9505012], [And1637129], [Gan1206.0528] for ${Ell}_T(G/B)$ and [HHH0409305], [CPZ0905.1341], [HKi0903.3926], [KKr1104.1089] for ${\Omega }_T(G/B)\text{.}$

The cobordism case specializes to the cases of cohomology $H_T$ and $K\text{-theory}$ $K_T$ by setting

$$F(x,y)=\{\begin{array}{cc}x+y,& \text{in}\hspace{0.17em}{H}_T,\\ x+y-xy,& \text{in}\hspace{0.17em}{K}_T,\end{array}\text{and}{x}_{\lambda }=\{\begin{array}{cc}{x}_{\lambda },& \text{<mo>∈</mo>}\hspace{0.17em}{H}_T,\\ 1-e^{\lambda },& \text{in}\hspace{0.17em}{K}_T\text{.}\end{array}$$

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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