The Schubert calculus framework

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 17 February 2013

The Schubert calculus framework

Flag an Schubert varieties

The basic data is

G a connected complex reductive algebraic group B a Borel subgroup T a maximal torus. (2.1)

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

W0=N(T)/T, 𝔥*= Hom(T,×) and𝔥=Hom (×,T), (2.2)

where Hom(H,K) is the abelian group of algebraic group homomorphisms from H to K with product given by pointwise multiplication, (ϕφ)(h)= ϕ(h)φ(h). Since the Weyl group acts on T, it also acts on 𝔥* and on 𝔥.

A standard parabolic subgroup of G is a subgroup PJB such that G/PJ is a projective variety. A parabolic subgroup of G is a conjugate of a standard parabolic subgroup.

Theflag varietyis G/Band G/PJare the partial flag varieties. (2.3)

These are studied via the Bruhat decomposition

G=wW0Bw BandG= uWJBu PJ (2.4)

where WJ= { vW0 vTPJ } and

WJ= { coset representativesu of cosets inW0/WJ } . (2.5)

The Schubert varieties are

Xw= BwBin G/Band XuJ= BuPJin G/PJ, (2.6)

and the Bruhat orders are the partial orders on W0 and WJ given by

Xw=BwB= vwBvB andXuJ= BuPJ= zuBzPJ. (2.7)

The T-fixed points

inG/Bare { wBwW0 } andinG/PJ are { uPJu WJ } . (2.8)

Let P1,,Pn be the minimal parabolic subgroups PiB. Then

Wi=W{i}= {1,si}and s1,,sn are thesimple reflections inW0. (2.9)

With respect to the action W0 on 𝔥*= 𝔥*, the si are reflections in the hyperplanes (𝔥*)si= { μ𝔥* siμ=μ } . An alternative description of the standard parabolic subgroups is to let J{1,2,,n} and let

WJ= sj jJ .ThenPJ= vWJ BvB. (2.10)

In particular, Pi=P{i}=B BsiB, for i=1,2,,n.

Theorem 2.1. (Coxeter) The group W0 is generated by s1,,sn with relations

si2=1and sisjsi mijfactors = sjsisj mijfactors

where π/mij= (𝔥*)si (𝔥*)sj is the angle between (𝔥*)si and (𝔥*)sj.

The definitions in (2.3), (2.8) and (2.6) provide T-equivariant maps

pJ: G/B G/PJ gB gPJ ιw: pt G/B pt wB σw: Xw G/B gB gB (2.11)


ιuJ: pt G/PJ pt uPJ σuJ: XuJ G/PJ gPJ gPJ (2.12)

for J{1,2,,}, wW0, and uWJ.

For example, in type G=GL3, with T and B the subgroups given by

T= { ( * 0 0 0 * 0 0 0 * ) } andB= { ( * * * 0 * * 0 0 * ) } ,

then W0= s1,s2 si2=1,s1 s2s1=s2s1 s2 , where

s1= ( 0 1 0 1 0 0 0 0 1 ) ands2= ( 0 1 0 1 0 0 0 0 1 ) .


Xw0=G/B Xs1s2 Xs2s1 Xs1 Xs2 pt=X1 and G/B G/P1G/P2 G/G=pt

where P1 and P2 are the subgroups of G=GL3() given by

P1= { ( * * * * * * 0 0 * ) } =BBs1Band P2= { ( * * * 0 * * 0 * * ) } =BBs2B.

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[[yλ]] of (3.4) the motivation comes from the formalism of generalized cohomology theories h. Model examples are: ordinary cohomology H, K-theory K, elliptic cohomology (see [MRa2330502, GKV9505012, Gro1994, And1637129, Lur2597740]) and complex and algebraic cobordism Ω (see [LMo2286826]). Key to our point of view is that if f:XY is a morphism of spaces, the contravariance of the cohomology theory provides

a pullbackf*:h(Y) h(X),anda pushforward f!:h(X)h(Y)

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(pt) such that

F ( c1h(1), c1h(2) ) =c1h (12),

where 1 and 2 are line bundles on X and c1h denotes the first Chern class in the cohomology theory h (see [LMo2286286, Cor. 4.1.8]). The Lazard ring 𝕃 is generated by symbols aij, for i,j>0, which satisfy the relations given by the equations

F(x,F(y,z))= F(F(x,y),z), F(x,y)=F (y,x),F (x,0)=x, (2.13)


F(x,y)=x+y+ a11xy+a12x y2+a21x2y +

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

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

  1. Normalization: specification of hT(pt),
  2. nice behaviour under products, smashes, suspensions: such as axioms for computing hG×K(M×N),
  3. functoriality/pullbacks: if f:XY then we have f*:hT(Y)hT(X)
  4. Thom isomorphism/orientability/pushforwards: For certain classes of maps f:XY there exists a pushforward f!:hT(X)hT(Y),
  5. Change of groups: For certain classes of groups G and K and group homomorphisms φ:GK there exist χφ:hGhK and χφ:hKhG.

The art of choosing appropriate categories of input "T-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-spaces

X=i>0 Xi,X1X2 X3,

where the successive quotients Xi/Xi-1 are homeomorphic to the Thom spaces Th(Vi) of some h-orientable T-vector bundles ViFi (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 Fi are points and the Vi are one dimensional representations of T. In particular, the assumptions of [HHH0409305, §3] hold for these cases.

The Borel model for hT(G/B)

The general combinatorial Schubert calculus uses 𝔥* and 𝔥 to build a C-algebra R with an action of W0 on R by C-algebra automorphisms (in favorite examples C may be , or the ring Th0 of holomorphic functions on the upper half plane, or the Lazard ring 𝕃, see the examples below). If

RW0= { fRwf=f forwW0 } is theinvariant ring,

then, conceptually,

R=hT(pt) andRW0=hG (pt), (2.14)

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

RRW0R= RCR f1-1f fRW0 , (2.15)

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

RRW0R is a good combinatorial model forhT (G/B), (2.16)

where the product on RRW0R is given by (f1g1) (f2g2)= f1f2g1g2.

There are four favorite examples:

Cohomology: hT=HT. Here

HT(pt)=S (𝔥*)= [x1,,xn] and HG(pt)=HT (pt)W0= [x1,,xn] W0 ,

where xi=xωi, where ω1,,ωn is a -basis of 𝔥*. Alternatively, HT(pt) is the ring

[ xλλ 𝔥* ] withxλ+μ= xλ+xμ,

for λ,μ𝔥* and with wxλ=xwλ for wW0 and λ𝔥*. Then

HT(G/B)=HT (pt)HG(pt) HT(pt)= [ y1,,yn, x1,,xn ] f(x1,,xn)- f(y1,,yn) f [x1,,xn] W0 .

K-theory: hT=KT. Here

KT(pt)= [𝔥*]= [ X1±1,, Xn±1 ] and KG(pt)=KT (pt)W0= [ X1±1,, Xn±1 ] W0 ,

where Xi=eωi, where ω1,,ωn is a -basis of 𝔥*. Alternatively, KT(pt) is the ring

[eλλ𝔥*] witheλ+μ= eλeμ,

for λ,μ𝔥* and with weλ=ewλ for wW0 and λ𝔥*. Then

KT(G/B)=KT (pt)KG(pt) KT(pt)= [ Y1±1,, Yn±1, X1±1,, Xn±1 ] f(X1,,Xn)- f(Y1,,Yn) f [ X1±1,, Xn±1 ] W0 .

Elliptic cohomology: hT=EllT. Here EllT(pt) is the structure sheaf of the abelian variety Aτ=𝔥*/ ( 𝔥*+ τ𝔥* ) . The homogeneous coordinate ring

forAτis Th= m0 Thm,and forAτ/W0 is ThW0.

Then the graded Th-module corresponding to

the sheafEllT(G/B) onAτis Th ThW0 Th.

Complex or algebraic cobordism: hT=ΩT. Algebraic cobordism is treated in the book of Levine-Morel [LMo2286826] and T-equivariant algebraic cobordism ΩT 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 aij with relations given by setting

F(x,y)=x+y+ i,j>0 aijxiyi in𝕃[[x,y]],

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).


ΩT(pt)=𝕃 [ [ xλ λ𝔥* ] ] withxλ+μ= xλ+Fxμ=F (xλ,xμ),

for λ,μ𝔥*. Then

ΩG(pt)= ΩT (pt)W0=𝕃 [ [ xλ λ𝔥* ] ] W0 ,wherewxλ= xwλ,

for wW0 and λ𝔥*, and

ΩT(G/B)= ΩT(pt) ΩG(pt) ΩT(pt)= 𝕃 [ [ yλ,xμ λ𝔥* ] ] f(x)-f(y) f𝕃 [ [ xλ λ𝔥* ] ] W0 .

Sample references for such identities are [KKu0866159] for the case of HT(G/B), [KKu0895705], [KLu0862716], [CGi1433132] for KT(G/B), [KPe0750341], [Gro1994], [GKV9505012], [And1637129], [Gan1206.0528] for EllT(G/B) and [HHH0409305], [CPZ0905.1341], [HKi0903.3926], [KKr1104.1089] for ΩT(G/B).

The cobordism case specializes to the cases of cohomology HT and K-theory KT by setting

F(x,y)= { x+y, inHT, x+y-xy, inKT, andxλ= { xλ, HT, 1-eλ, inKT.

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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