## Schubert classes $\left[{X}_{w}\right]$

Last update: 17 February 2013

## Schubert classes $\left[{X}_{w}\right]$

Now we consider the inclusions ${\sigma }_{w}:{X}_{w}\to G/B$ of the Schubert varieties into the flag variety. For $w\in {W}_{0},$ define the Schubert classes

$[Xw]= (σw)! (1),where (σw)!: ΩT (Xw)→ ΩT (G/B). (5.1)$

If ${X}_{w}$ is not smooth then, as discussed further below, it is not clear that ${\left({\sigma }_{w}\right)}_{!}$ is well defined. Though we consider various approaches to the analysis of $\left[{X}_{w}\right]={\left({\sigma }_{w}\right)}_{!}\left(1\right)$ below, w have not yet found a definition of ${\left({\sigma }_{w}\right)}_{!}$ which is fully satisfying (at least to us) in the singular case.

In generalized cohomology

$the Schubert class [Xw] is not always equal to [Zw→]$

for a reduced word $\stackrel{\to }{w}$ of $w,$ although, in equivariant cohomology and equivariant $K\text{-theory,}$ $\left[{X}_{w}\right]=\left[{Z}_{\stackrel{\to }{w}}\right]$ if $\stackrel{\to }{w}$ is a reduced word for $w\text{.}$ We consider various approaches to the analysis of $\left[{X}_{w}\right]={\left({\sigma }_{w}\right)}_{!}\left(1\right):$

1. Defining ${\left({\sigma }_{w}\right)}_{!}\left(1\right)$ by (3.3);
2. Comparing $\left[{X}_{w}\right]={\left({\sigma }_{w}\right)}_{!}\left(1\right)$ and the Bott-Samelson class $\left[{Z}_{\stackrel{\to }{w}}\right]$ via the diagram $ΩT (Γw→) (γ∼w→)! ↓ ΩT (Xw) ⟶(σw)! ΩT (G/B) {\left({\gamma }_{w}\right)}_{!} (5.2)$
3. Combinatorial forcing by support conditions, normalization and/or $\left(S,S\right)\text{-bimodule}$ structure of the cohomology.

(a) Is ${\left({\sigma }_{w}\right)}_{!}\left(1\right)$ given by (3.3)? As pointed out in [Tym0604578, Proposition 2.7], since ${X}_{w}$ is filtered by Schubert cells $BvB$ with $v\le w$ and $BvB\cong {ℂ}^{\ell \left(v\right)}$ has even real dimension, the Schubert variety ${X}_{w}$ has no odd-dimensional cohomology, and thus, by [GKM1489894, Theorem 14], the Schubert variety ${X}_{w}$ is ‘equivariantly formal’ (i.e., is a GKM-space) and the moment graph theory applies. The moment graph of ${X}_{w}$ is the subgraph of the moment graph of $G/B$ with vertices $\left\{v\in {W}_{0} \mid v\le w\right\}\text{.}$ If ${X}_{w}$ is smooth then there are no challenges in defining the pushforward ${\left({\sigma }_{w}\right)}_{!}$ and the pushforward formula in (3.3) gives that

$if Xw is smooth, then[Xw]v= yR- ∏ β∈R+ vsβ≤w y-β ,for v∈W0 such that v≤w, (5.3)$

as found, for example, in [BLa1782635, Theorem 7.2.1] (the notation $f={\sum }_{w\in {W}_{0}}{f}_{w}{b}_{w}$ for elements of ${\Omega }_{T}\left(G/B\right)$ is as (3.17)). For example, the inclusion ${\sigma }_{{s}_{2}{s}_{1}}:{X}_{{s}_{2}{s}_{1}}\to G/B$ for $G=G{L}_{3}$ corresponds to the inclusion of moment graphs

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

so that

$yR- y-α1 y-α2 [Xs2s1] = yR- y-α1 y-s1α2 0 yR- y-α1 y-α2 yR- y-α1 y-s1α2 0$

The following example illustrates that this procedure does not work well when ${X}_{w}$ is not smooth. From [Kum1383959, Prop. 6.1], the singular Schubert varieties for $G$ of rank 2 are

$Type Singular Locus B2 Xs1s2s1 Xs1 G2 Xs1s2s1 Xs1 G2 Xs1s2s1s2 Xs1s2 G2 Xs2s1s2s1 Xs2s1 G2 Xs1s2s1s2s1 Xs1s2s1 G2 Xs2s1s2s1s2 Xs2$

The inclusion ${\sigma }_{{s}_{1}{s}_{2}{s}_{1}}:{X}_{{s}_{1}{s}_{2}{s}_{1}}\to G/B$ for $G=S{p}_{4}$ (Type ${B}_{2}\text{)}$ corresponds to the inclusion of moment graphs

$1 {s}_{1} {s}_{2} {s}_{1}{s}_{2} {s}_{2}{s}_{1} {s}_{1}{s}_{2}{s}_{1} {y}_{-{\alpha }_{1}} {y}_{-{\alpha }_{2}} {y}_{-{s}_{1}{\alpha }_{2}} {y}_{-{s}_{1}{\alpha }_{2}} {y}_{-{s}_{2}{\alpha }_{1}} {y}_{-{\alpha }_{2}} {y}_{-{\alpha }_{1}} {y}_{-{s}_{2}{\alpha }_{1}} {y}_{-{\alpha }_{1}} 1 {s}_{1} {s}_{2} {s}_{1}{s}_{2} {s}_{2}{s}_{1} {s}_{1}{s}_{2}{s}_{1} {s}_{2}{s}_{1}{s}_{2} {s}_{1}{s}_{2}{s}_{1}{s}_{2} {y}_{-{\alpha }_{1}} {y}_{-{\alpha }_{2}} {y}_{-{s}_{1}{\alpha }_{2}} {y}_{-{s}_{2}{\alpha }_{1}} {y}_{-{s}_{1}{\alpha }_{2}} {y}_{-{s}_{2}{\alpha }_{1}} {y}_{-{\alpha }_{2}} {y}_{-{\alpha }_{1}} {y}_{-{s}_{1}{\alpha }_{2}} {y}_{-{s}_{2}{\alpha }_{1}} {y}_{-{\alpha }_{1}} {y}_{-{\alpha }_{2}} {y}_{-{s}_{1}{\alpha }_{2}} {y}_{-{s}_{2}{\alpha }_{1}} {y}_{-{\alpha }_{1}} {y}_{-{\alpha }_{2}}$

but the direct "naive" application of the pushforward formula in (3.3) produces

$[Xs1s2s1]? =? y-s2α1 y-s2α1 y-s1α2 y-s1α2 y-α2 y-α2 0 0 = y-(α1+α2) y-(α1+α2) y-(2α1+α2) y-(2α1+α2) y-α2 y-α2 0 0 (5.4)$

which cannot be correct for $\left[{X}_{{s}_{1}{s}_{2}{s}_{1}}\right]$ since the right hand side does not satisfy the condition to be in im $\Phi$ (the difference across the edge $1\to {s}_{2}$ is not divisible by ${y}_{-{\alpha }_{2}}\text{).}$ This answer needs to be corrected by finding $N$ so that

$[Xs1s2s1] = Ny-(α1+α2) Ny-(α1+α2) y-(2α1+α2) y-(2α1+α2) y-α2 y-α2 0 0$

where the correction factor $N$ appears on vertices corresponding to the singular locus.

In the example in (5.4) we see that the moment graph knows that ${X}_{{s}_{1}{s}_{2}{s}_{1}}$ is not smooth! It is interesting to contrast (5.4) with the same analysis for ${\sigma }_{{s}_{2}{s}_{1}{s}_{2}}:{X}_{{s}_{2}{s}_{1}{s}_{2}}\to G/B,$ where the pushforward formula gives

$[Xs2s1s2] = y-s1α2 y-s2α1 y-s1α2 y-α1 y-s2α1 0 y-α1 0 = y-(2α1+α2) y-(α1+α2) y-(2α1+α2) y-α1 y-(α1+α2) 0 y-α1 0$

which is in im $\Phi$ (this case works out well since ${X}_{{s}_{2}{s}_{1}{s}_{2}}$ is smooth).

(b) Using (5.2) to compare $\left[{X}_{w}\right]$ and $\left[{Z}_{\stackrel{\to }{w}}\right]\text{.}$ Working in rank 2, use notations ${y}_{R}^{-},{\Delta }_{121}$ and ${\Delta }_{212}$ as in (7.2), so that (see (4.8) and Figure 1)

$[Z212] = Δ212 yR- y-α1 y-α2 y-s1α2 Δ212 yR- y-α2 y-s2α1 y-s2s1α2 yR- y-α1 y-α2 y-s1α2 0 yR- y-α2 y-s2α1 y-s2s1α2 0$

Since ${X}_{{s}_{1}{s}_{2}{s}_{1}}$ is smooth it is reasonable to apply the pushforward formula in (3.3) which gives

$[Xs2s1s2] = yR- y-α1 y-α2 y-s2α1 yR- y-α1 y-α2 y-s1α2 yR- y-α1 y-α2 y-s2α1 yR- y-α2 y-s2α1 y-s2s1α2 yR- y-α1 y-α2 y-s1α2 0 yR- y-α2 y-s2α1 y-s2s1α2 0$

Using these and computing with the formulas (3.10)-(3.12) gives the formula

$[Z212] = [Xs2s1s2]+ ( Δ212- yR- y-α1 y-α2 y-s2α1 ) y-α2 yR- [Xs2] = [Xs2s1s2]+ yR- y-α1 y-α2 y-s2α1 ( y-s2α1 -y-α1 y-α2 +p ( yα2, y-α2 ) y-α1-1 ) y-α2 yR- [Xs2] = [Xs2s1s2]+ yR- y-α1 y-α2 y-s2α1 ( 1-p ( y-α1, y-α2 ) y-α1+p ( yα2, y-α2 ) y-α1-1 ) y-α2 yR- [Xs2] = [Xs2s1s2]+ 1y-s2α1 ( p ( yα2, y-α2 ) -p ( y-α1, y-α2 ) ) y-α1 y-α2 yR- [Xs2]$

which is reflected in [CPZ0905.1341, 17.3, first equation] and [HKi0903.3926, §5.2]. Similarly, with our conjectured correction factor $N$ as in (7.3), we get a formula which would provide $\left[{Z}_{121}\right]-\left[{X}_{{s}_{1}{s}_{2}{s}_{1}}\right]=0$ in cohomology and K-theory but have $\left[{Z}_{121}\right]-\left[{X}_{{s}_{1}{s}_{2}{s}_{1}}\right]\ne 0$ in complex or algebraic cobordism:

$[Z121]- [Xs1s2s1] = ( Δ121- NyR- y-α1 y-α2 y-s1α2 ) y-α1 yR- [Xs1] = yR- y-α1 y-α2 y-s1α2 ( y-s1α2- y-α2 y-α1 +p ( yα1, y-α1 ) y-α2-N ) y-α1 yR- [Xs1] = yR- y-α1 y-α2 y-s1α2 ( ( 1-p ( y-α2, y-jα1 ) y-α2 ) ( 1+ ∑k=1j-1 ( 1-p ( y-α1, y-kα1 ) y-kα1 ) ) +p ( yα1, y-α1 ) y-α2-N ) y-α1 yR- [Xs1] = yR- y-α1 y-α2 y-s1α2 ( p ( yα1, y-α1 ) -p ( y-α2, y-jα1 ) ) y-α2 y-α1 yR- [Xs1] = 1y-s1α2 ( p ( yα1, y-α1 ) -p ( y-α2, y-jα1 ) ) [Xs1].$

(c) Combinatorial forcing: The Schubert classes satisfy

1. (normalization) ${\left[{X}_{w}\right]}_{w}={\prod }_{\alpha \in R\left(w\right)}{y}_{-\alpha },$ where $R\left(w\right)=\left\{\alpha \in {R}^{+} \mid w\alpha \notin {R}^{+}\right\}\text{.}$
2. If ${W}_{J}u={W}_{J}z$ then ${\left[{X}_{{w}_{J}u}\right]}_{v}={\left[{X}_{{w}_{J}u}\right]}_{z},$
3. (support) ${\left[{X}_{w}\right]}_{v}=0$ unless $v\le w\text{.}$

These properties do not characterize the Schubert classes; the Bott-Samelson classes also satisfy these properties. As observed, for example, in [HHH0409305, Proposition 4.3], in equivariant cohomology a degree condition can be imposed to get uniqueness. It is not clear to us how to generalize the degree \ condition to equivariant K-theory and/or equivariant cobordism. It seems plausible that in generalized equivariant cohomology the Schubert classes might be characterized by positivity properties, or by using the $\left(S,S\right)\text{-bimodule}$ structure of ${\Omega }_{T}\left({X}_{w}\right)$ and ${\Omega }_{T}\left({Z}_{\stackrel{\to }{w}}\right)$ as in the theory of Soergel bimodules (see [Soe2329762] and [EWi2012]).

## 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.