Schubert classes $\left[X_w\right]$

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

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

$$\begin{array}{cc}\left[X_w\right]={\left({\sigma }_w\right)}_{!}\left(1\right),\text{where}{\left({\sigma }_w\right)}_{!}:{\Omega }_T\left(X_w\right)\to {\Omega }_T(G/B)\text{.}& \text{(5.1)}\end{array}$$

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

$$\text{the Schubert class}\hspace{0.17em}\left[X_w\right]\hspace{0.17em}\text{is}\hspace{0.17em}\text{not always}\hspace{0.17em}\text{equal to}\hspace{0.17em}\left[Z_{\overrightarrow{w}}\right]$$

for a reduced word $\overrightarrow{w}$ of $w,$ although, in equivariant cohomology and equivariant $K\text{-theory,}$ $\left[X_w\right]=\left[Z_{\overrightarrow{w}}\right]$ if $\overrightarrow{w}$ 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 $\left[Z_{\overrightarrow{w}}\right]$ via the diagram $$\begin{array}{cc} ΩT (Γw→) (γ∼w→)! ↓ ΩT (Xw) ⟶(σw)! ΩT (G/B) (γw)! & \text{(5.2)}\end{array}$$
3. Combinatorial forcing by support conditions, normalization and/or $(S,S)\text{-bimodule}$ 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 $\{v\in W_0\hspace{0.17em}\mid \hspace{0.17em}v\le w\}\text{.}$ 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

$$\begin{array}{cc}\text{if}\hspace{0.17em}{X}_w\hspace{0.17em}\text{is smooth,}\text{then}{\left[X_w\right]}_v=\frac{y_{R^{-}}}{\prod _{\genfrac{}{}{0ex}{}{\beta \in R^{+}}{v{s}_{\beta }\le w}}{y}_{-\beta }},\text{for}\hspace{0.17em}v\in W_0\hspace{0.17em}\text{such that}\hspace{0.17em}v\le w,& \text{(5.3)}\end{array}$$

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 ${\Omega }_T(G/B)$ 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

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

so that

$$\begin{array}{cccc}& & & \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}\\ \left[X_{s_2{s}_1}\right]& =& \begin{array}{c}\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}}\\ 0\end{array}& & \begin{array}{c}\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}\\ \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}}\end{array}\\ & & & 0\end{array}$$

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

$$\begin{array}{ccc}\text{Type}& \text{Singular}& \text{Locus}\\ \\ B_2& X_{s_1{s}_2{s}_1}& X_{s_1}\\ \\ G_2& X_{s_1{s}_2{s}_1}& X_{s_1}\\ G_2& X_{s_1{s}_2{s}_1{s}_2}& X_{s_1{s}_2}\\ G_2& X_{s_2{s}_1{s}_2{s}_1}& X_{s_2{s}_1}\\ G_2& X_{s_1{s}_2{s}_1{s}_2{s}_1}& X_{s_1{s}_2{s}_1}\\ G_2& X_{s_2{s}_1{s}_2{s}_1{s}_2}& X_{s_2}\end{array}$$

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

$$\begin{array}{cc} 1 s1 s2 s1s2 s2s1 s1s2s1 y-α1 y-α2 y-s1α2 y-s1α2 y-s2α1 y-α2 y-α1 y-s2α1 y-α1 & 1 s1 s2 s1s2 s2s1 s1s2s1 s2s1s2 s1s2s1s2 y-α1 y-α2 y-s1α2 y-s2α1 y-s1α2 y-s2α1 y-α2 y-α1 y-s1α2 y-s2α1 y-α1 y-α2 y-s1α2 y-s2α1 y-α1 y-α2 \end{array}$$

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

$$\begin{array}{cc}\begin{array}{ccccc}\left[X_{s_1{s}_2{s}_1}\right]?& =?& \begin{array}{cc}& y_{-s_2{\alpha }_1}\\ y_{-s_2{\alpha }_1}& & y_{-s_1{\alpha }_2}\\ y_{-s_1{\alpha }_2}& & y_{-{\alpha }_2}\\ y_{-{\alpha }_2}& & 0\\ & 0\end{array}& =& \begin{array}{cc}& y_{-({\alpha }_1+{\alpha }_2)}\\ y_{-({\alpha }_1+{\alpha }_2)}& & y_{-(2{\alpha }_1+{\alpha }_2)}\\ y_{-(2{\alpha }_1+{\alpha }_2)}& & y_{-{\alpha }_2}\\ y_{-{\alpha }_2}& & 0\\ & 0\end{array}\end{array}& \text{(5.4)}\end{array}$$

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

$$\begin{array}{ccc}\left[X_{s_1{s}_2{s}_1}\right]& =& \begin{array}{cc}& N{y}_{-({\alpha }_1+{\alpha }_2)}\\ N{y}_{-({\alpha }_1+{\alpha }_2)}& & y_{-(2{\alpha }_1+{\alpha }_2)}\\ y_{-(2{\alpha }_1+{\alpha }_2)}& & y_{-{\alpha }_2}\\ y_{-{\alpha }_2}& & 0\\ & 0\end{array}\end{array}$$

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

$$\begin{array}{ccccc}\left[X_{s_2{s}_1{s}_2}\right]& =& \begin{array}{cc}& y_{-s_1{\alpha }_2}\\ y_{-s_2{\alpha }_1}& & y_{-s_1{\alpha }_2}\\ y_{-{\alpha }_1}& & y_{-s_2{\alpha }_1}\\ 0& & y_{-{\alpha }_1}\\ & 0\end{array}& =& \begin{array}{cc}& y_{-(2{\alpha }_1+{\alpha }_2)}\\ y_{-({\alpha }_1+{\alpha }_2)}& & y_{-(2{\alpha }_1+{\alpha }_2)}\\ y_{-{\alpha }_1}& & y_{-({\alpha }_1+{\alpha }_2)}\\ 0& & y_{-{\alpha }_1}\\ & 0\end{array}\end{array}$$

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 $\left[Z_{\overrightarrow{w}}\right]\text{.}$ 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)

$$\begin{array}{ccc}\left[Z_{212}\right]& =& \begin{array}{cc}& {\Delta }_{212}\\ \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}& & {\Delta }_{212}\\ \frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}}& & \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}\\ 0& & \frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}}\\ & 0\end{array}\end{array}$$

Since $X_{s_1{s}_2{s}_1}$ is smooth it is reasonable to apply the pushforward formula in (3.3) which gives

$$\begin{array}{ccc}\left[X_{s_2{s}_1{s}_2}\right]& =& \begin{array}{cc}& \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}\\ \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}& & \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}\\ \frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}}& & \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}\\ 0& & \frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}}\\ & 0\end{array}\end{array}$$

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

$$\begin{array}{ccc}\left[Z_{212}\right]& =& \left[X_{s_2{s}_1{s}_2}\right]+({\Delta }_{212}-\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}})\frac{y_{-{\alpha }_2}}{y_{R^{-}}}\left[X_{s_2}\right]\\ & =& \left[X_{s_2{s}_1{s}_2}\right]+\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}(\frac{y_{-s_2{\alpha }_1}-y_{-{\alpha }_1}}{y_{-{\alpha }_2}}+p(y_{{\alpha }_2},y_{-{\alpha }_2})y_{-{\alpha }_1}-1)\frac{y_{-{\alpha }_2}}{y_{R^{-}}}\left[X_{s_2}\right]\\ & =& \left[X_{s_2{s}_1{s}_2}\right]+\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}(1-p(y_{-{\alpha }_1},y_{-{\alpha }_2})y_{-{\alpha }_1}+p(y_{{\alpha }_2},y_{-{\alpha }_2})y_{-{\alpha }_1}-1)\frac{y_{-{\alpha }_2}}{y_{R^{-}}}\left[X_{s_2}\right]\\ & =& \left[X_{s_2{s}_1{s}_2}\right]+\frac{1}{y_{-s_2{\alpha }_1}}(p(y_{{\alpha }_2},y_{-{\alpha }_2})-p(y_{-{\alpha }_1},y_{-{\alpha }_2}))y_{-{\alpha }_1}\frac{y_{-{\alpha }_2}}{y_{R^{-}}}\left[X_{s_2}\right]\end{array}$$

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:

$$\begin{array}{ccc}\left[Z_{121}\right]-\left[X_{s_1{s}_2{s}_1}\right]& =& ({\Delta }_{121}-\frac{N{y}_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}})\frac{y_{-{\alpha }_1}}{y_{R^{-}}}\left[X_{s_1}\right]\\ & =& \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}(\frac{y_{-s_1{\alpha }_2}-y_{-{\alpha }_2}}{y_{-{\alpha }_1}}+p(y_{{\alpha }_1},y_{-{\alpha }_1})y_{-{\alpha }_2}-N)\frac{y_{-{\alpha }_1}}{y_{R^{-}}}\left[X_{s_1}\right]\\ & =& \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}\left(\begin{array}{c}(1-p(y_{-{\alpha }_2},y_{-j{\alpha }_1})y_{-{\alpha }_2})(1+{\sum }_{k=1}^{j-1}(1-p(y_{-{\alpha }_1},y_{-k{\alpha }_1})y_{-k{\alpha }_1}))\\ +p(y_{{\alpha }_1},y_{-{\alpha }_1})y_{-{\alpha }_2}-N\end{array}\right)\frac{y_{-{\alpha }_1}}{y_{R^{-}}}\left[X_{s_1}\right]\\ & =& \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}(p(y_{{\alpha }_1},y_{-{\alpha }_1})-p(y_{-{\alpha }_2},y_{-j{\alpha }_1}))y_{-{\alpha }_2}\frac{y_{-{\alpha }_1}}{y_{R^{-}}}\left[X_{s_1}\right]\\ & =& \frac{1}{y_{-s_1{\alpha }_2}}(p(y_{{\alpha }_1},y_{-{\alpha }_1})-p(y_{-{\alpha }_2},y_{-j{\alpha }_1}))\left[X_{s_1}\right]\text{.}\end{array}$$

(c) Combinatorial forcing: The Schubert classes satisfy

1. (normalization) ${\left[X_w\right]}_w={\prod }_{\alpha \in R\left(w\right)}{y}_{-\alpha },$ where $R\left(w\right)=\{\alpha \in R^{+}\hspace{0.17em}\mid \hspace{0.17em}w\alpha \notin R^{+}\}\text{.}$
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 $(S,S)\text{-bimodule}$ structure of ${\Omega }_T\left(X_w\right)$ and ${\Omega }_T\left(Z_{\overrightarrow{w}}\right)$ 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.

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