Products with Schubert classes

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

Last update: 17 February 2013

Products with Schubert classes

For $w\in W_0$ define Schubert classes $\left[X_w\right]$ by $\left[X_w\right]={\left({\sigma }_w\right)}_{!}\left(1\right)$ as in (5.1). Continue to use notations $f={\sum }_{w\in W_0}{f}_w{b}_w$ for elements of ${\Omega }_T(G/B),$ as in (3.17).

The Schubert product problem: Find a combinatorial description of the $c_{uv}^w\in R$ given by

$$\begin{array}{cc}\left[X_u\right]\left[X_v\right]=\sum _{w\in W_0}{c}_{uv}^w\left[X_w\right]\text{.}& \text{(6.1)}\end{array}$$

As is visible from the formula (6.3) below and the formulas at the end of this section, if $v\le u$ in Bruhat order then

$$\begin{array}{cc}\left[X_u\right]\left[X_v\right]={\left[X_u\right]}_v\left[X_v\right]+\sum _{w<v}{c}_{uv}^w\left[X_w\right],& \text{(6.2)}\end{array}$$

and so the determination of the moment graph values ${\left[X_u\right]}_v$ is a subproblem of the Schubert product problem. The other coefficients $c_{uv}^w$ are determined by the ${\left[X_u\right]}_v$ in an intricate but, perhaps, controllable fashion. Furthermore, our computations of products in the rank two cases display a certain amount of positivity, indicating that there may be a positivity statement for equivariant cobordism analogous to that which holds for equivariant cohomology and equivariant K-theory (see [Gra9908172] and [AGM0808.2785]).

Properties (a) and (c) are already enough to provide an algorithm for expanding an element $f={\sum }_{w\in W_0}{f}_w{b}_w$ in terms of Schubert classes. If $f$ has support on $w$ with $\ell \left(w\right)\le k$ then

$$f=\sum _{\ell \left(w\right)=k}{f}_w\frac{1}{{\left[X_w\right]}_w}\left[X_w\right]=\sum _{\ell \left(v\right)\le k-1}(f_v-\sum _{\ell \left(w\right)=k}{f}_w\frac{{\left[X_w\right]}_v}{{\left[X_w\right]}_w})b_v$$

has support on $v$ with $\ell \left(v\right)\le k-1\text{.}$ Then

$$\begin{array}{c}f-\sum _{\ell \left(w\right)=k}{f}_w\frac{1}{{\left[X_w\right]}_w}\left[X_w\right]-\sum _{\ell \left(v\right)=k-1}(f_v-\sum _{\genfrac{}{}{0ex}{}{\ell \left(v\right)=k-1}{\ell \left(w\right)=k}}{f}_w\frac{{\left[X_w\right]}_v}{{\left[X_w\right]}_w})\frac{1}{{\left[Xv\right]}_v}\left[X_v\right]\\ =\sum _{\ell \left(z\right)\le k-2}(f_z-\sum _{\ell \left(w\right)=k}{f}_w\frac{{\left[X_w\right]}_z}{{\left[X_w\right]}_w}-\sum _{\ell \left(v\right)=k-1}{f}_v\frac{{\left[X_v\right]}_z}{{\left[X_v\right]}_v}+\sum _{\genfrac{}{}{0ex}{}{\ell \left(v\right)=k-1}{\ell \left(w\right)=k}}{f}_w\frac{{\left[X_w\right]}_v}{{\left[X_w\right]}_w}\frac{{\left[X_v\right]}_z}{{\left[X_v\right]}_v})b_z\end{array}$$

and induction gives that

$$\begin{array}{cc}f=\sum _{z\in W_0}\left(\sum _{k=1}^{\ell \left(w_0\right)}\sum _{w_1>\dots >w_k=z}{(-1)}^{k-1}{f}_{w_1}\frac{{\left[X{w}_1\right]}_{w_2}}{{\left[X{w}_1\right]}_{w_1}}\frac{{\left[X{w}_2\right]}_{w_3}}{{\left[X{w}_2\right]}_{w_2}}\dots \frac{{\left[X{w}_{k-1}\right]}_{w_k}}{{\left[X{w}_{k-1}\right]}_{w_{k-1}}}\frac{1}{{\left[X_{w_k}\right]}_{w_k}}\right)\left[X_z\right]& \text{(6.3)}\end{array}$$

with the terms in the sum naturally indexed by chains in the Bruhat order (compare to, for example, [BSo9703001]).

For example, in rank 2 using notations as in Section 7, if $f={\sum }_{w\le s_1{s}_2{s}_1{s}_2}{f}_2{b}_w$ then

$$\begin{array}{ccc}f& =& f_{s_1{s}_2{s}_1{s}_2}\frac{1}{{\left[X_{s_1{s}_2{s}_1{s}_2}\right]}_{s_1{s}_2{s}_1{s}_2}}\left[X_{s_1{s}_2{s}_1{s}_2}\right]\\ & & +(f_{s_1{s}_2{s}_1}-f_{s_1{s}_2{s}_1{s}_2})\frac{1}{{\left[X_{s_1{s}_2{s}_1}\right]}_{s_1{s}_2{s}_1}}\left[X_{s_1{s}_2{s}_1}\right]+(f_{s_2{s}_1{s}_2}-f_{s_1{s}_2{s}_1{s}_2})\frac{1}{{\left[X_{s_2{s}_1{s}_2}\right]}_{s_2{s}_1{s}_2}}\left[X_{s_2{s}_1{s}_2}\right]\\ & & +((f_{s_1{s}_2}-f_{s_2{s}_1{s}_2})+(f_{s_1{s}_2{s}_1{s}_2}-f_{s_1{s}_2{s}_1})\frac{{\left[X_{s_1{s}_2{s}_1}\right]}_{s_1{s}_2}}{{\left[X_{s_1{s}_2{s}_1}\right]}_{s_1{s}_2{s}_1}})\frac{1}{{\left[X_{s_1{s}_2}\right]}_{s_1{s}_2}}\left[X_{s_1{s}_2}\right]\\ & & +((f_{s_2{s}_1}-f_{s_1{s}_2{s}_1})+(f_{s_1{s}_2{s}_1{s}_2}-f_{s_2{s}_1{s}_{12}})\frac{{\left[X_{s_2{s}_1{s}_2}\right]}_{s_2{s}_1}}{{\left[X_{s_2{s}_1{s}_2}\right]}_{s_2{s}_1{s}_2}})\frac{1}{{\left[X_{s_2{s}_1}\right]}_{s_2{s}_1}}\left[X_{s_2{s}_1}\right]\\ & & +\left(\begin{array}{c}(f_{s_1}-f_{s_2{s}_1})+(f_{s_2{s}_1{s}_2}-f_{s_1{s}_2})\frac{{\left[X_{s_1{s}_2}\right]}_{s_1}}{{\left[X_{s_1{s}_2}\right]}_{s_1{s}_2}}\\ +(f_{s_1{s}_2{s}_1{s}_2}-f_{s_1{s}_2{s}_1})(\frac{{\left[X_{s_1{s}_2{s}_1}\right]}_{s_1}}{{\left[X_{s_1{s}_2{s}_1}\right]}_{s_1{s}_2{s}_1}}-\frac{{\left[X_{s_1{s}_2{s}_1}\right]}_{s_1{s}_2}}{{\left[X_{s_1{s}_2{s}_1}\right]}_{s_1{s}_2{s}_1}}\frac{{\left[X_{s_1{s}_2}\right]}_{s_1}}{{\left[X_{s_1{s}_2}\right]}_{s_1{s}_2}}-1)\end{array}\right)\frac{1}{{\left[X_{s_1}\right]}_{s_1}}\left[X_{s_1}\right]\\ & & +\left(\begin{array}{c}(f_{s_2}-f_{s_1{s}_2})+(f_{s_1{s}_2{s}_1}-f_{s_2{s}_1})\frac{{\left[X_{s_2{s}_1}\right]}_{s_2}}{{\left[X_{s_2{s}_1}\right]}_{s_2{s}_1}}\\ +(f_{s_2{s}_1{s}_2{s}_1}-f_{s_2{s}_1{s}_2})(\frac{{\left[X_{s_2{s}_1{s}_2}\right]}_{s_2}}{{\left[X_{s_2{s}_1{s}_2}\right]}_{s_2{s}_1{s}_2}}-\frac{{\left[X_{s_2{s}_1{s}_2}\right]}_{s_2{s}_1}}{{\left[X_{s_2{s}_1{s}_2}\right]}_{s_2{s}_1{s}_2}}\frac{{\left[X_{s_2{s}_1}\right]}_{s_2}}{{\left[X_{s_2{s}_1}\right]}_{s_2{s}_1}}-1)\end{array}\right)\frac{1}{{\left[X_{s_2}\right]}_{s_2}}\left[X_{s_2}\right]\\ & & +(f_1-f_{s_1}-f_{s_2}+f_{s_1{s}_2}+f_{s_2{s}_1}-f_{s_1{s}_2{s}_1}-f_{s_2{s}_1{s}_2}+f_{s_1{s}_2{s}_1{s}_2})\frac{1}{{\left[X_1\right]}_1}\left[X_1\right]\end{array}$$

and we may use the explicit values of ${\left[X_w\right]}_v$ given in Figure 2 to derive

$$\begin{array}{ccc}f& =& f_{s_1{s}_2{s}_1{s}_2}\frac{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}{y}_{-s_1{s}_2{\alpha }_1}{y}_{-s_1{s}_1{s}_1{\alpha }_2}}{y_{R^{-}}}\left[X_{s_1{s}_2{s}_1{s}_2}\right]\\ & & +(f_{s_1{s}_2{s}_1}-f_{s_1{s}_2{s}_1{s}_2})\frac{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}{y}_{-s_1{s}_2{\alpha }_1}}{y_{R^{-}}}\left[X_{s_1{s}_2{s}_1}\right]+(f_{s_2{s}_1{s}_2}-f_{s_1{s}_2{s}_1{s}_2})\frac{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}}{y_{R^{-}}}\left[X_{s_2{s}_1{s}_2}\right]\\ & & +((f_{s_1{s}_2}-f_{s_2{s}_1{s}_2})+(f_{s_1{s}_2{s}_1{s}_2}-f_{s_1{s}_2{s}_1})\frac{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}{y}_{-s_1{s}_2{\alpha }_1}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}})\frac{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}{y_{R^{-}}}\left[X_{s_1{s}_2}\right]\\ & & +((f_{s_2{s}_1}-f_{s_1{s}_2{s}_1})+(f_{s_1{s}_2{s}_1{s}_2}-f_{s_2{s}_1{s}_2})\frac{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}})\frac{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}}{y_{R^{-}}}\left[X_{s_2{s}_1}\right]\\ & & +\left(\begin{array}{c}(f_{s_1}-f_{s_2{s}_1})+(f_{s_2{s}_1{s}_2}-f_{s_1{s}_2})\frac{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}\\ +(f_{s_1{s}_2{s}_1{s}_2}-f_{s_1{s}_2{s}_1})(\frac{N{y}_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}{y}_{-s_1{s}_2{\alpha }_1}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}-\frac{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}{y}_{-s_1{s}_2{\alpha }_1}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}\frac{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}-1)\end{array}\right)\frac{y_{-{\alpha }_1}}{y_{R^{-}}}\left[X_{s_1}\right]\\ & & +\left(\begin{array}{c}(f_{s_2}-f_{s_1{s}_2})+(f_{s_1{s}_2{s}_1}-f_{s_2{s}_1})\frac{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}\\ +(f_{s_2{s}_1{s}_2{s}_1}-f_{s_2{s}_1{s}_2})(\frac{N{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}-\frac{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}\frac{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}-1)\end{array}\right)\frac{y_{-{\alpha }_2}}{y_{R^{-}}}\left[X_{s_2}\right]\\ & & +(f_1-f_{s_1}-f_{s_2}+f_{s_1{s}_2}+f_{s_2{s}_1}-f_{s_1{s}_2{s}_1}-f_{s_2{s}_1{s}_2}+f_{s_1{s}_2{s}_1{s}_2})\frac{1}{y_{R^{-}}}\left[X_1\right]\end{array}$$

which simplifies to

$$\begin{array}{ccc}{y}_{R^{-}}f& =& f_{s_1{s}_2{s}_1{s}_2}{y}_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}{y}_{-s_1{s}_2{\alpha }_1}{y}_{-s_1{s}_1{s}_1{\alpha }_2}\left[X_{s_1{s}_2{s}_1{s}_2}\right]\\ & & +(f_{s_1{s}_2{s}_1}-f_{s_1{s}_2{s}_1{s}_2})y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}{y}_{-s_1{s}_2{\alpha }_1}\left[X_{s_1{s}_2{s}_1}\right]\\ & & +(f_{s_2{s}_1{s}_2}-f_{s_1{s}_2{s}_1{s}_2})y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}\left[X_{s_2{s}_1{s}_2}\right]\\ & & +((f_{s_1{s}_2}-f_{s_2{s}_1{s}_2})y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}+(f_{s_1{s}_2{s}_1{s}_2}-f_{s_1{s}_2{s}_1})y_{-s_1{\alpha }_2}{y}_{-s_1{s}_2{\alpha }_1})\left[X_{s_1{s}_2}\right]\\ & & +((f_{s_2{s}_1}-f_{s_1{s}_2{s}_1})y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}+(f_{s_1{s}_2{s}_1{s}_2}-f_{s_2{s}_1{s}_2})y_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2})\left[X_{s_2{s}_1}\right]\\ & & +\left(\begin{array}{c}(f_{s_1}-f_{s_2{s}_1})y_{-{\alpha }_1}+(f_{s_2{s}_1{s}_2}-f_{s_1{s}_2})y_{-s_2{\alpha }_1}\\ +(f_{s_1{s}_2{s}_1{s}_2}-f_{s_1{s}_2{s}_1})(\frac{N{y}_{-s_1{s}_2{\alpha }_1}}{y_{-{\alpha }_2}}-\frac{y_{-s_1{\alpha }_2}{y}_{-s_1{s}_2{\alpha }_1}}{y_{-{\alpha }_2}}-y_{-{\alpha }_1})\end{array}\right)\left[X_{s_1}\right]\\ & & +\left(\begin{array}{c}(f_{s_2}-f_{s_1{s}_2})y_{-{\alpha }_2}+(f_{s_1{s}_2{s}_1}-f_{s_2{s}_1})y_{-s_1{\alpha }_2}\\ +(f_{s_2{s}_1{s}_2{s}_1}-f_{s_2{s}_1{s}_2})(\frac{y_{-s_2{s}_1{\alpha }_2}}{y_{-{\alpha }_1}}-\frac{y_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}}{y_{-{\alpha }_1}}-y_{-{\alpha }_2})\end{array}\right)\left[X_{s_2}\right]\\ & & +(f_1-f_{s_1}-f_{s_2}+f_{s_1{s}_2}+f_{s_2{s}_1}-f_{s_1{s}_2{s}_1}-f_{s_2{s}_1{s}_2}+f_{s_1{s}_2{s}_1{s}_2})\left[X_1\right]\end{array}$$

this last formulas allows for quick computation of products with Schubert classes in rank 2 for low dimensional Schubert varieties. In particular, for $g={\sum }_{w\in W_0}{g}_w{b}_w$ in ${\Omega }_T(G/B),$

$$\begin{array}{ccc}g\left[X_1\right]& =& g_1\left[X_1\right],\\ g\left[X_{s_1}\right]& =& g_{s_1}\left[X_{s_1}\right]+g_{1,s_1}\left[X_1\right],\text{where}{g}_{1,s_1}=\frac{g_1-g_{s_1}}{y_{-{\alpha }_1}},\\ g\left[X_{s_2}\right]& =& g_{s_2}\left[X_{s_2}\right]+g_{1,s_2}\left[X_1\right],\text{where}{g}_{1,s_2}=\frac{g_1-g_{s_2}}{y_{-{\alpha }_2}},\\ g\left[X_{s_1{s}_2}\right]& =& g_{s_1{s}_2}\left[X_{s_1{s}_2}\right]+g_{s_1,s_1{s}_2}\left[X_{s_1}\right]+g_{s_2,s_1{s}_2}\left[X_{s_2}\right]+\frac{g_{1,s_1}-g_{s_2,s_1{s}_2}}{y_{-{\alpha }_2}}\left[X_1\right],\\ g\left[X_{s_2{s}_1}\right]& =& g_{s_2{s}_1}\left[X_{s_2{s}_1}\right]+g_{s_1,s_2{s}_1}\left[X_{s_1}\right]+g_{s_2,s_2{s}_1}\left[X_{s_2}\right]+\frac{g_{1,s_2}-g_{s_2,s_2{s}_1}}{y_{-{\alpha }_1}}\left[X_1\right],\end{array}$$

where

$$g_{s_1,s_1{s}_2}=\frac{g_{s_1}-g_{s_1{s}_2}}{y_{-{\alpha }_2}},g_{s_2,s_1{s}_2}=\frac{g_{s_2}-g_{s_1{s}_2}}{y_{-s_2{\alpha }_1}},g_{s_1,s_2{s}_1}=\frac{g_{s_1}-g_{s_2{s}_1}}{y_{-s_1{\alpha }_2}},g_{s_2,s_2{s}_1}=\frac{g_{s_2}-g_{s_2{s}_1}}{y_{-{\alpha }_1}}\text{.}$$

Using (3.19), Pieri-Chevalley rules giving the expansions of products $x_{\lambda }\left[X_w\right]$ in terms of Schubert classes are directly determined from these formulas.

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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