## Products with Schubert classes

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 ${\Omega }_{T}\left(G/B\right),$ as in (3.17).

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

$[Xu][Xv]= ∑w∈W0 cuvw [Xw]. (6.1)$

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

$[Xu][Xv]= [Xu]v [Xv]+ ∑w

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

$f=∑ℓ(w)=k fw1[Xw]w [Xw]= ∑ℓ(v)≤k-1 ( fv-∑ℓ(w)=k fw [Xw]v [Xw]w ) bv$

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

$f-∑ℓ(w)=k fw1[Xw]w [Xw]- ∑ℓ(v)=k-1 ( fv- ∑ ℓ(v)=k-1 ℓ(w)=k fw [Xw]v [Xw]w ) 1[Xv]v [Xv] =∑ℓ(z)≤k-2 ( fz-∑ℓ(w)=k fw [Xw]z [Xw]w - ∑ℓ(v)=k-1 fv [Xv]z [Xv]v + ∑ ℓ(v)=k-1 ℓ(w)=k fw [Xw]v [Xw]w [Xv]z [Xv]v ) bz$

and induction gives that

$f=∑z∈W0 ( ∑k=1ℓ(w0) ∑w1>…>wk=z (-1)k-1 fw1 [Xw1]w2 [Xw1]w1 [Xw2]w3 [Xw2]w2 … [Xwk-1]wk [Xwk-1]wk-1 1 [Xwk]wk ) [Xz] (6.3)$

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

$f = fs1s2s1s2 1 [Xs1s2s1s2] s1s2s1s2 [Xs1s2s1s2] + ( fs1s2s1- fs1s2s1s2 ) 1 [Xs1s2s1] s1s2s1 [Xs1s2s1]+ ( fs2s1s2- fs1s2s1s2 ) 1 [Xs2s1s2] s2s1s2 [Xs2s1s2] + ( ( fs1s2- fs2s1s2 ) + ( fs1s2s1s2 -fs1s2s1 ) [Xs1s2s1] s1s2 [Xs1s2s1] s1s2s1 ) 1[Xs1s2]s1s2 [Xs1s2] + ( ( fs2s1- fs1s2s1 ) + ( fs1s2s1s2 -fs2s1s12 ) [Xs2s1s2] s2s1 [Xs2s1s2] s2s1s2 ) 1[Xs2s1]s2s1 [Xs2s1] + ( ( fs1- fs2s1 ) + ( fs2s1s2 -fs1s2 ) [Xs1s2]s1 [Xs1s2]s1s2 + ( fs1s2s1s2 -fs1s2s1 ) ( [Xs1s2s1] s1 [Xs1s2s1] s1s2s1 - [Xs1s2s1] s1s2 [Xs1s2s1] s1s2s1 [Xs1s2] s1 [Xs1s2] s1s2 -1 ) ) 1[Xs1]s1 [Xs1] + ( ( fs2- fs1s2 ) + ( fs1s2s1 -fs2s1 ) [Xs2s1]s2 [Xs2s1]s2s1 + ( fs2s1s2s1 -fs2s1s2 ) ( [Xs2s1s2] s2 [Xs2s1s2] s2s1s2 - [Xs2s1s2] s2s1 [Xs2s1s2] s2s1s2 [Xs2s1] s2 [Xs2s1] s2s1 -1 ) ) 1[Xs2]s2 [Xs2] + ( f1-fs1- fs2+ fs1s2+ fs2s1- fs1s2s1- fs2s1s2+ fs1s2s1s2 ) 1[X1]1 [X1]$

and we may use the explicit values of ${\left[{X}_{w}\right]}_{v}$ given in Figure 2 to derive

$f = fs1s2s1s2 y-α1 y-s1α2 y-s1s2α1 y-s1s1s1α2 yR- [Xs1s2s1s2] + ( fs1s2s1- fs1s2s1s2 ) y-α1 y-s1α2 y-s1s2α1 yR- [Xs1s2s1]+ ( fs2s1s2- fs1s2s1s2 ) y-α2 y-s2α1 y-s2s1α2 yR- [Xs2s1s2] + ( ( fs1s2- fs2s1s2 ) + ( fs1s2s1s2 -fs1s2s1 ) y-α1 y-s1α2 y-s1s2α1 y-α1 y-α2 y-s2α1 ) y-α2 y-s2α1 yR- [Xs1s2] + ( ( fs2s1- fs1s2s1 ) + ( fs1s2s1s2 -fs2s1s2 ) y-α2 y-s2α1 y-s2s1α2 y-α1 y-α2 y-s1α2 ) y-α1 y-s1α2 yR- [Xs2s1] + ( ( fs1- fs2s1 ) + ( fs2s1s2 -fs1s2 ) y-α2 y-s2α1 y-α1 y-α2 + ( fs1s2s1s2 -fs1s2s1 ) ( Ny-α1 y-s1α2 y-s1s2α1 y-α1 y-α2 y-s1α2 - y-α1 y-s1α2 y-s1s2α1 y-α1 y-α2 y-s2α1 y-α2 y-s2α1 y-α1 y-α2 -1 ) ) y-α1 yR- [Xs1] + ( ( fs2- fs1s2 ) + ( fs1s2s1 -fs2s1 ) y-α1 y-s1α2 y-α1 y-α2 + ( fs2s1s2s1 -fs2s1s2 ) ( Ny-α2 y-s2α1 y-s2s1α2 y-α1 y-α2 y-s2α1 - y-α2 y-s2α1 y-s2s1α2 y-α1 y-α2 y-s1α2 y-α1 y-s1α2 y-α1 y-α2 -1 ) ) y-α2 yR- [Xs2] + ( f1-fs1- fs2+ fs1s2+ fs2s1- fs1s2s1- fs2s1s2+ fs1s2s1s2 ) 1yR- [X1]$

which simplifies to

$yR-f = fs1s2s1s2 y-α1 y-s1α2 y-s1s2α1 y-s1s1s1α2 [Xs1s2s1s2] + ( fs1s2s1- fs1s2s1s2 ) y-α1 y-s1α2 y-s1s2α1 [Xs1s2s1] + ( fs2s1s2- fs1s2s1s2 ) y-α2 y-s2α1 y-s2s1α2 [Xs2s1s2] + ( ( fs1s2- fs2s1s2 ) y-α2 y-s2α1+ ( fs1s2s1s2 -fs1s2s1 ) y-s1α2 y-s1s2α1 ) [Xs1s2] + ( ( fs2s1- fs1s2s1 ) y-α1 y-s1α2+ ( fs1s2s1s2 -fs2s1s2 ) y-s2α1 y-s2s1α2 ) [Xs2s1] + ( (fs1-fs2s1) y-α1+ ( fs2s1s2- fs1s2 ) y-s2α1 + ( fs1s2s1s2 -fs1s2s1 ) ( Ny-s1s2α1 y-α2 - y-s1α2 y-s1s2α1 y-α2 -y-α1 ) ) [Xs1] + ( (fs2-fs1s2) y-α2+ ( fs1s2s1- fs2s1 ) y-s1α2 + ( fs2s1s2s1 -fs2s1s2 ) ( y-s2s1α2 y-α1 - y-s2α1 y-s2s1α2 y-α1 -y-α2 ) ) [Xs2] + ( f1-fs1- fs2+fs1s2 +fs2s1- fs1s2s1- fs2s1s2+ fs1s2s1s2 ) [X1]$

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 ${\Omega }_{T}\left(G/B\right),$

$g[X1] = g1[X1], g[Xs1] = gs1 [Xs1]+ g1,s1 [X1],where g1,s1= g1-gs1 y-α1 , g[Xs2] = gs2 [Xs2]+ g1,s2 [X1],where g1,s2= g1-gs2 y-α2 , g[Xs1s2] = gs1s2 [Xs1s2]+ gs1,s1s2 [Xs1]+ gs2,s1s2 [Xs2]+ g1,s1- gs2,s1s2 y-α2 [X1], g[Xs2s1] = gs2s1 [Xs2s1]+ gs1,s2s1 [Xs1]+ gs2,s2s1 [Xs2]+ g1,s2- gs2,s2s1 y-α1 [X1],$

where

$gs1,s1s2= gs1- gs1s2 y-α2 , gs2,s1s2= gs2- gs1s2 y-s2α1 , gs1,s2s1= gs1- gs2s1 y-s1α2 , gs2,s2s1= gs2- gs2s1 y-α1 .$

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.