## Partial flag varieties and Bott-Samelson classes $\left[{Z}_{\stackrel{\to }{w}}\right]$

Last update: 17 February 2013

## Partial flag varieties and Bott-Samelson classes $\left[{Z}_{\stackrel{\to }{w}}\right]$

In this section we review the formulas for the Bott-Samelson classes as established in, for example, [HKi0903.3926, CPZ0905.1341, BEv0968883, BEv1044959]. Though some of these references are not considering the equivariant case, the same machinery applies to define these classes in ${\Omega }_{T}\left(G/B\right)\text{.}$ In particular, this is the place in the theory where the BGG/Demazure operators are derived from the geometry. These operators play a fundamental role in the combinatorial study of ${\Omega }_{T}\left(G/B\right)\text{.}$

### Pushforwards to partial flag varieties: BGG/Demazure operators

Using the notation for parabolic subgroups and partial flag varieties as in (2.3), if $J\subseteq \left\{1,2,\dots ,n\right\}$ and

$πJ: G/B → G/PJ gB ↦ gPJ thenπJ (wB)=uPJ, where wWJ=u PJ.$

Then, in the setting on Theorem 3.1,

$S⊗S0W SWJ≅ΩT (G/PJ),$

and ${\pi }_{J}^{*}:{\Omega }_{T}\left(G/{P}_{J}\right)\to {\Omega }_{T}\left(G/B\right)$ and ${\left({\pi }_{J}\right)}_{!}:{\Omega }_{T}\left(G/B\right)\to {\Omega }_{T}\left(G/{P}_{J}\right)$ correspond to

$πJ*:S ⊗SW0 SWJ↪S ⊗SW0S and (πJ)!:S ⊗SW0S⟶ S⊗SW0 SWJ (4.1)$

where ${\left({\pi }_{J}\right)}_{!}$ is given by the operator in the nil affine Hecke algebra given by

$(πJ)!= ( ∑v∈WJ tv ) 1xJ,where xJ=∏α∈RJ+ x-α.$

with ${R}_{J}^{+}$ the set of positive roots for ${P}_{J}\supseteq B\supseteq T\text{.}$ A special case is when $J=\left\{i\right\},$ for which

$WJ={1,si} andπi* (πi)!=Ai =(1+tsi) 1x-αi, (4.2)$

is the BGG-Demazure operator (see [BEv0968883, Cor.-Def. 1.9]). The calculus of the operators ${A}_{i}$ is controlled via the identities in Section 8.

### Bott-Samelson classes

For a sequence $\stackrel{\to }{w}=\left({i}_{1},\dots ,{i}_{\ell }\right)$ with $1\le {i}_{1},\dots ,{i}_{\ell }\le n$ define the Bott-Samelson class

$[Zw→]= [Zi1i2…iℓ] =Ai1Ai2. Aiℓ[Zpt], (4.3)$

where, in the notation of (3.17),

$[Zpt]v= { ∏α∈R+ y-α, if v=1, 0, if v≠1. (4.4)$

Theorem 4.1. [BEv1044959, Prop. 1], [HKi0903.3926, Prop. 3.1], [KKu0895705, Lemma 3.15], see also [HHH0409305, Proposition 4.1]) The generalized cohomology

$hT(G/B) has hT(pt)-basis { [Zw→]= [ γw→: Γw→→ G/B ] ∣ w∈W0 } ,$

where, for each $w\in {W}_{0},$ $\stackrel{\to }{w}={s}_{{i}_{1}}\dots {s}_{{i}_{\ell }}$ is a fixed reduced word for $w\text{.}$

Let us explain where this comes from. Let $X$ be a $T\text{-variety.}$ Following [Ful1644323, Example 1.9.1], or [CGi1433132, §5.5], a cellular decomposition of $X$ is a filtration

$∅=X-1⊆ X0⊆X1⊆…⊆ Xd=X$

by closed subvarieties such that ${X}_{i}={X}_{i-1}$ are isomorphic to a disjoint union of affine spaces ${𝔸}^{{\ell }_{i}}$ for $i=1,2,\dots ,d\text{.}$ The "cells" of $X$ are the ${X}_{i}-{X}_{i-1}\text{.}$

Theorem 4.2. (see [Gro1958, Prop. 7]; [Ful1644323, Example 1.9.1] who refers to [Cho0078006]; [CGi1433132, Lemma 5.5.1]; [BEv1044959, Proposition 1]; [HKi0903.3926, Theorem 2.5]) Let $X$ be a $T\text{-variety}$ with a cellular decomposition. Then ${h}_{T}\left(X\right)$ has an ${h}_{T}\left(\text{pt}\right)\text{-basis}$ given by resolutions of cell closures (choose one resolution for each cell).

For $X=G/B,$ the Bruhat decomposition

$G=⨆w∈W0BwB provides the desired cell decomposition$

and the Schubert varieties ${X}_{w}=\stackrel{‾}{BwB}$ are the closures of the Schubert cells. Let ${P}_{1},\dots ,{P}_{n}$ be the minimal parabolics of $G$ (with ${P}_{i}\supseteq B$ and ${P}_{i}\ne B\text{)}$ and let ${s}_{1},\dots ,{s}_{n}$ be the corresponding simple reflections in ${W}_{0}\text{.}$ The group ${W}_{0}$ is generated by ${s}_{1},\dots ,{s}_{n}\text{.}$ Let $\stackrel{\to }{w}={s}_{{i}_{1}}\dots {s}_{{i}_{\ell }}$ be a reduced word for $w\text{.}$ Then the Bott-Samelson variety ${\Gamma }_{{i}_{1},\dots ,{i}_{\ell }}={P}_{{i}_{1}}{×}_{B}{P}_{{i}_{2}}{×}_{B}\dots {×}_{B}{P}_{{i}_{\ell }}/B$ provides a resolution of ${X}_{w},$

$γi1,…,iℓ: Pi1 ×B Pi2 ×B… ×B Piℓ ×Bpt ⟶ Xw ↪ G/B [g1,…,gℓ] ⟼ g1…gℓB (4.5)$

Then following, for example, the proof of [BEv1044959, Prop. 2], since the diagram

$Pi1 ×B… ×B Piℓ ×B Piℓ+1 ×Bpt ⟶ γ i1… iℓ+1 G/B τ ↓ ↓ πiℓ+1 Pi1 ×B… ×B Piℓ ×Bpt ⟶γi1…iℓ G/B ⟶πiℓ+1 G/Piℓ+1 (4.6)$

1. commutes, and
2. has both vertical maps fibrations with fibre ${P}_{{i}_{\ell +1}}/B,$

it is a pullback square. Thus

$(γi1…ii+1)! (ι*(1)) = πiℓ+1* ( πiℓ+1∘ γi1…iℓ ) ! (1) = πiℓ+1* (πiℓ+1)! (γi1…iℓ)! (1) = Aiℓ+1 (γi1…iℓ)! (1). (4.7)$

The following result then follows by induction.

Theorem 4.3. ([HKi0903.3926, Theorem 3.2], [BEv1044959, Proposition 2]) If $U=\left({i}_{1},\dots ,{i}_{\ell }\right)$ is a sequence in $\left\{1,\dots ,n\right\}$ and ${\gamma }_{{i}_{1}\dots {i}_{\ell }}$ is as in (4.5) then

$[Zi1…iℓ]= [ (γi1…iℓ)! (1) ] =Ai1… Aiℓ[Zpt], where [Zpt] is the class of a point.$

Theorem 4.3 says that the values on the vertices of the element $\left[{Z}_{{i}_{1}\dots {i}_{\ell }}\right]$ on the moment graph of ${\Gamma }_{{i}_{1},\dots ,{i}_{\ell }}$ are exactly the coefficients of the ${2}^{\ell }$ terms in the expansion of

$Ai1…Aiℓ= (1+tsi1) 1x-αi1… (1+tsiℓ) 1x-αiℓ.$

For example, in type $G{L}_{3},$

$[Z121]= ( y-(α1+α2) y-α1 1·1·1 + y-α2 yα1 ts1·1·1+1· ts2·1+ y-(α1+α2) y-α1 1·1·ts1 +ts1· ts2·1+ y-α2 yα1 ts1·1· ts1+1· ts2· ts1 +ts1·ts2 ·ts1 ) b1$

provides the expansion of $\left[{Z}_{121}\right]=\left(1+{t}_{{s}_{1}}\right)\frac{1}{{x}_{-{\alpha }_{1}}}\left(1+{t}_{{s}_{1}}\right)\frac{1}{{x}_{-{\alpha }_{1}}}\left(1+{t}_{{s}_{1}}\right)\frac{1}{{x}_{-{\alpha }_{1}}}{y}_{{R}^{-}}{b}_{1}$ in the basis $\left\{{b}_{w} \mid w\in {W}_{0}\right\}\text{.}$ An example of the pushpull in (4.6) in the case of type $G{L}_{3}$

$P1 ×B P2 ×B P1 ×B pt ⟶γ121 GL3/B τ↓ ↓π1 P1 ×B P2 ×B pt ⟶γ12 GL3/B ⟶π1 GL3/P1 (4.8)$

has moment graphs as in Figure 1, and the computation in (4.7) for this example is

$1 111 111 1 ⟶(γ121)! Δ121 Δ1211 11 1 ↑τ* ↑π1* 1 11 1 ⟶(γ12)! y-(α1+α2) y-α2 y-(α1+α2) 0 y-α2 0 ⟶(π1)! Δ121 1 1$

where ${\Delta }_{121}=\frac{{y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)}}{{y}_{-{\alpha }_{1}}}+\frac{{y}_{-{\alpha }_{2}}}{{y}_{{\alpha }_{1}}}\text{.}$

### Change of groups morphisms across $\iota :B↪{P}_{J}$

In the same way that Theorem 3.1 provides $S{\otimes }_{S}^{{W}_{0}}S\cong {\Omega }_{T}\left(G/B\right)$ one can obtain

$SWJ ⊗S0WS ≅ΩPJ (G/B),$

and, if $\iota :B↪{P}_{J}$ is the inclusion then the change of group homomorphisms

$ιJ:ΩPJ (G/B)→ΩT (G/B)and ιJ:ΩT (G/B)→ΩPJ (G/B)$

are given, combinatorially, by

$ιJ:SWJ ⊗S0WS↪S ⊗S0WSand ιJ:S⊗S0WS ⟶SWJ ⊗S0WS,$

with

$ιJ(f⊗g)= ∑w∈WJw (1yJf)⊗g, whereyJ= ∏α∈RJ+ y-α,$

with ${R}_{J}^{}$ the set of positive roots for ${P}_{J}\supseteq B\supseteq T\text{.}$ The pushforward ${\iota }^{J}$ is similar to the pushforward operator ${\left({\pi }_{J}\right)}_{!}$ appearing in (4.1) except acting on the left factor of $S{\otimes }_{{S}^{{W}_{0}}}S$ (see, for example, the definition of ${\delta }_{i}$ in [Kaj2010, §7]).

$1·1·1 {s}_{1}·1·1 1·{s}_{2}·1 1·1·{s}_{1} {s}_{1}·{s}_{2}·1 {s}_{1}·1·{s}_{1} 1·{s}_{2}·{s}_{1} {s}_{1}·{s}_{2}·{s}_{1} {y}_{-{\alpha }_{1}} {y}_{-{\alpha }_{1}} {y}_{-{\alpha }_{1}} {y}_{-{\alpha }_{1}} {y}_{-{s}_{1}{\alpha }_{2}} {y}_{-{s}_{1}{\alpha }_{2}} {y}_{-{\alpha }_{2}} {y}_{-{\alpha }_{2}} {y}_{-{s}_{1}{s}_{2}{\alpha }_{1}} {y}_{-{s}_{1}{\alpha }_{1}} {y}_{-{s}_{2}{\alpha }_{1}} {y}_{-{\alpha }_{1}} ⟶(γ121)! 1 {s}_{1} {s}_{2} {s}_{1}{s}_{2} {s}_{2}{s}_{1} {s}_{1}{s}_{2}{s}_{1}={s}_{2}{s}_{1}{s}_{2} {y}_{-{\alpha }_{2}} {y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)} {y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)} {y}_{-{\alpha }_{1}} {y}_{-{\alpha }_{1}} {y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)} {y}_{-{\alpha }_{2}} {y}_{-{\alpha }_{2}} {y}_{-{\alpha }_{1}} ↑τ* ↑π1* 1·1 {s}_{1}·1 1·{s}_{2} {s}_{1}·{s}_{2} {y}_{-{s}_{2}{\alpha }_{1}} {y}_{-{\alpha }_{2}} {y}_{-{\alpha }_{2}} {y}_{-{\alpha }_{1}} ⟶(γ12)! 1 {s}_{1} {s}_{2} {s}_{1}{s}_{2} {s}_{2}{s}_{1} {s}_{1}{s}_{2}{s}_{1}={s}_{2}{s}_{1}{s}_{2} {y}_{-{\alpha }_{2}} {y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)} {y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)} {y}_{-{\alpha }_{1}} {y}_{-{\alpha }_{1}} {y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)} {y}_{-{\alpha }_{2}} {y}_{-{\alpha }_{2}} {y}_{-{\alpha }_{1}} ⟶(π1)! 1 {s}_{2} {s}_{2}{s}_{1} {y}_{-{\alpha }_{2}} {y}_{-{\alpha }_{1}} {y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)} Figure 1: An example of the moment graphs for the diagram (4.8)$

## 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.