Partial flag varieties and Bott-Samelson classes [Zw]

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 17 February 2013

Partial flag varieties and Bott-Samelson classes [Zw]

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 ΩT(G/B). 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 ΩT(G/B).

Pushforwards to partial flag varieties: BGG/Demazure operators

Using the notation for parabolic subgroups and partial flag varieties as in (2.3), if J{1,2,,n} and

πJ: G/B G/PJ gB gPJ thenπJ (wB)=uPJ, wherewWJ=u PJ.

Then, in the setting on Theorem 3.1,


and πJ*:ΩT (G/PJ) ΩT (G/B) and (πJ)!: ΩT(G/B) ΩT(G/PJ) correspond to

πJ*:S SW0 SWJS SW0S and (πJ)!:S SW0S SSW0 SWJ (4.1)

where (πJ)! is given by the operator in the nil affine Hecke algebra given by

(πJ)!= ( vWJ tv ) 1xJ,where xJ=αRJ+ x-α.

with RJ+ the set of positive roots for PJBT. A special case is when J={i}, 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 Ai is controlled via the identities in Section 8.

Bott-Samelson classes

For a sequence w=(i1,,i) with 1i1,,in define the Bott-Samelson class

[Zw]= [Zi1i2i] =Ai1Ai2. Ai[Zpt], (4.3)

where, in the notation of (3.17),

[Zpt]v= { αR+ y-α, ifv=1, 0, ifv1. (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 ] wW0 } ,

where, for each wW0, w=si1si is a fixed reduced word for w.

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

=X-1 X0X1 Xd=X

by closed subvarieties such that Xi=Xi-1 are isomorphic to a disjoint union of affine spaces 𝔸i for i=1,2,,d. The "cells" of X are the Xi-Xi-1.

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-variety with a cellular decomposition. Then hT(X) has an hT(pt)-basis given by resolutions of cell closures (choose one resolution for each cell).

For X=G/B, the Bruhat decomposition

G=wW0BwB provides the desired cell decomposition

and the Schubert varieties Xw=BwB are the closures of the Schubert cells. Let P1,,Pn be the minimal parabolics of G (with PiB and PiB) and let s1,,sn be the corresponding simple reflections in W0. The group W0 is generated by s1,,sn. Let w=si1si be a reduced word for w. Then the Bott-Samelson variety Γi1,,i =Pi1×B Pi2×B ×BPi/B provides a resolution of Xw,

γi1,,i: Pi1 ×B Pi2 ×B ×B Pi ×Bpt Xw G/B [g1,,g] g1gB (4.5)

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

Pi1 ×B ×B Pi ×B Pi+1 ×Bpt γ i1 i+1 G/B τ πi+1 Pi1 ×B ×B Pi ×Bpt γi1i G/B πi+1 G/Pi+1 (4.6)

  1. commutes, and
  2. has both vertical maps fibrations with fibre Pi+1/B,

it is a pullback square. Thus

(γi1ii+1)! (ι*(1)) = πi+1* ( πi+1 γi1i ) ! (1) = πi+1* (πi+1)! (γi1i)! (1) = Ai+1 (γi1i)! (1). (4.7)

The following result then follows by induction.

Theorem 4.3. ([HKi0903.3926, Theorem 3.2], [BEv1044959, Proposition 2]) If U=(i1,,i) is a sequence in {1,,n} and γi1i is as in (4.5) then

[Zi1i]= [ (γi1i)! (1) ] =Ai1 Ai[Zpt], where [Zpt] is the class of a point.

Theorem 4.3 says that the values on the vertices of the element [Zi1i] on the moment graph of Γi1,,i are exactly the coefficients of the 2 terms in the expansion of

Ai1Ai= (1+tsi1) 1x-αi1 (1+tsi) 1x-αi.

For example, in type GL3,

[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 [Z121]= (1+ts1) 1x-α1 (1+ts1) 1x-α1 (1+ts1) 1x-α1 yR-b1 in the basis {bwwW0}. An example of the pushpull in (4.6) in the case of type GL3

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 Δ121= y-(α1+α2) y-α1 + y-α2 yα1 .

Change of groups morphisms across ι:BPJ

In the same way that Theorem 3.1 provides SSW0S ΩT (G/B) one can obtain


and, if ι:BPJ 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(fg)= wWJw (1yJf)g, whereyJ= αRJ+ y-α,

with RJ+ the set of positive roots for PJBT. The pushforward ιJ is similar to the pushforward operator (πJ)! appearing in (4.1) except acting on the left factor of SSW0S (see, for example, the definition of δi in [Kaj2010, §7]).

1·1·1 s1·1·1 1·s2·1 1·1·s1 s1·s2·1 s1·1·s1 1·s2·s1 s1·s2·s1 y-α1 y-α1 y-α1 y-α1 y-s1α2 y-s1α2 y-α2 y-α2 y-s1s2α1 y-s1α1 y-s2α1 y-α1 (γ121)! 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 τ* π1* 1·1 s1·1 1·s2 s1·s2 y-s2α1 y-α2 y-α2 y-α1 (γ12)! 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 (π1)! 1 s2 s2s1 y-α2 y-α1 y-(α1+α2) 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.

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