Schubert Products G2

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

Last update: 2 September 2012

Examples

Type G2

Here we will use

W0= s1,s2 si2=1, s1s2s1s2s1 s2=s2s1s2 s1s2s1

with

s1α1=-α1, s1α2=3α1+ α2, s2α1=α1+ α2, s2α2=-α2,

The same in moment graph pictures is

b1 bs1 bs2 bs1s2 bs2s1 bs1s2s1 bs2s1s2 bs1s2s1s2 bs2s1s2s1 bs1s2s1s2s1 bs2s1s2s1s2 bw0 bwbasis D-α1 Dα1 D-(α1+α2) Dα1+α2 D-(2α1+α2) D2α1+α2 D-(2α1+α2) D2α1+α2 D-(α1+α2) Dα1+α2 D-α1 Dα1 1D-α1 D-α2 D-(3α1+α2) Dα2 D-(3α1+2α2) D3α1+α2 D-(3α1+2α2) D3α1+2α2 D-(3α1+α2) D3α1+2α2 D-α2 D3α1+α2 Dα2 1D-α2

or, alternatively,

D-α1 Dα1 D-s2α1 Ds2α1 D-s1s2α1 Ds1s2α1 D-s2s1s2α1 Ds2s1s2α1 D-s1s2s1s2α1 Ds1s2s1s2α1 D-s2s1s2s1s2α1 Dα1 1D-α1 D-α2 D-s1α2 Dα2 D-s2s1α2 Ds1α2 D-s1s2s1α2 Ds2s1α2 D-s2s1s2s1α2 Ds1s2s1α2 D-s1s2s1s2s1α2 Ds2s1s2s1α2 Dα2 1D-α2

Let

DR-=D-α1 D-α2 D-(α2+α1) D-(α2+2α1) D-(α2+3α1) D-(2α2+3α1)

DR- 0 0 0 0 0 0 0 0 0 0 0 Zpt Δ1 Δ1 0 0 0 0 0 0 0 0 0 0 Z1 Δ2 0 D2 0 0 0 0 0 0 0 0 0 Z2 Δ12 Δ12 Γ12 Γ12 0 0 0 0 0 0 0 0 Z12 Δ21 Γ21 Δ21 0 Γ21 0 0 0 0 0 0 0 Z21 Δ121 Δ121 Γ121 Γ121 K121 K121 0 0 0 0 0 0 Z121 Δ212 Γ212 Δ212 K212 Γ212 0 K212 0 0 0 0 0 Z212 Δ1212 Δ1212 Γ1212 Γ1212 K1212 K1212 L1212 L1212 0 0 0 0 Z1212 Δ2121 Γ2121 Δ2121 K2121 Γ2121 L2121 K2121 0 L2121 0 0 0 Z2121 Δ12121 Δ12121 Γ12121 Γ12121 K12121 K12121 L12121 L12121 M12121 M12121 0 0 Z12121 Δ21212 Γ21212 Δ21212 K21212 Γ21212 L21212 K21212 M21212 L21212 0 M21212 0 Z21212

Where

Δ1 = DR- 1 D-α1 Δ12 = DR- 1 D-α1 D-α2 ,and Γ12=Δ2 1 D-s2α1 =DR- 1 D-α2 D-s2α1 Δ121 = Δ21 1 D-α1 +Γ21 1 D-s1α1 = DR- ( 1 D-α12 D-α2 + 1 D-α1 D-s1α2 Dα1 ) , = DR- D-α1 ( 1 D-α1 D-α2 + 1 D-s1α1 D-s1α2 ) , Γ121 = Δ21 1 D-s2α1 = DR- 1 D-α1 D-α2 D-s2α1 , K121 = Γ21 1 D-s1s2α1 =DR- 1 D-α1 D-s1α2 D-s1s2α1 , Δ1212 = Δ212 1 D-α1 + Γ212 1 D-s1α1 = DR- ( 1 D-α22 D-α12 + 1 D-α2 D-s2α1 D-α1 Dα2 + 1 D-α2 D-s1α2 D-α1 Dα1 ) = DR- D-α1 D-α2 ( 1 D-α2 D-α1 + 1 D-s2α1 D-s2α2 + 1 D-s1α2 D-s1α1 ) , Γ1212 = Δ212 1 D-s2α1 + K212 1 D-s2s1α1 = DR- ( 1 D-α22 D-α1 D-s2α1 + 1 D-α2 D-s2α12 Dα2 + 1 D-α2 D-s2α1 D-s2s1α2 Ds2α1 ) = DR- D-α2 D-s2α1 ( 1 D-α2 D-α1 + 1 D-s2α1 D-s2α2 + 1 D-s2s1α2 D-s2s1α1 ) , K1212 = Γ212 1 D-s1s2α1 = DR- 1 D-α2 D-α1 D-s1α2 D-s1s2α1 , L1212 = K212 1 D-s2s1s2α1 = DR- 1 D-α2 D-s2α1 D-s2s1α2 D-s2s1s2α1 , Δ12121 = Δ2121 1 D-α1 + Γ2121 1 D-s1α1 = DR- ( 1 D-α12 D-α2 D-s1α2 Dα1 + 1 D-α1 D-s1α22 Dα12 + 1 D-α1 D-s1α2 D-s1s2α1 Ds1α2 Dα1 ) = DR- D-α1 D-s1α2 D-s1α1 ( 1 D-α1 D-α2 + 1 D-s1α2 D-s1α1 + 1 D-s1s2α1 D-s1s2α2 ) , Γ12121 = Δ2121 1 D-s2α1 + K2121 1 D-s2s1α1 = DR- ( 1 D-α12 D-α22 D-s2α1 + 1 D-α1 D-s1α2 D-α2 Dα1 D-s2α1 + 1 D-α1 D-s2α12 D-α2 Dα2 + 1 D-α1 D-α2 D-s2α1 D-s2s1α2 Ds2α1 ) = DR- D-α1 D-α2 D-s2α1 ( 1 D-α1 D-α2 + 1 D-s1α2 D-s1α1 + 1 D-s2α1 D-s2α2 + 1 D-s2s1α2 D-s2s1α1 ) , K12121 = Γ2121 1 D-s1s2α1 + L2121 1 D-s1s2s1α1 = DR- ( 1 D-α12 D-α2 D-s1α2 D-s1s2α1 + 1 D-α1 D-s1α22 Dα1 D-s1s2α1 + 1 D-α1 D-s1α2 D-s1s2α1 Ds1α2 D-s1s2α1 + 1 D-α1 D-s1α2 D-s1s2α1 D-s1s2α1 D-s1s2s1α2 Ds1s2α1 ) , = DR- D-α1 D-s1α2 D-s1s2α1 ( 1 D-α1 D-α2 + 1 D-s1α2 D-s1α1 + 1 D-s1s2α2 D-s1s2α1 + 1 D-s1s2s1α2 D-s1s2s1α1 ) L12121 = K2121 1 D-s2s1s2α1 = DR- 1 D-α1 D-α2 D-s2α1 D-s2s1α2 D-s2s1s2α1 , M12121 = L2121 1 D-s1s2s1s2α1 = DR- 1 D-α1 D-s1α2 D-s1s2α1 D-s1s2s1α2 D-s1s2s1s2α1

and the corresponding expressions for reduced words beginning with 2 are obtained by switching 1s and 2s in the above expressions. these formulas are pleasant enough that one can almost guess what happens for all rank 2 cases: Types I2(m) and Type A1(1).

This formulation allows for efficient computations of products. For example

Z12Z21 = Δ12 Γ21 Δ1 Z1 + Δ21 Γ12 Δ2 Z2+ Δ12 Δ21- Δ12 Γ21- Δ21 Γ12 DR- Zpt = DR- D-α1 D-α2 D-s1α2 Z1+ DR- D-α1 D-α2 D-s2α1 Z2 + DR- D-α1 D-α2 D-s1α2 D-s2α1 ( D-s2α1 D-s1α2 - D-α1 D-s1α2 - D-α2 D-s2α1 ) D-α1 D-α2 Zpt = DR- D-α1 D-α2 D-s1α2 Z1+ DR- D-α1 D-α2 D-s2α1 Z2 + DR- D-α1 D-α2 D-s1α2 D-s2α1 ( ( D-s2α1 -D-α1 D-α2 ) ( D-s1α2 -D-α2 D-α1 ) -1 ) Zpt,

simultaneously generalizing 3 formulas for equivariant cohomology and K-theory given in [GR],

[s1s2] [s2s1]= { {-[1]}+ [s1]+ [s2] , in TypeA2 , ( {a11}+ y21 ) [1]-α11 [s1]-α21 [s2] , in TypeB2 , α21 α32 ( y11+ y21+ α31 ) [1]-α11 α21 α32 [s1]- α21 α31 α32 [s2] , in TypeG2 .

Another example is

Z212 = Γ21 Z21+ Δ212- Γ21 Δ21 Δ2 Z2 = DR- D-α1 D-s1α2 Z21+ DR- D-α1 D-α2 D-s1α2 ( D-s1α2 -D-α2 D-α1 ) Z2,

simultaneously generalizing 3 formulas for equivariant cohomology and K-theory given in [GR],

[s1s2]2= { -α01 [s1s2]+ y01 [s2] , in TypeA2 , α01 α11 [s1s2]- α11 ( y01+ y11 ) [s2] , in TypeB2 , α01 α11 α21 α32 [s1s2] - α11 α21 α32 ( y01+ y11+ y21 ) [s2] , in TypeG2 .

Switching 1s and 2s in the product Z212 generalizes 3 more formulas from [GR]:

[s2s1]2= { -α10 [s1s2]+ y10 [s1] , in TypeA2 , α10 α21 [s2s1]- α21 y10 [s1] , in TypeB2 , α10 α21 α31 α32 [s2s1] - α21 α31 α32 y10 [s1] , in TypeG2 .

Products

Zpt2=DR- Zpt,ZptZ1= DR-D-10 Zpt,ZptZ2 =DR-D0-1 Zpt,

ZptZ12= DR- D-10 D0-1 Zpt,ZptZ21 = DR- D-10 D0-1 Zpt,

Z12=DR-D-10 Z1,Z1Z2= DR- D-10 D0-1 Zpt, Z22= DR-D0-1Zpt,

ZptZ12= DR- D-10 D0-1 Zpt,ZptZ21 = DR- D-10 D0-1 Zpt

Z1Z12= DR- D-10 D0-1 Z1,Z1Z21= DR- D-10 D-j-1 Z1+ ( DR- D-102 D0-1 - DR- D-102 D-j-1 ) Zpt

and this second formula is rewritten as

Z1Z21= DR- D-10 D-j-1 Z1+ DR- D-10 D-j-1 D0-1 ( D-j-1- D0-1 D-10 ) Zpt

since

DR- D-102 D0-1 - DR- D-102 D-j-1 = DR- D-102 ( 1D0-1 - 1D-j-1 ) = DR- D-10 D0-1 D-j-1 ( D-j-1- D0-1 D-10 ) .

Next

Z2Z12= DR- D0-1 D-1-1 Z2+ ( DR- D0-12 D-10 - DR- D0-12 D-1-1 ) Zpt, Z2Z21= DR- D-10 D0-1 Z2

and the first of these can be rewritten as

Z2Z12= DR- D0-1 D-1-1 Z2+ DR- D0-1 D-1-1 D-10 ( D-1-1- D-10 D0-1 ) Zpt.

Then

Z122 = DR- D0-1 D-1-1 Z12+ DR- D0-12 D-10 Z1- DR- D0-1 D-1-1 D0-1 Z1 = DR- D0-1 D-1-1 Z12+ DR- D0-12 ( 1D-10- 1D-1-1 ) Z1 = DR- D0-1 D-1-1 Z12+ DR- D0-1 D-10 D-1-1 ( D-1-1- D-10 D0-1 ) Z1,

and

Z212 = DR- D-10 D-j-1 Z21+ DR- D-102 D0-1 Z2- DR- D-10 D-j-1 D-10 Z2 = DR- D-10 D-j-1 Z21+ DR- D-102 ( 1D0-1- 1D-j-1 ) Z2 = DR- D-10 D-j-1 Z21 + DR- D-10 D0-1 D-j-1 ( D-j-1- D0-1 D-10 ) Z2.

Then

Z12Z21 = DR- D0-1 D-10 D-j-1 Z1+ DR- D0-1 D-10 D-1-1 Z2 + ( DR- D0-12 D-102 - DR- D0-1 D-102 D-j-1 - DR- D0-12 D-10 D-1-1 ) Zpt = DR- D0-1 D-10 D-j-1 Z1+ DR- D0-1 D-10 D-1-1 Z2 + DR- D0-1 D-10 ( 1 D0-1 D-10 - 1 D-10 D-j-1 - 1 D0-1 D-1-1 ) Zpt = DR- D0-1 D-10 D-j-1 Z1+ DR- D0-1 D-10 D-1-1 Z2 DR- D0-1 D-10 ( D-j-1 D-1-1 - D0-1 D-1-1 - D-10 D-j-1 D0-1 D-10 D-1-1 D-j-1 ) Zpt = DR- D0-1 D-10 D-j-1 Z1+ DR- D0-1 D-10 D-1-1 Z2 + DR- D0-1 D-10 D-1-1 D-j-1 ( D-j-1 D-1-1 - D0-1 D-1-1 - D-10 D-j-1 D0-1 D-10 ) Zpt

From [Ku, Prop. ???], the singular Schubert varieties in rank 2 groups are

Type Singular Locus B2 Xs1s2s1 Xs1 G2 Xs1s2s1 Xs1 G2 Xs1s2s1s2 Xs1s2 G2 Xs2s1s2s1 Xs2s1 G2 Xs1s2s1s2s1 Xs1s2s1 G2 Xs2s1s2s1s2 Xs2

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