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GKMVarieties :: KClass ^ ZZ

KClass ^ ZZ -- computes powers of an equivariant K-classes

Synopsis

Description

This method computes the $n$-th power of an equivariant $K$-class $C$.

i1 : Gr24 = generalizedFlagVariety("A",3,{2}); --the Grassmannian of projective lines in projective 3-space
i2 : O1 = ampleKClass Gr24 -- the O(1) bundle on Gr24 as an equivariant K-class

o2 = an equivariant K-class on a GKM variety 

o2 : KClass
i3 : O2 = O1^2

o3 = an equivariant K-class on a GKM variety 

o3 : KClass
i4 : peek O2

o4 = KClass{variety => a GKM variety with an action of a 4-dimensional torus}
                                                       2 2
            KPolynomials => HashTable{{set {0, 1}} => T T }
                                                       0 1
                                                       2 2
                                      {set {0, 2}} => T T
                                                       0 2
                                                       2 2
                                      {set {0, 3}} => T T
                                                       0 3
                                                       2 2
                                      {set {1, 2}} => T T
                                                       1 2
                                                       2 2
                                      {set {1, 3}} => T T
                                                       1 3
                                                       2 2
                                      {set {2, 3}} => T T
                                                       2 3
i5 : Oneg1 = O1^(-1)

o5 = an equivariant K-class on a GKM variety 

o5 : KClass
i6 : peek Oneg1

o6 = KClass{variety => a GKM variety with an action of a 4-dimensional torus}
                                                       -1 -1
            KPolynomials => HashTable{{set {0, 1}} => T  T  }
                                                       0  1
                                                       -1 -1
                                      {set {0, 2}} => T  T
                                                       0  2
                                                       -1 -1
                                      {set {0, 3}} => T  T
                                                       0  3
                                                       -1 -1
                                      {set {1, 2}} => T  T
                                                       1  2
                                                       -1 -1
                                      {set {1, 3}} => T  T
                                                       1  3
                                                       -1 -1
                                      {set {2, 3}} => T  T
                                                       2  3

Caveat

$n$ is allowed to be negative only when $C$ is a line bundle, or a direct sum of copies of a line bundle.

See also