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

- Operator: ^
- Usage:
    `C^n`
- Inputs:
  - C, an equivariant K-class,
  - n, an integer,
- Outputs:
  - an equivariant K-class, the n-th power of C

## 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

- makeKClass -- constructs an equivariant K-class
- KClass * KClass -- computes the product of two equivariant K-classes
- KClass + KClass -- computes the sum of two equivariant K-classes

## Ways to use this method:

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

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GKMVarieties/Documentation_GKMVarieties.m2:740:0.