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GKMVariety ** GKMVariety -- product of GKM varieties

Synopsis

Description

Given two GKM varieties $X$ and $Y$ with an action of a common torus $T$, the product is $X \times Y$ with the structure of a GKM variety given by the diagonal action of $T$. This method constructs $X \times Y$ as a GKMVariety. To speed up computation, this method does not automatically cache the moment graph of $X \times Y$. The user can cache this using the method MomentGraph ** MomentGraph.

The following example exhibits the product of $\mathbb P^1$ with the Lagrangian Grassmannian SpGr(2,4).

i1 : R = makeCharacterRing 2;
 
i2 : X = projectiveSpace(1,R);
 
i3 : Y = generalizedFlagVariety("C",2,{2},R);
 
i4 : Z = X ** Y;
 
i5 : peek Z

o5 = GKMVariety{cache => CacheTable{}                                                                                                                                                                                                     }
                characterRing => R
                charts => HashTable{(set {0}, {set {0, 1*}}) => {{-1, 1}, {0, 2}, {-1, 1}, {-2, 0}} }
                                    (set {0}, {set {0, 1}}) => {{-1, 1}, {0, -2}, {-2, 0}, {-1, -1}}
                                    (set {0}, {set {1*, 0*}}) => {{-1, 1}, {1, 1}, {2, 0}, {0, 2}}
                                    (set {0}, {set {1, 0*}}) => {{-1, 1}, {2, 0}, {1, -1}, {0, -2}}
                                    (set {1}, {set {0, 1*}}) => {{1, -1}, {0, 2}, {-1, 1}, {-2, 0}}
                                    (set {1}, {set {0, 1}}) => {{1, -1}, {0, -2}, {-2, 0}, {-1, -1}}
                                    (set {1}, {set {1*, 0*}}) => {{1, -1}, {1, 1}, {2, 0}, {0, 2}}
                                    (set {1}, {set {1, 0*}}) => {{1, -1}, {2, 0}, {1, -1}, {0, -2}}
                points => {(set {0}, {set {1, 0*}}), (set {0}, {set {1*, 0*}}), (set {1}, {set {0, 1*}}), (set {1}, {set {0, 1}}), (set {0}, {set {0, 1*}}), (set {0}, {set {0, 1}}), (set {1}, {set {1*, 0*}}), (set {1}, {set {1, 0*}})}

We can cache the moment graph of $Z$ as follows:

i6 : G = momentGraph X;
 
i7 : H = momentGraph Y;
 
i8 : momentGraph(Z, G** H);
 
i9 : peek Z

o9 = GKMVariety{cache => CacheTable{}                                                                                                                                                                                                     }
                characterRing => R
                charts => HashTable{(set {0}, {set {0, 1*}}) => {{-1, 1}, {0, 2}, {-1, 1}, {-2, 0}} }
                                    (set {0}, {set {0, 1}}) => {{-1, 1}, {0, -2}, {-2, 0}, {-1, -1}}
                                    (set {0}, {set {1*, 0*}}) => {{-1, 1}, {1, 1}, {2, 0}, {0, 2}}
                                    (set {0}, {set {1, 0*}}) => {{-1, 1}, {2, 0}, {1, -1}, {0, -2}}
                                    (set {1}, {set {0, 1*}}) => {{1, -1}, {0, 2}, {-1, 1}, {-2, 0}}
                                    (set {1}, {set {0, 1}}) => {{1, -1}, {0, -2}, {-2, 0}, {-1, -1}}
                                    (set {1}, {set {1*, 0*}}) => {{1, -1}, {1, 1}, {2, 0}, {0, 2}}
                                    (set {1}, {set {1, 0*}}) => {{1, -1}, {2, 0}, {1, -1}, {0, -2}}
                momentGraph => a "moment graph" on 8 vertices with 17 edges 
                points => {(set {0}, {set {1, 0*}}), (set {0}, {set {1*, 0*}}), (set {1}, {set {0, 1*}}), (set {1}, {set {0, 1}}), (set {0}, {set {0, 1*}}), (set {0}, {set {0, 1}}), (set {1}, {set {1*, 0*}}), (set {1}, {set {1, 0*}})}

See also

Ways to use this method: