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MomentGraph ** MomentGraph -- the product of two moment graphs

Synopsis

Description

Let $G$ and $H$ be moment graphs associated to the GKM varieties $X$ and $Y$, respectively. This function produces the moment graph of $X ** Y$; the latter is a GKM variety via the diagonal action of the torus.

i1 : G = momentGraph projectiveSpace 1;
 
i2 : H = momentGraph generalizedFlagVariety("C",2,{2}); -- The isotropic Grassmannian SpGr(2,4)
 
i3 : J = G ** H;
 
i4 : peek J

o4 = MomentGraph{cache => CacheTable{}                                                                                                                                                                                                       }
                 edges => HashTable{{(set {0}, {set {0, 1*}}), (set {0}, {set {0, 1}})} => {0, 2}    }
                                    {(set {0}, {set {0, 1*}}), (set {0}, {set {1*, 0*}})} => {-2, 0}
                                    {(set {0}, {set {0, 1*}}), (set {0}, {set {1, 0*}})} => {-1, 1}
                                    {(set {0}, {set {0, 1*}}), (set {1}, {set {0, 1*}})} => {-1, 1}
                                    {(set {0}, {set {0, 1*}}), (set {1}, {set {1, 0*}})} => {-1, 1}
                                    {(set {0}, {set {0, 1}}), (set {0}, {set {1*, 0*}})} => {-1, -1}
                                    {(set {0}, {set {0, 1}}), (set {0}, {set {1, 0*}})} => {-2, 0}
                                    {(set {0}, {set {0, 1}}), (set {1}, {set {0, 1}})} => {-1, 1}
                                    {(set {0}, {set {1*, 0*}}), (set {1}, {set {1*, 0*}})} => {-1, 1}
                                    {(set {0}, {set {1, 0*}}), (set {0}, {set {1*, 0*}})} => {0, -2}
                                    {(set {0}, {set {1, 0*}}), (set {1}, {set {1, 0*}})} => {-1, 1}
                                    {(set {1}, {set {0, 1*}}), (set {1}, {set {0, 1}})} => {0, 2}
                                    {(set {1}, {set {0, 1*}}), (set {1}, {set {1*, 0*}})} => {-2, 0}
                                    {(set {1}, {set {0, 1*}}), (set {1}, {set {1, 0*}})} => {-1, 1}
                                    {(set {1}, {set {0, 1}}), (set {1}, {set {1*, 0*}})} => {-1, -1}
                                    {(set {1}, {set {0, 1}}), (set {1}, {set {1, 0*}})} => {-2, 0}
                                    {(set {1}, {set {1, 0*}}), (set {1}, {set {1*, 0*}})} => {0, -2}
                 HTpt => QQ[t ..t ]
                             0   1
                 vertices => {(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: