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flagMap -- creates equivariant maps between generalized flag varieties

Synopsis

Description

Let $L =\{k_1,\dots,k_m\}$ be a set of ranks of linear subscpaces of $\mathbb C^n$ and consider a subset $L' \subseteq L$. Let $X = Fl(L; n)$ and $Y=Fl(L';n)$ be the associated generalized flag varieties (if they exist). This method produces the canonical projection from $X$ to $Y$ that forgets the linear subspaces having ranks $L \setminus L'$.

i1 : R = makeCharacterRing 3

o1 = R

o1 : PolynomialRing
 
i2 : X = generalizedFlagVariety("B",3,{1,2},R);
 
i3 : Y1 = generalizedFlagVariety("B",3,{2},R);
 
i4 : Y2 = generalizedFlagVariety("B",3,{1},R);
 
i5 : peek flagMap(X,Y1)

o5 = EquivariantMap{cache => CacheTable{}                                            }
                    ptsMap => HashTable{{set {0*}, set {1*, 0*}} => {set {1*, 0*}}}
                                        {set {0*}, set {1, 0*}} => {set {1, 0*}}
                                        {set {0*}, set {2*, 0*}} => {set {2*, 0*}}
                                        {set {0*}, set {2, 0*}} => {set {2, 0*}}
                                        {set {0}, set {0, 1*}} => {set {0, 1*}}
                                        {set {0}, set {0, 1}} => {set {0, 1}}
                                        {set {0}, set {0, 2}} => {set {0, 2}}
                                        {set {0}, set {2*, 0}} => {set {2*, 0}}
                                        {set {1*}, set {0, 1*}} => {set {0, 1*}}
                                        {set {1*}, set {1*, 0*}} => {set {1*, 0*}}
                                        {set {1*}, set {1*, 2}} => {set {1*, 2}}
                                        {set {1*}, set {2*, 1*}} => {set {2*, 1*}}
                                        {set {1}, set {0, 1}} => {set {0, 1}}
                                        {set {1}, set {1, 0*}} => {set {1, 0*}}
                                        {set {1}, set {1, 2}} => {set {1, 2}}
                                        {set {1}, set {2*, 1}} => {set {2*, 1}}
                                        {set {2*}, set {2*, 0*}} => {set {2*, 0*}}
                                        {set {2*}, set {2*, 0}} => {set {2*, 0}}
                                        {set {2*}, set {2*, 1*}} => {set {2*, 1*}}
                                        {set {2*}, set {2*, 1}} => {set {2*, 1}}
                                        {set {2}, set {0, 2}} => {set {0, 2}}
                                        {set {2}, set {1*, 2}} => {set {1*, 2}}
                                        {set {2}, set {1, 2}} => {set {1, 2}}
                                        {set {2}, set {2, 0*}} => {set {2, 0*}}
                    source => a "GKM variety" with an action of a 3-dimensional torus
                    target => a "GKM variety" with an action of a 3-dimensional torus
 
i6 : peek flagMap(X,Y2)

o6 = EquivariantMap{cache => CacheTable{}                                            }
                    ptsMap => HashTable{{set {0*}, set {1*, 0*}} => {set {0*}}}
                                        {set {0*}, set {1, 0*}} => {set {0*}}
                                        {set {0*}, set {2*, 0*}} => {set {0*}}
                                        {set {0*}, set {2, 0*}} => {set {0*}}
                                        {set {0}, set {0, 1*}} => {set {0}}
                                        {set {0}, set {0, 1}} => {set {0}}
                                        {set {0}, set {0, 2}} => {set {0}}
                                        {set {0}, set {2*, 0}} => {set {0}}
                                        {set {1*}, set {0, 1*}} => {set {1*}}
                                        {set {1*}, set {1*, 0*}} => {set {1*}}
                                        {set {1*}, set {1*, 2}} => {set {1*}}
                                        {set {1*}, set {2*, 1*}} => {set {1*}}
                                        {set {1}, set {0, 1}} => {set {1}}
                                        {set {1}, set {1, 0*}} => {set {1}}
                                        {set {1}, set {1, 2}} => {set {1}}
                                        {set {1}, set {2*, 1}} => {set {1}}
                                        {set {2*}, set {2*, 0*}} => {set {2*}}
                                        {set {2*}, set {2*, 0}} => {set {2*}}
                                        {set {2*}, set {2*, 1*}} => {set {2*}}
                                        {set {2*}, set {2*, 1}} => {set {2*}}
                                        {set {2}, set {0, 2}} => {set {2}}
                                        {set {2}, set {1*, 2}} => {set {2}}
                                        {set {2}, set {1, 2}} => {set {2}}
                                        {set {2}, set {2, 0*}} => {set {2}}
                    source => a "GKM variety" with an action of a 3-dimensional torus
                    target => a "GKM variety" with an action of a 3-dimensional torus

See also

Ways to use flagMap :

For the programmer

The object flagMap is a method function.