# flagMap -- creates equivariant maps between generalized flag varieties

## Synopsis

• Usage:
f = flagMap(X,Y)
• Inputs:
• X, , the source generalized flag variety
• Y, , the target generalized flag variety
• Outputs:
• f, ,

## 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 {0*, 1*}} => {set {0*, 1*}}} {set {0*}, set {0*, 1}} => {set {0*, 1}} {set {0*}, set {0*, 2*}} => {set {0*, 2*}} {set {0*}, set {0*, 2}} => {set {0*, 2}} {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 {0, 2}} => {set {0, 2}} {set {1*}, set {0*, 1*}} => {set {0*, 1*}} {set {1*}, set {0, 1*}} => {set {0, 1*}} {set {1*}, set {1*, 2*}} => {set {1*, 2*}} {set {1*}, set {1*, 2}} => {set {1*, 2}} {set {1}, set {0*, 1}} => {set {0*, 1}} {set {1}, set {0, 1}} => {set {0, 1}} {set {1}, set {1, 2*}} => {set {1, 2*}} {set {1}, set {1, 2}} => {set {1, 2}} {set {2*}, set {0*, 2*}} => {set {0*, 2*}} {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 {0*, 2}} => {set {0*, 2}} {set {2}, set {0, 2}} => {set {0, 2}} {set {2}, set {1*, 2}} => {set {1*, 2}} {set {2}, set {1, 2}} => {set {1, 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 i6 : peek flagMap(X,Y2) o6 = EquivariantMap{cache => CacheTable{} } ptsMap => HashTable{{set {0*}, set {0*, 1*}} => {set {0*}}} {set {0*}, set {0*, 1}} => {set {0*}} {set {0*}, set {0*, 2*}} => {set {0*}} {set {0*}, set {0*, 2}} => {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 {0, 2}} => {set {0}} {set {1*}, set {0*, 1*}} => {set {1*}} {set {1*}, set {0, 1*}} => {set {1*}} {set {1*}, set {1*, 2*}} => {set {1*}} {set {1*}, set {1*, 2}} => {set {1*}} {set {1}, set {0*, 1}} => {set {1}} {set {1}, set {0, 1}} => {set {1}} {set {1}, set {1, 2*}} => {set {1}} {set {1}, set {1, 2}} => {set {1}} {set {2*}, set {0*, 2*}} => {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 {0*, 2}} => {set {2}} {set {2}, set {0, 2}} => {set {2}} {set {2}, set {1*, 2}} => {set {2}} {set {2}, set {1, 2}} => {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

## Ways to use flagMap :

• "flagMap(GKMVariety,GKMVariety)"

## For the programmer

The object flagMap is .