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GKMVarieties :: makeKClass(GKMVariety,FlagMatroid)

makeKClass(GKMVariety,FlagMatroid) -- the equivariant K-class of a flag matroid

Synopsis

Description

A flag matroid of whose constituent matroids have ranks $r_1, \ldots, r_k$ and ground set size $n$ defines a KClass on the (partial) flag variety $Fl(r_1,\ldots, r_k;n)$. When the flag matroid arises from a matrix representing a point on the (partial) flag variety, this equivariant K-class coincides with that of the structure sheaf of its torus orbit closure. See [CDMS18] or [DES20].

i1 : X = generalizedFlagVariety("A",2,{1,2})

o1 = a GKM variety with an action of a 3-dimensional torus

o1 : GKMVariety
i2 : A = matrix{{1,2,3},{0,2,3}}

o2 = | 1 2 3 |
     | 0 2 3 |

              2        3
o2 : Matrix ZZ  <--- ZZ
i3 : FM = flagMatroid(A,{1,2})

o3 = a flag matroid with rank sequence {1, 2} on 3 elements 

o3 : FlagMatroid
i4 : C1 = makeKClass(X,FM)

o4 = an equivariant K-class on a GKM variety 

o4 : KClass
i5 : C2 = orbitClosure(X,A)

o5 = an equivariant K-class on a GKM variety 

o5 : KClass
i6 : C1 === C2

o6 = true

See also