Let $X$ be a generalized flag variety parameterizing flags of linear subspaces of dimensions $\{r_1, ... , r_k\}$ in $\mathbb C^n$ with $1 <= r_1 < \cdots < r_k$. Then a point $p$ of $X$ can be identified with a matrix $M$ of size $r_k \times n$ such that the first $r_i$ rows of $M$ spans a subspace of dimension $r_i$. Given $X$ and such a matrix $M$ representing the point $p$, this method computes the equivariant K-class of the closure of the torus orbit of $p$.
i1 : M = matrix(QQ,{{1,0,1,2},{0,1,2,1}})
o1 = | 1 0 1 2 |
| 0 1 2 1 |
2 4
o1 : Matrix QQ <-- QQ
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i2 : X1 = generalizedFlagVariety("A",3,{2})
o2 = a "GKM variety" with an action of a 4-dimensional torus
o2 : GKMVariety
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i3 : X2 = generalizedFlagVariety("C",2,{2})
o3 = a "GKM variety" with an action of a 2-dimensional torus
o3 : GKMVariety
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i4 : C1 = orbitClosure(X1,M)
o4 = an "equivariant K-class" on a GKM variety
o4 : KClass
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i5 : C2 = orbitClosure(X2,M)
o5 = an "equivariant K-class" on a GKM variety
o5 : KClass
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i6 : peek C1
o6 = KClass{variety => a "GKM variety" with an action of a 4-dimensional torus}
-1 -1
KPolynomials => HashTable{{set {0, 1}} => 1 - T T T T }
0 1 2 3
-1 -1
{set {0, 2}} => 1 - T T T T
0 1 2 3
-1 -1
{set {0, 3}} => 1 - T T T T
0 1 2 3
-1 -1
{set {1, 2}} => - T T T T + 1
0 1 2 3
-1 -1
{set {1, 3}} => - T T T T + 1
0 1 2 3
-1 -1
{set {2, 3}} => - T T T T + 1
0 1 2 3
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i7 : peek C2
o7 = KClass{variety => a "GKM variety" with an action of a 2-dimensional torus}
-2 2
KPolynomials => HashTable{{set {0, 1*}} => 1 - T T }
0 1
-2 -2
{set {0, 1}} => 1 - T T
0 1
2 2
{set {1*, 0*}} => - T T + 1
0 1
2 -2
{set {1, 0*}} => - T T + 1
0 1
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i8 : X = generalizedFlagVariety("A",3,{1,2})
o8 = a "GKM variety" with an action of a 4-dimensional torus
o8 : GKMVariety
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i9 : Mat = random(QQ^2,QQ^4)
o9 = | 9/2 9/4 1 3/2 |
| 1/2 1/2 3/4 3/4 |
2 4
o9 : Matrix QQ <-- QQ
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i10 : C = orbitClosure(X,Mat)
o10 = an "equivariant K-class" on a GKM variety
o10 : KClass
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i11 : FM = flagMatroid(Mat,{1,2})
o11 = a "flag matroid" with rank sequence {1, 2} on 4 elements
o11 : FlagMatroid
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i12 : C' = makeKClass(X,FM)
o12 = an "equivariant K-class" on a GKM variety
o12 : KClass
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i13 : C === C'
o13 = true
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In type "D", the orthogonal Grassmannian $SOGr(n,2n)$ has two connected components. To compute the torus orbit closure of a point $p$ it suffices to restrict to either $SOGr(n,n;2n)$ or $SOGr(n-1,n-1;2n)$, depending on which component $p$ is located in; see the last example in Example: generalized flag varieties for more details. Here is an example with $n=4$:
i14 : R = makeCharacterRing 4
o14 = R
o14 : PolynomialRing
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i15 : X1 = generalizedFlagVariety("D",4,{4,4},R)
o15 = a "GKM variety" with an action of a 4-dimensional torus
o15 : GKMVariety
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i16 : X2 = generalizedFlagVariety("D",4,{3,3},R)
o16 = a "GKM variety" with an action of a 4-dimensional torus
o16 : GKMVariety
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i17 : A = matrix{{1,3,-2,-1/4},{-1,-1,19,-61/4},{0,1,19,-73/4},{2,0,22,-89/4}};
4 4
o17 : Matrix QQ <-- QQ
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i18 : B = matrix(QQ,{{1,2,3,4},{5,6,7,8},{9,10,11,12},{13,14,15,16}});
4 4
o18 : Matrix QQ <-- QQ
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i19 : M = A | B
o19 = | 1 3 -2 -1/4 1 2 3 4 |
| -1 -1 19 -61/4 5 6 7 8 |
| 0 1 19 -73/4 9 10 11 12 |
| 2 0 22 -89/4 13 14 15 16 |
4 8
o19 : Matrix QQ <-- QQ
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i20 : assert(A* transpose(B) + B *transpose(A) == 0) -- verifying that M is isotropic
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i21 : C1 = orbitClosure(X1,M)
o21 = an "equivariant K-class" on a GKM variety
o21 : KClass
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i22 : C2 = orbitClosure(X2,M)
o22 = an "equivariant K-class" on a GKM variety
o22 : KClass
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i23 : peek C1
o23 = KClass{variety => a "GKM variety" with an action of a 4-dimensional torus }
-1 -1 -1 -1 -2 -2
KPolynomials => HashTable{{set {0, 1*, 2, 3*}} => 1 - T T T T - T T + T T T T }
0 1 2 3 0 2 0 1 2 3
-1 -1 -1 -1 -2 -2 -2 -2
{set {0, 1, 2, 3}} => 1 - 2T T T T + T T T T
0 1 2 3 0 1 2 3
-1 -1 -1 -1 -2 -2
{set {0, 2*, 1*, 3}} => 1 - T T T T - T T + T T T T
0 1 2 3 0 3 0 1 2 3
-1 -1 -1 -1 -2 -2
{set {0, 2*, 1, 3*}} => 1 - T T T T - T T + T T T T
0 1 2 3 0 1 0 1 2 3
-1 -1 -2 -2 -1 -1
{set {1*, 0*, 2, 3}} => - T T T T + 1 + T T T T - T T
0 1 2 3 0 1 2 3 2 3
-1 -1 -2 -2 -1 -1
{set {1, 0*, 2, 3*}} => - T T T T + 1 + T T T T - T T
0 1 2 3 0 1 2 3 1 2
{set {2*, 1*, 0*, 3*}} => 0
-1 -1 -2 -2 -1 -1
{set {2*, 1, 0*, 3}} => - T T T T + 1 + T T T T - T T
0 1 2 3 0 1 2 3 1 3
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i24 : peek C2 -- since the point corresponding to M lies in X1, C2 is just the empty class i.e. 0
o24 = KClass{KPolynomials => HashTable{{set {0, 1*, 2, 3}} => 0 } }
{set {0, 1, 2, 3*}} => 0
{set {0, 2*, 1*, 3*}} => 0
{set {0, 2*, 1, 3}} => 0
{set {1*, 0*, 2, 3*}} => 0
{set {1, 0*, 2, 3}} => 0
{set {2*, 1*, 0*, 3}} => 0
{set {2*, 1, 0*, 3*}} => 0
variety => a "GKM variety" with an action of a 4-dimensional torus
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