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 closue of the torus orbit of $p$.
The following example computes the torus orbit closure of a point in the standard Grassmannian $Gr(2,4)$ and in the Lagrangian Grassmannian $SpGr(2,4)$.
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
|
i2 : X1 = generalizedFlagVariety("A",3,{2})
o2 = a GKM variety with an action of a 4-dimensional torus
o2 : GKMVariety
|
i3 : X2 = generalizedFlagVariety("C",2,{2})
o3 = a GKM variety with an action of a 2-dimensional torus
o3 : GKMVariety
|
i4 : C1 = orbitClosure(X1,M) o4 = an equivariant K-class on a GKM variety o4 : KClass |
i5 : C2 = orbitClosure(X2,M) o5 = an equivariant K-class on a GKM variety o5 : KClass |
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
|
i7 : peek C2
o7 = KClass{variety => a GKM variety with an action of a 2-dimensional torus}
2 2
KPolynomials => HashTable{{set {0*, 1*}} => - T T + 1}
0 1
2 -2
{set {0*, 1}} => - T T + 1
0 1
-2 2
{set {0, 1*}} => 1 - T T
0 1
-2 -2
{set {0, 1}} => 1 - T T
0 1
|
In type "A", the equivariant K-class of the orbit closure of a point coincides with that of its flag matroid.
i8 : X = generalizedFlagVariety("A",3,{1,2})
o8 = a GKM variety with an action of a 4-dimensional torus
o8 : GKMVariety
|
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
|
i10 : C = orbitClosure(X,Mat) o10 = an equivariant K-class on a GKM variety o10 : KClass |
i11 : FM = flagMatroid(Mat,{1,2})
o11 = a flag matroid with rank sequence {1, 2} on 4 elements
o11 : FlagMatroid
|
i12 : C' = makeKClass(X,FM) o12 = an equivariant K-class on a GKM variety o12 : KClass |
i13 : C === C' o13 = true |
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 |
i15 : X1 = generalizedFlagVariety("D",4,{4,4},R)
o15 = a GKM variety with an action of a 4-dimensional torus
o15 : GKMVariety
|
i16 : X2 = generalizedFlagVariety("D",4,{3,3},R)
o16 = a GKM variety with an action of a 4-dimensional torus
o16 : GKMVariety
|
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
|
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
|
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
|
i20 : assert(A* transpose(B) + B *transpose(A) == 0) -- verifying that M is isotropic |
i21 : C1 = orbitClosure(X1,M) o21 = an equivariant K-class on a GKM variety o21 : KClass |
i22 : C2 = orbitClosure(X2,M) o22 = an equivariant K-class on a GKM variety o22 : KClass |
i23 : peek C1
o23 = KClass{KPolynomials => HashTable{{set {0*, 1*, 2*, 3*}} => 0 }}
-1 -1 -2 -2 -1 -1
{set {0*, 1*, 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 {0*, 1, 2*, 3}} => - T T T T + 1 + T T T T - T T
0 1 2 3 0 1 2 3 1 3
-1 -1 -2 -2 -1 -1
{set {0*, 1, 2, 3*}} => - T T T T + 1 + T T T T - T T
0 1 2 3 0 1 2 3 1 2
-1 -1 -1 -1 -2 -2
{set {0, 1*, 2*, 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, 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
{set {0, 1, 2*, 3*}} => 1 - T T T T - T T + T T T T
0 1 2 3 0 1 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
variety => a GKM variety with an action of a 4-dimensional torus
|
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*, 1, 2*, 3*}} => 0
{set {0*, 1, 2, 3}} => 0
{set {0, 1*, 2*, 3*}} => 0
{set {0, 1*, 2, 3}} => 0
{set {0, 1, 2*, 3}} => 0
{set {0, 1, 2, 3*}} => 0
variety => a GKM variety with an action of a 4-dimensional torus
|
By default the option RREFMethod is set to false. In this case the method produces the torus orbit closure by only computing the minors of the matrix. If the option RREFMethod is set to true, the method row reduces the matrix instead of computing its minors.
i25 : X = generalizedFlagVariety("A",3,{1,2,3})
o25 = a GKM variety with an action of a 4-dimensional torus
o25 : GKMVariety
|
i26 : Mat = random(QQ^3,QQ^4)
o26 = | 7/4 1/2 7 6/7 |
| 7/9 7/10 3/7 2/3 |
| 7/10 7/3 5/2 1 |
3 4
o26 : Matrix QQ <--- QQ
|
i27 : time C = orbitClosure(X,Mat)
-- used 0.811603 seconds
o27 = an equivariant K-class on a GKM variety
o27 : KClass
|
i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
-- used 1.7072 seconds
o28 = an equivariant K-class on a GKM variety
o28 : KClass
|
The object orbitClosure is a method function with options.