# orbitClosure -- computes the equivariant K-class of a torus orbit closure of a point in a generalized flag variety

## Synopsis

• Usage:
C = orbitClosure(X,M)
• Inputs:
• X, ,
• M, , representing a point in a generalized flag variety
• Optional inputs:
• RREFMethod => ..., default value false
• Outputs:
• C, ,

## Description

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