For a torus acting on the vector space $V$ this function returns the equivariant Hilbert series of the coordinate ring $K[V]$ as a rational function of $z_0, \ldots, z_{r-1}, t$ where $r$ is the rank of the torus. The series in $t$ which is the coefficient of $z_0^0\cdots z_{r-1}^0$ gives the ordinary Hilbert series of $K[V]^T$ . The option Order => N can be used to compute the series up to the $t$ -degree N-1.
Here is an example of a rank 2 torus acting on a polynomial ring in 3 variables, whose invariant ring is generated by the single element $x_1 x_2 x_3$ .
i1 : R = QQ[x_1..x_3] o1 = R o1 : PolynomialRing |
i2 : W = matrix{{-1,0,1},{0,-1,1}}
o2 = | -1 0 1 |
| 0 -1 1 |
2 3
o2 : Matrix ZZ <--- ZZ
|
i3 : T = diagonalAction(W, R)
* 2
o3 = R <- (QQ ) via
| -1 0 1 |
| 0 -1 1 |
o3 : DiagonalAction
|
i4 : equivariantHilbertSeries T
1
o4 = -------------------------------
-1 -1
(1 - z T)(1 - z T)(1 - z z T)
0 1 0 1
o4 : Expression of class Divide
|
i5 : S = equivariantHilbertSeries(T, Order => 7)
-1 -1 2 2 -2 -1 -1 -2 2
o5 = 1 + (z z + z + z )T + (z z + z + z + z + z z + z )T +
0 1 1 0 0 1 0 1 1 0 1 0
------------------------------------------------------------------------
3 3 2 2 -1 -3 -1 -1 -2 -2 -1 -3 3
(z z + z z + z z + z z + 1 + z + z z + z z + z z + z )T
0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0
------------------------------------------------------------------------
4 4 3 2 2 3 2 -2 2 -1 -4 -1
+ (z z + z z + z z + z + z z + z z + z + z + z + z +
0 1 0 1 0 1 0 0 1 0 1 1 1 1 0
------------------------------------------------------------------------
-1 -3 -2 -2 -2 -3 -1 -4 4 5 5 4 3 3 4 3
z z + z z + z z + z z + z )T + (z z + z z + z z + z z +
0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1
------------------------------------------------------------------------
2 2 2 -1 3 -3 -2 -5 -1 2 -1 -1
z z + z z + z z + z + z z + z + z + z + z z + z z +
0 1 0 1 0 1 0 0 1 1 1 1 0 1 0 1
------------------------------------------------------------------------
-1 -4 -2 -2 -3 -3 -3 -2 -4 -1 -5 5 6 6 5 4
z z + z + z z + z z + z z + z z + z )T + (z z + z z
0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1
------------------------------------------------------------------------
4 5 4 2 3 3 3 2 4 2 2 -2 2 -1 -4
+ z z + z z + z z + z + z z + z z + z z + z z + z z + z z +
0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1
------------------------------------------------------------------------
3 -3 -6 -1 -1 -2 -1 -5 -2 2 -2 -1 -2 -4
z + 1 + z + z + z z + z z + z z + z z + z z + z z +
1 1 1 0 1 0 1 0 1 0 1 0 1 0 1
------------------------------------------------------------------------
-3 -3 -3 -4 -4 -2 -5 -1 -6 6
z + z z + z z + z z + z z + z )T
0 0 1 0 1 0 1 0 1 0
o5 : ZZ[z ..z ][T]
0 1
|
i6 : sub(S, {z_0 => 0, z_1 => 0})
3 6
o6 = 1 + T + T
o6 : ZZ[z ..z ][T]
0 1
|
The object equivariantHilbertSeries is a method function with options.