equivariantHilbertSeries -- equivariant Hilbert series for a diagonal action

Synopsis

• Usage:
equivariantHilbertSeries D
• Inputs:
• Optional inputs:
• Order (missing documentation) => an integer, default value infinity, stopping degree of the equivariant Hilbert series
• Outputs:
• , the equivariant Hilbert series

Description

This function is provided by the package InvariantRing.

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

Ways to use equivariantHilbertSeries :

• "equivariantHilbertSeries(DiagonalAction)"

For the programmer

The object equivariantHilbertSeries is .