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InvariantRing :: equivariantHilbertSeries

equivariantHilbertSeries -- equivariant Hilbert series for a diagonal action

Synopsis

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 :

For the programmer

The object equivariantHilbertSeries is a method function with options.