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InvariantRing :: equivariantHilbert

equivariantHilbert -- stores equivariant Hilbert series expansions

Description

This key is created in the cache table of a DiagonalAction upon calling equivariantHilbertSeries with the Order option. It stores partial expansions of the equivariant Hilbert series to avoid computing them again.

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 : T.cache.?equivariantHilbert

o4 = false
i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
 -- 0.0453672 seconds elapsed

                  -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
     z  z   + z  z  + z  z   + z  z   + z  )T
      0  1     0  1    0  1     0  1     0

o5 : ZZ[z ..z ][T]
         0   1
i6 : T.cache.?equivariantHilbert

o6 = true
i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
 -- 0.000668953 seconds elapsed

For the programmer

The object equivariantHilbert is a symbol.