diagInvariants -- ring of invariants of a diagonalizable group action

Synopsis

• Usage:

  diagInvariants(T,U,R)

• Inputs:
  • a matrix, whose rows are the values of the indeterminates under the torus action
  • a matrix, whose rows are the values of the indeterminates under action of the finite group
  • a ring, the basering
• Outputs:
  • an instance of the type MonomialSubalgebra, the ring of invariants

Description

This function computes the ring of invariants of a diagonalizable group D = T\times G where T is a torus and G is a finite abelian group, both acting diagonally on the polynomial ring K[X_1,\ldots,X_n]. The group actions are specified by the input matrices M and N. The first matrix specifies the torus action, the second the action of the finite group. See torusInvariants or finiteDiagInvariants for more detail. The output is the monomial subalgebra of invariants.

This method can be used with the options allComputations and grading.

i1 : R=QQ[x,y,z,w];

i2 : T=matrix({{-1,-1,2,0},{1,1,-2,-1}});

              2        4
o2 : Matrix ZZ  <--- ZZ

i3 : U=matrix{{1,1,1,1,5},{1,0,2,0,7}}

o3 = | 1 1 1 1 5 |
     | 1 0 2 0 7 |

              2        5
o3 : Matrix ZZ  <--- ZZ

i4 : diagInvariants(T,U,R)

         70 35     19 10   4 6 5   15 5 10   26 4 15   37 3 20   48 2 25   59   30   70 35
o4 = QQ[y  z  , x*y  z  , x y z , x  y z  , x  y z  , x  y z  , x  y z  , x  y*z  , x  z  ]

o4 : monomial subalgebra of R

See also

• torusInvariants -- ring of invariants of torus action
• finiteDiagInvariants -- ring of invariants of a finite group action