finiteDiagInvariants -- ring of invariants of a finite group action

Synopsis

• Usage:

  finiteDiagInvariants(U,R)

• Inputs:
  • a matrix, whose rows are the values of the indeterminates under the action of a 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 finite abelian group G acting diagonally on the surrounding polynomial ring K[X_1,\ldots,X_n]. The group is the direct product of cyclic groups generated by finitely many elements g_1,\ldots,g_w. The element g_i acts on the indeterminate X_j by g_i(X_j)= \lambda_i^{u_{ij}}X_jwhere \lambda_i is a primitive root of unity of order equal to ord(g_i). The ring of invariants is generated by all monomials satisfying the system u_{i1}a_1+...+u_{in} a_n \equiv \ 0 mod ord(g_i), i=1,\ldots,w. The input to the function is the w\times (n+1) matrix U with rows u_{i1} \ldots u_{in} ord(g_i), i=1,\ldots,w. The output is the monomial subalgebra of invariants R^G=\{f\in R : g_i f= f for all i=1,\ldots,w\}.

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

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

i2 : U=matrix{{1,1,1,1,5},{1,0,2,0,7}}

o2 = | 1 1 1 1 5 |
     | 1 0 2 0 7 |

              2        5
o2 : Matrix ZZ  <--- ZZ

i3 : finiteDiagInvariants(U,R)

         5   7 3   14    35     4     7 2     14   2 3   2 7    3 2   3 7   4    5     3      24       3   2 13   3 2   5   4   5     3   5 2   2   5 3      5 4    7 3   7   2   7 2    7 3   12   2   12        12 2    14    14    19    35
o3 = QQ[w , z w , z  w, z  , y*w , y*z w , y*z  , y w , y z w, y w , y z , y w, y , x*z w, x*z  , x*y*z , x z  , x z , x z*w , x y*z*w , x y z*w , x y z*w, x y z, x w , x y*w , x y w, x y , x  z*w , x  y*z*w, x  y z, x  w, x  y, x  z, x  ]

o3 : monomial subalgebra of R

See also

• diagInvariants -- ring of invariants of a diagonalizable group action
• torusInvariants -- ring of invariants of torus action