# finiteDiagInvariants -- ring of invariants of a finite group action

## Synopsis

• Usage:
finiteDiagInvariants(U,R)
• Inputs:
• , whose rows are the values of the indeterminates under the action of a finite group
• a ring, the basering
• Outputs:

## 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