This function computes the ring of invariants of a finite abelian group G acting diagonally on the surrounding polynomial ring K[X
1,...,X
n]. The group is the direct product of cyclic groups generated by finitely many elements g
1,...,g
w. The element g
i acts on the indeterminate X
j by g
i(X
j)= λ
iuijX
jwhere λ
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
i1a
1+...+u
in a
n ≡ 0 mod ord(g
i), i=1,...,w. The input to the function is the w×(n+1) matrix U with rows u
i1 ...u
in ord(g
i), i=1,...,w. The output is the monomial subalgebra of invariants R
G={f∈R : g
i f= f for all i=1,...,w}.
R=QQ[x,y,z,w]; |
U=matrix{{1,1,1,1,5},{1,0,2,0,7}} |
finiteDiagInvariants(U,R) |