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Normaliz :: finiteDiagInvariants

finiteDiagInvariants -- ring of invariants of a finite group action

Synopsis

Description

This function computes the ring of invariants of a finite abelian group G acting diagonally on the surrounding polynomial ring K[X1,...,Xn]. The group is the direct product of cyclic groups generated by finitely many elements g1,...,gw. The element gi acts on the indeterminate Xj by gi(Xj)= λiuijXjwhere λi is a primitive root of unity of order equal to ord(gi). The ring of invariants is generated by all monomials satisfying the system ui1a1+...+uin an ≡ 0 mod ord(gi), i=1,...,w. The input to the function is the w×(n+1) matrix U with rows ui1 ...uin ord(gi), i=1,...,w. The output is the monomial subalgebra of invariants RG={f∈R : gi f= f for all i=1,...,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)

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

o3 : monomial subalgebra of R

See also

Ways to use finiteDiagInvariants :