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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)

         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

Ways to use finiteDiagInvariants :