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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}.

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)

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

o3 : monomial subalgebra of R

See also

Ways to use finiteDiagInvariants :