Algebraic Group Actions on Polynomials

In the invariant theory package of Magma, a linear algebraic group G is given by polynomials, say in variables t₁, ..., t_m, defined over some field K that is representable in Magma, as the affine variety over the algebraic closure bar(K) of K given by these polynomials. A G-module is given by a matrix A ∈K[t₁, ..., t_m]^{n x n} such that a group element (η₁, ..., η_m) ∈G acts on bar(K)ⁿ by the matrix obtained by substituting (η₁, ..., η_m) into the polynomials occurring in the matrix A.

G then also acts on the ring of polynomials on bar(K)ⁿ by

σ(f) = f σ ^{- 1}

for σ ∈G and f ∈bar(K)[x₁, ..., x_n]. Since the algorithms in Magma do not work with the algebraic closure, single group elements are never dealt with. In fact, all relevant algorithms ony involve field elements of K, the field of definition. [Next][Prev] [Right] [Left] [Up] [Index] [Root]