Package invarFC0

This article is about a function from ApCoCoA-2.

This page is about the InvFC0 package, which is about invariant theory for finite groups in characteristic 0. For a complete function list, see Category:Package invarFC0.

Mathematical Definitions

Let $P=\mathbb {Q} [x_{1},\ldots ,x_{n}]$ be the polynomial ring over the rational numbers in n indeterminates and let ${\bf {x}}=(x_{1},\ldots ,x_{n})^{\rm {tr}}$ be the column vector containing the indeterminates of P. Then a matrix group $G\subseteq {\rm {GL}}_{n}(\mathbb {Q} )$ acts on $P$ via

$G\times P\to P,({\mathcal {A}},f)\mapsto {\mathcal {A}}f:=f({\mathcal {A}}\cdot {\textbf {x}})$

where

$f({\mathcal {A}}\cdot {\bf {x}})=f(a_{11}x_{1}+\cdots +a_{1n}x_{n},\ldots ,a_{n1}x_{1}+\cdots +a_{nn}x_{n})$

for ${\mathcal {A}}=(a_{ij})\in G$. David Hilbert has proven in 1890 that the set

$P^{G}=\{f\in P\mid {\mathcal {A}}f=f{\text{ for all }}f\in P\}$

is a finitely generated graded Q-subalgebra of P as long as G is a finite group. This package provides a function for computing a minimal generating set of this subalgebra and a function for computing a SAGBI basis of it.

For details about the algorithms, see Harm Derksen, Gregor Kemper: Computational invariant theory or B. Andraschko, Computational Invariant Theory For Finite Matrix Groups.

Package Description

Given a list G of matrices over a polynomial ring P that form a multiplicative group and a positive integer d, one can call the functions

InvFC0.SAGBI(G);
InvFC0.SAGBI(G,d);
InvFC0.MinGens(G);
InvFC0.MinGens(G,d);

In the first case, a SAGBI basis of $P^{G}$ is computed if a finite one exists. If no finite SAGBI basis exists, this function does not terminate, therefore one can use the second call to specify a degree d such that a d-truncated SAGBI basis is computed. Note that as it is the case for Gröbner bases, the term ordering is given by the ring P.

The last two calls work in the same way, but produce a minimal homogeneous system of generators for $P^{G}$. By Hilbert's finiteness theorem, both of them terminate, but it is also possible to specify a degree to obtain a truncated homogeneous generating system.