Introduction
The invariant theory part of OSCAR provides functionality for computing polynomial invariants of group actions, focusing on finite groups, tori, and linearly reductive groups, respectively.
The basic setting in this context consists of a group , a field , a vector space over of finite dimension and a representation of on . The induced right action on the dual vector space ,
extends to a right action of on the graded symmetric algebra
which preserves the grading.
In OSCAR, group actions are by convention assumed to be right actions and we follow this convention with our definition above. Note, however, that the left action given by is quite common in the literature.
The invariants of are the fixed points of the action defined above, its invariant ring is the graded subalgebra
Explicitly, fixing a basis of and its dual basis, say, of , we may identify and . Then the action of an element with on a polynomial is given as follows:
Accordingly, may be regarded as a graded subalgebra of :
The main objective of invariant theory in OSCAR is the computation of -algebra generators for invariant rings.
If is finitely generated as a -algebra, then any minimal system of homogeneous generators is called a fundamental system of invariants for . By Nakayama's lemma, the number of elements in such a system is uniquely determined as the embedding dimension of . Similarly, the degrees of these elements are uniquely determined.
If is finitely generated as a -algebra, then admits a graded Noether normalization, that is, a Noether normalization with homogeneous. Given any such Noether normalization, is called a homogeneous system of parameters or a system of primary invariants for , and any minimal system of homogeneous generators of as a -module is called a system of secondary invariants for with respect to . A secondary invariant is called irreducible if it cannot be written as a polynomial expression in the primary invariants and the other secondary invariants. The irreducible secondary invariants form a minimal system of homogeneous generators for as a -algebra. Somewhat abusing notation, we call every minimal system of homogeneous generators for as a -algebra a system of irreducible secondary invariants.
For the invariant rings handled by OSCAR, the assumption that is finitely generated as a -algebra will be guaranteed by theoretical results. In addition, where not mentioned otherwise, the following will hold:
- There exists a Reynolds operator . That is, is a -linear graded map which projects onto , and which is a -module homomorphism.
- The ring is Cohen-Macaulay. Equivalently, is a free module (of finite rank) over any of its graded Noether normalizations.
The textbook
- [DK15]
and the survey article
- [DJ98]
provide details on theory and algorithms as well as references.
Contact
Please direct questions about this part of OSCAR to the following people:
You can ask questions in the OSCAR Slack.
Alternatively, you can raise an issue on github.