Introduction

The invariant theory part of OSCAR provides functionality for computing polynomial invariants of group actions, focusing on finite and linearly reductive groups, respectively.

The basic setting in this context consists of a group $G$, a field $K$, a vector space $V$ over $K$ of finite dimension $n,$ and a representation $\rho: G \to \text{GL}(V)$ of $G$ on $V$. The induced action on the dual vector space $V^\ast$,

\[V^\ast \times G \rightarrow V^\ast, (f, \pi)\mapsto f \;\! . \;\! \pi := f\circ \rho(\pi),\]

extends to an action of $G$ on the graded symmetric algebra

\[K[V]:=S(V^*)=\bigoplus_{d\geq 0} S^d V^*\]

which preserves the grading.

The invariants of $G$ are the fixed points of this action, its invariant ring is the graded subalgebra

\[K[V]^G:=\{f\in K[V] \mid f \;\! . \;\! \pi =f {\text { for any }} \pi\in G\} \subset K[V].\]

Explicitly, the choice of a basis of $V$ and its dual basis, say, $\{x_1, \dots, x_n\}$ of $V^*$ gives rise to isomorphisms $\text{GL}(V) \cong \text{GL}_n(K)$ and $K[V]\cong K[x_1, \dots, x_n]$. After identifying $\text{GL}(V)$ with $\text{GL}_n(K)$ and $K[V]$ with $K[x_1, \dots, x_n]$ by means of these isomorphisms, the action of $G$ on $K[V]$ is given as follows:

\[(f \;\! . \;\! \pi) (x_1, \dots, x_n) = f((x_1, \dots, x_n) \cdot \rho(\pi)).\]

Accordingly, $K[V]^G$ may be regarded as a graded subalgebra of $K[x_1, \dots, x_n]$:

\[K[V]^G \cong K[x_1, \dots, x_n]^G :=\{f\in K[x_1, \dots, x_n] \mid f \;\! . \;\! \pi =f {\text { for any }} \pi\in G\}.\]

The main objective of invariant theory in OSCAR is the computation of $K$-algebra generators for invariant rings.

Note

If $K[V]^G$ is finitely generated as a $K$-algebra, then any minimal system of homogeneous generators is called a fundamental system of invariants for $K[V]^G$. By Nakayama's lemma, the number of elements in such a system is uniquely determined as the embedding dimension of $K[V]^G$. Similarly, the degrees of these elements are uniquely determined.

Note

If $K[V]^G$ is finitely generated as a $K$-algebra, then $K[V]^G$ admits a graded Noether normalization, that is, a Noether normalization $K[p_1, \dots, p_m] \subset K[V]^G$ with $p_1, \dots, p_m$ homogeneous. Given any such Noether normalization, $p_1, \dots, p_m$ is called a homogeneous system of parameters or a system of primary invariants for $K[V]^G$, and any minimal system $s_0=1, s_1,\dots, s_l$ of homogeneous generators of $K[V]^G$ as a $K[p_1, \dots, p_m]$-module is called a system of secondary invariants for $K[V]^G$ with respect to $p_1, \dots, p_m$. A secondary invariant $s_i\neq 1$ 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 $K[V]^G$ as a $K[p_1, \dots, p_m]$-algebra. Somewhat abusing notation, we call every minimal system of homogeneous generators for $K[V]^G$ as a $K[p_1, \dots, p_m]$-algebra a system of irreducible secondary invariants.

Note

For the invariant rings handled by OSCAR, the assumption that $K[V]^G$ is finitely generated as a $K$-algebra will be guaranteed by theoretical results. In addition, where not mentioned otherwise, the following will hold:

  • There exists a Reynolds operator $\mathcal R: K[V] \to K[V]$. That is, $\mathcal R$ is a $K$-linear graded map which projects $K[V]$ onto $K[V]^G$, and which is a $K[V]^G$-module homomorphism.
  • The ring $K[V]^G$ is Cohen-Macaulay. Equivalently, $K[V]^G$ is a free module (of finite rank) over any of its graded Noether normalizations.

The textbook

and the survey article

provide details on theory and algorithms as well as references.