Properties of Invariant Rings#

The following functions return non-trivial structural properties of invariant rings of finite groups.

HilbertSeries(R): RngInvar -> FldFunUElt#: The Hilbert series of the invariant ring \(R=K[V]^G\), returned as an element of the rational function field \({\mathbb{Z}}(t)\). The Molien series of \(G\) will be used if possible; otherwise (the modular matrix group case) secondary invariants for \(R\) will be constructed to determine the result.

HilbertSeriesApproximation(R, n): RngInvar, RngIntElt -> RngSerLaurElt#: The Hilbert series of the invariant ring \(R=K[V]^G\), returned as a Laurent series with \(n\) known terms. The conjugacy classes of \(G\) will be used to compute the approximation.

IsCohenMacaulay(R): RngInvar -> BoolElt#: Given the invariant ring \(R=K[V]^G\) of the group \(G\) over the field \(K\), return true iff \(R\) is Cohen-Macaulay. This is always true in the non-modular case. Otherwise, secondary invariants for \(R\) will be constructed to determine the result.

FreeResolution(R): RngInvar -> [ ModMPol ]#: Given the invariant ring \(R=K[V]^G\) of the group \(G\) over the field \(K\), return a free resolution of (the module of) \(R\). This is just the same as the invocation FreeResolution(Module(R)). The free resolution is returned as a sequence \(F\) such that \(F[1]\) is \(M\), \(F[i+1]\) is the syzygy module of \(F[i]\) for \(i<\#F\), and the last element of \(F\) is free (its basis has no syzygies).

MinimalFreeResolution(R): RngInvar -> [ ModMPol ]#: Given the invariant ring \(R=K[V]^G\) of the group \(G\) over the field \(K\), return a minimal free resolution of (the module of) \(R\). This is just the same as the invocation MinimalFreeResolution(Module(R)).

HomologicalDimension(R): RngInvar -> RngInt#: Given the invariant ring \(R=K[V]^G\) of the group \(G\) over the field \(K\), return the homological dimension of \(R\). This is just the length of a minimal free resolution of \(R\) minus 1 (taking account of the fact that the module \(M\) of \(R\) is always included in the free resolution).

Depth(R): RngInvar -> RngIntElt#: Given the invariant ring \(R=K[V]^G\) of the group \(G\) over the field \(K\), return the depth of \(R\). This is \(n-d\) by the Auslander-Buchsbaum formula, where \(n\) is the rank of \(R\) and \(d\) is the homological dimension of \(R\).

Example: Depth (ex-6d79cb)#

We construct a minimal free resolution of the invariant ring of the group generated by the degree-5 Jordan block over \({\bf F}_{2}\) and verify that the depth is 3.

> K:=GF(2);
> G := MatrixGroup<5,K | [1,0,0,0,0, 1,1,0,0,0, 0,1,1,0,0,
>                         0,0,1,1,0, 0,0,0,1,1]>;
> R := InvariantRing(G);
> time F := MinimalFreeResolution(R);
Time: 0.690
> F;
Chain complex with terms of degree 3 down to -1
Dimensions of terms: 0 1 7 22 0
> Depth(R);
3
> HomologicalDimension(R);
2

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Sections Steenrod Operations and Minimalization and Homogeneous Module Testing present functions whose scope is not limited to the context of invariant theory.