Rings of invariants for modular representations of the Klein four group (Q2790708)

The present paper studies the rings of invariants of the indecomposable representations of the Klein four group over fields of characteristic 2. For each such representation \(V\) over a field \(\mathbb F\) the authors compute the Noether number and give minimal generating sets for the Hilbert ideal and the field of fractions. Recall that the maximal degree of a polynomial in a minimal homogeneous generating set for \(\mathbb F[V]^G\) is known as the Noether number of \(V\) (where \(\mathbb F[V]^G\) is the ring of invariant polynomials regarding a given group \(G\)). The ideal in \(\mathbb F[V]\) generated by the homogeneous invariants of positive degree is the Hilbert ideal of \(V\). If the characteristic of \(F\) divides \(|G|\), then \(V\) is called a modular representation. Computing rings of invariants for modular representations can be difficult even in basic cases. This paper investigates the rings of invariants of modular representations of the Klein four group \(G\cong \mathbb Z/2\times \mathbb Z/2\). In Section 3 is proven that the Noether number for the even dimensional representations is either \(3m-2\lfloor m/2\rfloor\) or \(3m-2\lceil m/2\rceil\). In Sections 4 and 5 is proven that the Noether number for the odd dimensional representations is either \(m+1\) or \(3m\). It is also shown that, with the exception of the regular representation, the Hilbert ideal for each of these representations is a complete intersection.