Invariants of Finite Groups and the Weak Lefschetz Property

The aim of this chapter is twofold. First, to investigate and to provide new examples of GT–systems and GT-surfaces with a non abelian finite group \(\Lambda \subset \mathrm {SL}(3,\mathbb {K})\) of order \({|\Lambda |}\) . In Blichfeldt et al. (Theory and Applications of Finite Groups. Wiley, New York, 1916) and Yau and Yu (Memoirs Am. Math. Soc. 505:1993), finite subgroups of \(\mathrm {SL}(3,\mathbb {K})\) are classified. Following this classification, we determine which groups \(\Lambda \subset \mathrm {SL}(3,\mathbb {K})\) give rise to a Togliatti system, i.e. the ideal generated by the invariants of \(\Lambda \) of degree \({|\Lambda |}\) is an artinian ideal and fails the WLP in degree \({|\Lambda |}-1\) . Second, to gather a family of monomial Togliatti systems directly related to the invariants of a finite abelian subgroup of \(\mathrm {SL}(3,\mathbb {K})\) , introduced in Colarte-Gómez et al. (Ann. Mat. Pura Appl. 200:1757–1780, 2021); and the first large family of non monomial GT-systems appearing in the literature and introduced in Colarte-Gómez et al. (Isr. J. Math. 247:195–215, 2022). We collect some results regarding the surfaces associated with these ideals as well.