Strong Lefschetz elements of the coinvariant rings of finite Coxeter groups
A graded \(\mathbb{R}\)-algebra \(\bigoplus_{d=0}^{m} R_{d}\) with a symmetric Hilbert function (i.e., \(\dim_{\mathbb{R}} (R_{k}) = \dim_{\mathbb{R}} (R_{m-k})\)) is said to have the \textit{Strong Lefschetz Property} (SLP) if there is a \(\ell \in R_{1}\) such that \( \bullet \ell : R_{i} \rightarrow R_{m-i} \) (multiplication) is an isomorphism for \(0 \leq i \leq \lfloor m/2 \rfloor\). Such an element \(\ell\) is called a Strong Lefschetz Element (SLE). If \(X\) is a compact Kähler manifold of complex dimension \(m\), define \(R_{d} := H^{2d}(X; \mathbb{R})\) then the graded ring \(R = \bigoplus_{0}^{m} R_{d} \) with cup-product has SLP. In particular, it is a consequence of the Hard Lefschetz theorem, the Kähler class \([\omega]\) is a SLE. Let \(W\) be any finite Coxeter group, without an irreducible \(H_{4}\) component, acting on a real vector space \(V\). The authors consider the ring of \(W\) co-invariants given by \(R = \mathrm{Sym}(V^{\star}) / J_{+}\) where (\(J_{+} =\) the ideal of non-constant \(W\) invariant polynomials) and show that \(R\) has SLP (the case when \(W\) has irreducible \(H_{4}\) component has been considered in [\textit{Y. Numta} and \textit{A. Wachi}, J. Algebra 318, No. 2, 1032--1038 (2007; Zbl 1134.13003)]). They also characterize the set of all SLE in \(R\). The description is conjectural in the case when \(W\) has irreducible components of type \(H_{4}\). When \(W\) is a Weyl group, the proof proceeds by analyzing the geometry of the flag variety. In the case when \(W\) is \(I_{2}(m)\) or \(H_{3}\) explicit computations are used. When \(W_{S}\) is a parabolic subgroup of \(W\), the authors also consider the ring \(R^{W_{S}}\), of \(W_{S}\) invariants in \(R\) and characterize the set of Strong Lefschetz Elements in \(R^{W_{S}}\).
scientific article; zbMATH DE number 6054175
scientific article