The strong Lefschetz property of the coinvariant ring of the Coxeter group of type H4

We prove that the coinvariant ring of the irreducible Coxeter group of type H4 has the strong Lefschetz property.Comment: 9 page

Graph presentations for moments of noncentral Wishart distributions and their applications

We provide formulas for the moments of the real and complex noncentral Wishart distributions of general degrees. The obtained formulas for the real and complex cases are described in terms of the undirected and directed graphs, respectively. By considering degenerate cases, we give explicit formulas for the moments of bivariate chi-square distributions and $2\times 2$ Wishart distributions by enumerating the graphs. Noting that the Laguerre polynomials can be considered to be moments of a noncentral chi-square distributions formally, we demonstrate a combinatorial interpretation of the coefficients of the Laguerre polynomials