We compute the representation zeta functions of some finitely generated nilpotent groups associated to unipotent group schemes over rings of integers in number fields. These group schemes are defined by Lie lattices whose presentations are modelled on certain prehomogeneous vector spaces. Our method is based on evaluating p-adic integrals associated to certain rank varieties of linear forms.

Suppose $G$ is a split connected‎ ‎reductive orthogonal or symplectic group over an infinite field‎ ‎$F,$ $P=MN$ is a maximal parabolic subgroup of $G,$ $frak{n}$ is‎ ‎the Lie algebra of the unipotent radical $N.$ Under the adjoint‎ ‎action of its stabilizer in $M,$ every maximal prehomogeneous‎ ‎subspaces of $frak{n}$ is determined‎.

Let k be an algebraically closed field and let G be a reductive linear algebraic group over k. Let P be a parabolic subgroup of G, Pu its unipotent radical and pu the Lie algebra of Pu. A fundamental result of R. Richardson says that P acts on pu with a dense orbit (see [9]). The analogous result for the coadjoint action of P on pu is already known for char k = 0 (see [5]). In this note we prov...

Anari, Gharan, and Vinzant proved (complete) log-concavity of the basis generating functions for all matroids. From viewpoint combinatorial Hodge theory, it is natural to ask whether these are “strictly” log-concave simple In this paper, we show strictness graphic matroids, that is, Kirchhoff polynomials graphs strictly log-concave. Our key observation polynomial a complete graph can be seen as...

Let G be a complex reductive group, V a G-module and f ∈ O(V )G a nonconstant homogeneous invariant. We investigate relations between the following properties: • df : V → V ∗ is dominant, • f is homaloidal, i.e., df induces a birational map P(V ) → P(V ∗), • V is stable, i.e., the generic G-orbit is closed. If f generates O(V )G, we show that the properties are equivalent, generalizing results ...

The main aim of this paper is to obtain a converse theorem for double Dirichlet series and use it to show that the Shintani zeta functions [13] which arise in the theory of prehomogeneous vector spaces are actually linear combinations of Mellin transforms of metaplectic Eisenstein series on GL(2). The converse theorem we prove will apply to a very general family of double Dirichlet series which...