On connectedness in the parametric geometry of numbers (Q6631405)

The paper uses multilinear algebra to reformulate a criterion by \textit{W. M. Schmidt} and \textit{L. Summerer} [Acta Arith. 140, No. 1, 67--91 (2009; Zbl 1236.11060)] for connectedness of \(L\)-series of certain unimodular lattices \(\Lambda \). For a given \((m,n)\)-weight vector \(\omega \) the authors define a one parameter diagonal flow \(\{ a_t^{\omega } \mid t\in \mathbb{R} \}\). If the eigenvalues of \(\Lambda ^k(a_1^{\omega })\) on the subset of the \(k\)-Grassmannian spanned by those \(k\)-dimensional subspaces having a basis in \(\Lambda \) are upper bounded by 1, then the corresponding \(L\)-series is connected at \(k\).