Root systems in number fields

The Multiplicities of Root Number Functions

We consider the $q$th root number function for the symmetric group. Our aim is to develop an asymptotic formula for the multiplicities of the $q$th root number function as $q$ tends to $\infty$. We use character theory, number theory and combinatorics.

Root lattices in number fields

We explore whether a root lattice may be similar to the lattice $\mathscr O$ of integers of a number field $K$ endowed with the inner product $(x, y):={\rm Trace}_{K/\mathbb Q}(x\cdotθ(y))$, where $θ$ is an involution of $K$. We classify all pairs $K$, $θ$ such that $\mathscr O$ is similar to either an even root lattice or the root lattice $\mathbb Z^{[K:\mathbb Q]}$. We also classify all pairs $K$, $θ$ such that $\mathscr O$ is a root lattice. In addition to this, we show that $\mathscr O$ is never similar to a positive-definite even unimodular lattice of rank $\leqslant 48$, in particular, $…

根簇与多簇在不完美希尔伯特域上的推广

Root Clusters and Multiclusters over Imperfect Hilbertian Fields

将根簇理论从完美域推广到一般域，引入根簇大小、多簇大小等概念，并在希尔伯特域上建立了这些广义概念的逆问题结果。