Some cyclic virtually irreducible lattices

Weight enumerator of some irreducible cyclic codes

In this article, we show explicitly all possible weight enumerators for every irreducible cyclic code of length n over a finite field Fq, in the case which each prime divisor of n is also a divisor of q − 1.

Rigidity of Commensurators and Irreducible Lattices

Superrigidity for Irreducible Lattices and Geometric Splitting

1.1. Superrigidity. In the early seventies, Margulis proved his celebrated superrigidity theorem for irreducible lattices in semisimple Lie and algebraic groups of higher rank. One of the motivations for this result is that it implies arithmeticity : a complete classification of higher rank lattices. In the case where the semisimple group is not almost simple, superrigidity reads as follows (se...

Irreducible elements in multi-adjoint concept lattices

One of the most important elements in a lattice are the irreducible elements. For example, when the lattice is finite, which is usual in the computational case, it forms a base from which the complete lattice is obtained. These elements are also important in Formal Concept Analysis, since they are the basic information of a relational system. In the general fuzzy framework of multi-adjoint conc...

Arithmeticity vs. Non-linearity for Irreducible Lattices

We establish an arithmeticity vs. non-linearity alternative for irreducible lattices in suitable product groups, for instance products of topologically simple groups. This applies notably to a (large class of) Kac-Moody groups. The alternative relies heavily on the superrigidity theorem we propose in [Md], since we follow Margulis’ reduction of arithmeticity to superrigidity.