Linear independence in commutative semigroups
Let be a connected complex reductive group where is a finite-dimensional complex vector space. Let and let . Following Raïs we say that the orbit is characteristic for if the identity component of is . If is semisimple, we say that is semi-characteristic for if the identity component of is an extension of by a torus. We classify the -orbits which are not (semi)-characteristic in many cases.
The maximal column rank of an m by n matrix is the maximal number of the columns of A which are linearly independent. We compare the maximal column rank with rank of matrices over a nonbinary Boolean algebra. We also characterize the linear operators which preserve the maximal column ranks of matrices over nonbinary Boolean algebra.
Given a finite-volume hyperbolic 3-manifold, we compose a lift of the holonomy in with the -dimensional irreducible representation of in . In this paper we give local coordinates of the -character variety around the character of this representation. As a corollary, this representation is isolated among all representations that are unipotent at the cusps.