Free modular lattices and their representations

Let be $L$ a modular lattice, and $V$ a finite-dimensional vector space over a field $k$. A representation of $L$ in $V$ is a morphism from $L$ into the lattice $\mathscr L(V)$ of all subspaces of $V$. In this paper we study representations of finitely generated free modular lattices $D^r$. An element $a$ of a lattice $L$ is called perfect if for every indecomposable representation $\rho\colon L\to\mathscr L(k^n)$ the subspace $\rho(a)$ of $V=k^n$ is such that $\rho(a)=0$ or $\rho(a)=V$. We construct and study certain important sublattices of $D^r$, called “cubicles”. All elements of the cubicles are perfect. There are indecomposable representations connected with the cubicles. It will be shown that almost all these representations, except the elementary ones, have the important property of complete irreducibility; here a representation $\rho$ of $L$ is called completely irreducible if the sublattice $\rho(L)\subset\mathscr L(k^n)$ is isomorphic to the lattice $\mathbf P(\mathbf Q, n-1)$ of linear submanifolds of projective space over the field $\mathbf Q$ of rational numbers.