A subset Y of the general linear group GL(n,q) is called
t-intersecting if rk(xβy)β€nβt for all x,yβY, or
equivalently x and y agree pointwise on a t-dimensional subspace of
Fqnβ for all x,yβY. We show that, if n is sufficiently large
compared to t, the size of every such t-intersecting set is at most that of
the stabiliser of a basis of a t-dimensional subspace of Fqnβ. In
case of equality, the characteristic vector of Y is a linear combination of
the characteristic vectors of the cosets of these stabilisers. We also give
similar results for subsets of GL(n,q) that intersect not
necessarily pointwise in t-dimensional subspaces of Fqnβ and for
cross-intersecting subsets of GL(n,q). These results may be
viewed as variants of the classical Erd\H{o}s-Ko-Rado Theorem in extremal set
theory and are q-analogs of corresponding results known for the symmetric
group. Our methods are based on eigenvalue techniques to estimate the size of
the largest independent sets in graphs and crucially involve the representation
theory of GL(n,q).Comment: 34 pages, minor change