Rank Distribution of Linear Maps and Proportion of Indecomposable Representations

Abstract

The distribution of the ranks of all linear maps between two nite-dimensional vector spaces over a eld of order q is determined both theoretically and experimen- tally. The distribution is derived by using a particular group action on the set of rectangular matrices with entries from a eld of order q and then applying the orbit- stabilizer theorem. In addition, the proportion of vectors in the kernel of a linear map based on its image dimension is determined and used to experimentally determine the rank of a linear map. Moreover, the proportion of the indecomposable representations in the functor category vect(K) ! is discussed. In fact, the proportion of the indecomposables is de ned in two ways and one must be precise about what is meant by a proportion of indecomposables of a representation. Certain computations are quite tedious based on the de nition used.