Rank Distribution of Linear Maps and Proportion of Indecomposable Representations

Cox, Kerry
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Faculty of Graduate Studies and Research, University of Regina

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.

A Thesis Submitted to the Faculty of Graduate Studies and Research In Partial Fulfillment of the Requirements for the Degree of Master of Science in Mathematics, University of Regina. v, 65 p.