Reduction for characters of finite algebra groups

Let J be a finite-dimensional nilpotent algebra over a finite field Fq. We formulate a procedure for analysing characters of the group 1+J. In particular, we study characters of the group Un(q) of unipotent triangular n×n matrices over Fq. Using our procedure, we compute the number of irreducible characters of Un(q) of each degree for n⩽13. Also, we explain and generalise a phenomenon concerning the group U13(2) discovered by Isaacs and Karagueuzian.