chapter  6
62 Pages

Nilpotent Representations

Having obtained some control over the quotient variety trissn A of a Cayleysmooth algebra A we turn to the study of the fibers of the quotient map

trepn A pi-- trissn A

If (Q•, α) is the local marked quiver setting of a point ξ ∈ trissn A then the GLn-structure of the fiber pi−1(ξ) is isomorphic to the GL(α)-structure of the nullcone Nullα Q• consisting of all nilpotent α-dimensional representations of Q•. In geometric invariant theory, nullcones are investigated by a refinement of the Hilbert criterion: Hesselink’s stratification. The main aim of the present chapter is to prove that the different strata

in the Hesselink stratification of the nullcone of quiver-representations can be studied via moduli spaces of semistable quiver-representations. We will illustrate the method first by considering nilpotent m-tuples of n × n matrices and generalize the results later to quivers and Cayley-smooth orders. The methods allow us to begin to attack the ”hopeless” problem of studying simultaneous conjugacy classes of matrices. We then turn to the description of representation fibers, which can be studied quite explicitly for low-dimensional Cayley-smooth orders, and investigate the fibers of the Brauer-Severi fibration. Before reading the last two sections on Brauer-Severi varieties, it may be helpful to glance through the final chapter where similar, but easier, constructions are studied.