Critical operators for the degree of the minimal polynomial of derivations restricted to Grassmann spaces

Abstract

Let V be a finite dimension vector space. For a linear operator on V, f, D(f) denotes the restriction of the derivation associated with f to the mth Grassmann space of V. In [Cyclic spaces for Grassmann derivatives and additive theory, Bull. London Math. Soc. 26 (1994) 140-146] Dias da Silva and Hamidoune obtained a lower bound for the degree of the minimal polynomial of D(f), over an arbitrary field. Over a field of zero characteristic that lower bound is given bydeg(PD(f))[greater-or-equal, slanted]m(deg(Pf)-m)+1.Using additive number theory results, results on the elementary divisors of D(f) and methods presented by Marcus and Ali in [Minimal polynomials of additive commutators and jordan products, J. Algebra 22 (1972) 12-33] we obtain a characterization of equality cases in the former inequality, over a field of zero characteristic, whenever m does not exceed the number of distinct eigenvalues of f.