Quasi-projective Brauer characters

We study p-Brauer characters of a finite group G which are restrictions of generalized characters vanishing on p-singular elements for a fixed prime p dividing the order of G. Such Brauer characters are called quasi-projective. We show that for each irreducible Brauer character phi there exists a minimal p-power, say p(a(phi)), such that p(a)(phi) phi is quasi-projective. The exponent a(phi) only depends on the Cartan matrix of the block to which yo belongs. Moreover p(a(phi)) is bounded by the vertex of the module affording phi, and equality holds in case that G is p-solvable. We give some evidence for the conjecture that a(phi) = 0 occurs if and only if phi belongs to a block of defect 0. Finally, we study indecomposable quasi-projective Brauer characters of a block B. This set is finite and corresponds to a minimal Hilbert basis of the rational cone defined by the Cartan matrix of B. (C) 2018 Elsevier Inc. All rights reserved.