Application of Full Quivers of Representations of Algebras,to Polynomial Identities

In [7 Belov , A. , Rowen , L. H. , Vishne , U. Full quivers of representations of algebras. To appear in Trans. Amer. Math. Soc.  [Google Scholar]] we introduced the notion of full quivers of representations of algebras, which are more explicit than quivers of algebras, and better suited for algebras over finite fields. Here, we consider full quivers as a combinatorial tool in order to describe PI-varieties of algebras. We apply the theory to clarify the proofs of diverse topics in the literature: Determining which relatively free algebras are weakly Noetherian, determining when relatively free algebras are finitely presented, presenting a quick proof for the rationality of the Hilbert series of a relatively free PI-algebra, and explaining counterexamples to Specht's conjecture for varieties of Lie algebras.