Minimal G-graded varieties of quadratic growth and small colength

The sequence of codimensions and the sequence of colengths of an algebra [Formula: see text] are closely related. In fact, it is known that [Formula: see text] has polynomial growth if and only if its sequence of colengths is bounded by a constant. In recent years, results concerning classification of varieties with colength bounded by a fixed constant have been presented in several contexts. In this paper, we classify the varieties of [Formula: see text]-graded colength equal to [Formula: see text] and show that there are infinitely many unitary [Formula: see text]-graded varieties under this situation. Some results obtained throughout the way allow us to conclude the paper with the classification of minimal [Formula: see text]-graded varieties of quadratic growth of codimensions.