Polynomial codimension growth of graded algebras

We study associative $G$-graded algebras with 1 of polynomial $G$-codimensiongrowth, where $G$ is a finite group. For any fixed $k\geq 1,$ weconstruct associative $G$-graded algebras of upper triangularmatrices whose $G$-codimension sequence is given asymptotically by apolynomial of degree $k$ whose leading coefficient is the largest orsmallest possible.