On the codimension growth of G-graded algebras

Let W be an associative PI-affine algebra over a field F of characteristic zero. Suppose W is G-graded where G is a finite group. Let exp(W) and exp(W_e) denote the codimension growth of W and of the identity component W_e, respectively. We prove: exp(W) \leq |G|^2 exp(W_e). This inequality had been conjectured by Bahturin and Zaicev.Comment: 9 page

On group gradings on PI-algebras

We show that there exists a constant K such that for any PI- algebra W and any nondegenerate G-grading on W where G is any group (possibly infinite), there exists an abelian subgroup U of G with $[G : U] \leq exp(W)^K$. A G-grading $W = \bigoplus_{g \in G}W_g$ is said to be nondegenerate if $W_{g_1}W_{g_2}... W_{g_r} \neq 0$ for any $r \geq 1$ and any $r$ tuple $(g_1, g_2,..., g_r)$ in $G^r$.Comment: 17 page

Norm formulas for finite groups and induction from elementary abelian subgroups

It is known that the norm map N_G for a finite group G acting on a ring R is surjective if and only if for every elementary abelian subgroup E of G the norm map N_E for E is surjective. Equivalently, there exists an element x_G in R with N_G(x_G) = 1 if and only for every elementary abelian subgroup E there exists an element x_E in R such that N_E(x_E) = 1. When the ring R is noncommutative, it is an open problem to find an explicit formula for x_G in terms of the elements x_E. In this paper we present a method to solve this problem for an arbitrary group G and an arbitrary group action on a ring.Using this method, we obtain a complete solution of the problem for the quaternion and the dihedral 2-groups,and for a group of order 27. We also show how to reduce the problem to the class of (almost) extraspecial p-groups.Comment: 31 pages. In Section 1 a universal ring and the proof of the existence of formulas for any finite group were adde

Multialternating graded polynomials and growth of polynomial identities

Let G be a finite group and A a finite dimensional G-graded algebra over a field of characteristic zero. When A is simple as a G-graded algebra, by mean of Regev central polynomials we construct multialternating graded polynomials of arbitrarily large degree non vanishing on A. As a consequence we compute the exponential rate of growth of the sequence of graded codimensions of an arbitrary G-graded algebra satisfying an ordinary polynomial identity. In particular we show it is an integer. The result was proviously known in case G is abelian.Comment: To appear in Proc. of AM

Hilbert series of PI relatively free G-graded algebras are rational functions

Let G be a finite group, (g_{1},...,g_{r}) an (unordered) r-tuple of G^{(r)} and x_{i,g_i}'s variables that correspond to the g_i's, i=1,...,r. Let F be the corresponding free G-graded algebra where F is a field of zero characteristic. Here the degree of a monomial is determined by the product of the indices in G. Let I be a G-graded T-ideal of F which is PI (e.g. any ideal of identities of a G-graded finite dimensional algebra is of this type). We prove that the Hilbert series of F/I is a rational function. More generally, we show that the Hilbert series which corresponds to any g-homogeneous component of F/I is a rational function.Comment: 14 page