Let \(A_n\) be the \(n\)-th alternating group, and consider its group algebra \(FA_n\) as a subalgebra of \(FS_n\) over an algebraically closed field \(F\) of characteristic 0. One identifies \(FS_n\) with the \(F\)-space of all multilinear polynomials in \(x_1,\dots,x_n\) in the free associative algebra over \(F\). In this way one applies the representation theory of \(S_n\) to the study of PI algebras. Such an approach has proved extremely useful and efficient. One may consider the subspace of this space that corresponds to \(A_n\) and try and relate it to polynomial identities of certain important algebras. The authors of the paper under review define the cocharacter and the codimension sequences of a PI algebra with respect to \(A_n\). Then they compare these cocharacters and codimensions with the ordinary ones. Furthermore they compute the \(A_n\)-cocharacter and the \(A_n\)-codimension of \(E\), the infinite dimensional Grassmann algebra.