chapter  3
24 Pages

Representations of Sn and Their Applications

In this chapter we turn to a sublime connection, first exploited by Regev, of PI-theory with classical representation theory (via the group algebra F [Sn]), leading to many interesting results (including Regev’s Exponential Bound Theorem). Throughout, we assume that C is a field F , and we take an algebra A over F . Perhaps the key result of this chapter, discovered independently by Amitsur-Regev and Kemer, shows in characteristic 0 that any PI-algebra satisfies a “sparse” identity. This is the key hypothesis in translating identities to affine superidentities in Chapter 7, leading to Kemer’s solution of Specht’s problem.