chapter  8
16 Pages

Trace Identities

Exercises for Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

Several times in this book we used results about traces. Now we shall develop these trace identities formally, showing in one crucial respect that they behave better than polynomial identities. Razmyslov, Procesi, and Helling proved that all trace identities of Mn(Q) formally are consequences of a single trace identity, which Razmyslov identified with the Cayley-Hamilton equation (rewritten as a trace identity via Newton’s formulas.) We prove this basic result in this chapter, and also provide tools for further investigation. In the process, following Regev [Reg84], we see that the trace identities have a cocharacter theory analogous to that of the PI-theory, and the two can be tied together quite neatly.