Let \({\mathbb{T}}_{m,n}\) be the trace ring of m generic \(n\times n\) matrices over a field F of characteristic 0. There are three main motivations to study \({\mathbb{T}}_{m,n}:\) (a) Representation theory. The centre \({\mathcal R}_{m,n}\) of \({\mathbb{T}}_{m,n}\) is an affine algebra. The Artin-Procesi theorem claims that the points of the variety related to \({\mathcal R}_{m,n}\) are in one-to-one correspondence with the equivalence classes of semi-simple n-dimensional representations of the free algebra \({\mathbb{F}}_ m=F<x_ 1,...,x_ m>\). (b) Invariant theory of \(n\times n\) matrices. The Gurevich-Siberskij-Procesi theorem identifies \({\mathcal R}_{m,n}\) with the ring of invariants of m copies of \(n\times n\) matrices under the action of \(GL_ n(F)\) by conjugation. (c) Deep relations with PI-algebras and modern structure ring theory.
The book is devoted to the trace ring \({\mathbb{T}}_{m,2}\) of m generic \(2\times 2\) matrices. It is reasonably self-contained and presents both old and new results from a unique point of view. The main result in Chapter 1 “Invariant Theory” is that \({\mathbb{T}}_{m,2}\) is the ring of matrix concomitants. Besides, a fairly precise description of \({\mathbb{T}}_{m,2}\) is obtained.
The aim of Chapter 2 “Quadratic Forms” is to clarify the connection between \({\mathbb{T}}_{m,2}\) and the algebraic theory of quadratic forms. It turns out that the subalgebra \({\mathbb{T}}^ 0_ m\) of \({\mathbb{T}}^ 0_{m,2}\) generated by the generic trace zero matrices is an epimorphic image of an iterated Ore extension \(\Lambda_ m\) of F; \(\Lambda_ m\) is the generic Clifford algebra \(Cl_ m\) of an m-ary quadratic form. The ring theoretical and the arithmetical properties of \(\Lambda_ m\) are studied in depth.
Chapter 3 “Homological Algebra” deals with the homological properties of quotients of the generic Clifford algebra \(Cl_ m\) and the trace ring \({\mathbb{T}}_{m,2}\). For many ideals, the quotients of \(Cl_ m\) are Cohen- Macaulay. The centre \({\mathcal R}_{m,2}\) of \({\mathbb{T}}_{m,2}\) is a unique factorization domain and the rings \({\mathbb{T}}^ 0_ m\) and \({\mathbb{T}}_{m,2}\) are Gorenstein.
The final Chapter 4 “Poincaré series” studies the Poincaré (or Hilbert) series \({\mathcal P}({\mathbb{T}}_{m,2},t)\) of \({\mathbb{T}}_{m,2}\). It is a rational function with residue in the pole \(t=1\) equal to the Krull dimension 4m-3 of \({\mathbb{T}}_{m,2}\). An algorithm to calculate \({\mathcal P}({\mathbb{T}}_{m,2},t)\) modulo a given prime number p is suggested. It is established that the Poincaré series satisfies the following functional equation \({\mathcal P}({\mathbb{T}}_{m,2},1/t)=-t^{4m}{\mathcal P}({\mathbb{T}}_{m,2},t)\). Three proofs based on different ideas are given. One of the main results of the chapter claims that \({\mathbb{T}}_{m,2}\) has finite global dimension if and only if \(m\leq 3.\)
The reviewer’s opinion is that the results included in the book have had their influence on the recent development of ring theory. We supply a few examples: Formanek and Teranishi have established that for all m,n the Poincaré series of \({\mathcal R}_{m,n}\) and \({\mathbb{T}}_{m,n}\) satisfy a similar functional equation; van der Bergh has given an explicit rational form for the Poincaré series of \({\mathbb{T}}_{m,n}\) and has proved that \({\mathbb{T}}_{m,n}\) is Cohen-Macaulay; Koshlukov and the reviewer have applied the construction of the generic Clifford algebra to Jordan algebras with polynomial identities; etc.