Rings of matrix invariants in positive characteristic
Let \(R_{n,m}\) be the ring of invariants of \(m\)-tuples of \(n\times n\) matrices, \(m,n\geq 2\), over an infinite field \(K\). In the present very interesting paper the authors focus on explicit presentations (by means of generators and defining relations) of \(R_{n,m}\) in positive characteristic in the spirit of the Razmyslov-Procesi theory of matrix invariants in characteristic 0 and find the exact frontiers for validity of the results of characteristic 0. First they establish the same relation as in characteristic 0 between the multilinear matrix invariants and the Nagata-Higman theorem for nilpotency of nil algebras (which is not more valid if the positive characteristic \(p\) is small with respect to the degree of the nil identity). In particular, the authors discover a new phenomenon: If \(0<\text{char }K\leq n\), then the degree of the generators of \(R_{n,m}\) goes to infinity with \(m\). Then the authors study in detail the case of \(2\times 2\) matrices. They show that \(R_{2,m}\) has a system of generators consisting of \(\det(X_i)\) and \(\text{tr}(X_{i_1}\cdots X_{i_s})\), where \(X_1,\dots,X_m\) are \(2\times 2\) generic matrices and \(1\leq i\leq m\), \(1\leq i_1<\cdots<i_s\leq m\). If \(\text{char }K>2\), then \(R_{2,m}\) is generated by the determinant and the traces for \(s\leq 3\). If \(\text{char }K=2\), then one needs the traces of all products for \(s\leq m\). Further, the authors study the Cohen-Macaulay property of \(R_{n,m}\). In particular, they give an explicit Cohen-Macaulay presentation of \(M_{2,4}\), i.e. a homogeneous system of parameters and a basis of \(R_{2,4}\) as a free module over the algebra generated by this system. (As the authors mention in a foot-note, recently [\textit{M. Hashimoto}, Math. Z. 236, No. 3, 605-623 (2001; Zbl 1034.13007)] showed that \(R_{n,m}\) is Cohen-Macaulay over any infinite field.) Then they show that \(R_{n,m}\) is a complete intersection in characteristic \(p\) exactly in the same cases as in characteristic 0. Finally, the authors obtain an interesting consequence for invariant theory of the special orthogonal group. Roughly speaking, the invariants in characteristic \(p>2\) behave as in characteristic 0 and the situation changes drastically in characteristic 2.
scientific article; zbMATH DE number 1864816
scientific article