Vol. 107, No. 1, 1983

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Some Poincaré series related to identities of 2 × 2 matrices

Patrick Ronald Halpin

Vol. 107 (1983), No. 1, 107–115

A partial solution to a problem of Procesi has recently been given by Formanek, Halpin, Li by determining the Poincaré series of the ideal of two variable identities of M2(k). Two related results are obtained in this article.

A weak identity of Mn(k) is a polynomial which vanishes identically on sln, the subspace of Mn(k) of matrices of trace zero. We show that the Poincaré series of the ideal of two variable weak identities of M2(k) is rational. In addition it is shown that the ideal of identities of upper triangular 2 × 2 matrices in an arbitrary finite number of variables has a rational Poincaré series. As an application we are able to determine this ideal precisely.

Mathematical Subject Classification 2000
Primary: 16A42, 16A42
Secondary: 18F25
Received: 24 August 1981
Published: 1 July 1983
Patrick Ronald Halpin