manuscripta math. 83. 365 -385 (1994) manuscripta mathematica Summary: 1 manuscripta math. 83. 365 - 385 (1994) manuscripta mathematica ©Springer-Verlag 1994 Generic 2 x 2 Matrices With Involution Helmer Aslaksen and Eng-Chye Tan Procesi has given a linear basis for the ring of m generic 2 X 2 matrices. We do the same for the ring of m generic 2 X 2 matrices with transpose involution. Introduction Let k be a field of characteristic 0 and let M2 = M2( k) be the set of 2 X 2 matrices over k. Consider an involution * on M2 of the first kind. It follows from [14] that it is sufficient to study the symplectic involution and the transpose involution. The case of symplectic involution is essentially treated in Section 7 of [12], so we will only consider the transpose involution. We will use the following notation: Mi is the set of symmetric matrices, M:; the skew-symmetric matrices, Mf the matrices of trace 0, Mf+ the symmetric traceless matrices and k the scalar matrices. Hence M2 = Mi EB M2- = k EB Mf = k EB Mf+ EB M:;. Collections: Mathematics