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14.37 (1999)

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Let $G(n)$ be one of the classical groups (special, orthogonal, or symplectic) of $(n \times n)$-matrices over an infinite field $K$ of non-zero characteristic, and $M(n)$ the space of all $(n \times n)$-matrices over $K$. The group $G(n)$ acts diagonally by conjugation on the space $M(n)^m = M(n) \oplus \dots \oplus M(n)$ ($m$ copies). Find generators of the algebra of invariants $K[M(n)^m]^{G(n)}$.

In characteristic 0 they were found in (C. Procesi, Adv. Math., 19 (1976), 306–381).

Contributor: A. N. Zubkov

Progress

Comment of 2001: in positive characteristic the problem is solved for all cases excepting the orthogonal groups in characteristic 2 and special orthogonal groups of even degree (A. N. Zubkov, Algebra and Logic, 38, no. 5 (1999), 299–318). Comment of 2009: ...and for special orthogonal groups of even degree over (infinite) fields of odd characteristic (A. A. Lopatin, J. Algebra, 321 (2009), 1079–1106).

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