SO(n)-invariants of several matrices and quivers
by
Artem Lopatin
Institute of Mathematics, Omsk

Suppose that a group G < GL(n) acts on the direct sum M(n)^d=M(n)⊕...⊕M(n) of n×n matrices by the diagonal conjugation. This action induces the action of G on the coordinate ring R=K[M(n)^d] in a natural way. Denote by R^G the algebra of invariants. We proved that over an infinite field of the characteristic different from 2 the algebra R^SO(n) is generated by the pfaffians and the coefficients of the characteristic polynomial of products of the generic matrices and the transpose generic matrices. Similar result we also obtained for quivers. So the problem of describing generators of the algebra of invariants of a quiver is solved for a product of any classical groups, i.e., for GL(n), O(n), Sp(n), SL(n), SO(n), where the characteristic of the base field is different from 2 wherever we talk about O(n) and SO(n).

Date received: February 12, 2006