Commuting Pairs in the Centralizers of 2-regular Matrices
نویسنده
چکیده
In Mn(k), k an algebraically closed field, we call a matrix l-regular if each eigenspace is at most l-dimensional. We prove that the variety of commuting pairs in the centralizer of a 2-regular matrix is the direct product of various affine spaces and various determinantal varieties Zl,m obtained from matrices over truncated polynomial rings. We prove that these varieties Zl,m are irreducible, and apply this to the case of the k-algebra generated by three commuting matrices: we show that if one of the three matrices is 2-regular, then the algebra has dimension at most n. We also show that such an algebra is always contained in a commutative subalgebra of Mn(k) of dimension exactly n.
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