Can one factor the classical adjoint of a generic matrix?

Details Can one factor the classical adjoint of a generic matrix? Can one factor the classical adjoint of a generic matrix?

2003-06-08

arxiv.org/abs/math/0306126v2

Transformation Groups, 11 (2006) 7-15

Mathematics

Commutative Algebra

Revised version contains answer to "even n" question left open in original version. (Answer due to Buchweitz & Leuschke; simpl

Scientific paper

10.1007/s00031-005-1101-x

Let k be a field, n a positive integer, X a generic nxn matrix over k (i.e., a matrix (x_{ij}) of n^2 independent indeterminates over the polynomial ring k[x_{ij}]), and adj(X) its classical adjoint. It is shown that if char k=0 and n is odd, then adj(X) is not the product of two noninvertible nxn matrices over k[x_{ij}]. If n is even and >2, a restricted class of nontrivial factorizations occur. The nonzero-characteristic case remains open. The operation adj on matrices arises from the (n-1)st exterior power functor on modules; the same question can be posed for matrix operations arising from other functors.

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