Some schemes related to the commuting variety

Details Some schemes related to the commuting variety Some schemes related to the commuting variety

2003-06-18

arxiv.org/abs/math/0306275v1

J. Algebraic Geom. 14 (2005) 283-294

Mathematics

Algebraic Geometry

9 pages

Scientific paper

The_commuting variety_ is the pairs of NxN matrices (X,Y) such that XY = YX. We introduce the_diagonal commutator scheme_, {(X,Y) : XY-YX is diagonal}, which we prove to be a reduced complete intersection, one component of which is the commuting variety. (We conjecture there to be only one other component.) The diagonal commutator scheme has a flat degeneration to the scheme {(X,Y) : XY lower triangular, YX upper triangular}, which is again a reduced complete intersection, this time with n! components (one for each permutation). The degrees of these components give interesting invariants of permutations.

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