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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Some schemes related to the commuting variety

Author: Allen Knutson
Journal: J. Algebraic Geom. 14 (2005), 283-294
Published electronically: October 26, 2004
MathSciNet review: 2123231
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Abstract: The commuting variety is the pairs of $n\times n$ matrices $(X,Y)$ such that $XY=YX$. We introduce the diagonal commutator scheme, $\big \{ (X,Y) : XY-YX$ is diagonal$\big \}$, 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 $\big \{ (X,Y) : XY$ lower triangular, $YX$ upper triangular$\big \}$, 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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Additional Information

Allen Knutson
Affiliation: Department of Mathematics, University of California, Berkeley, 1033 Evans Hall, Berkeley, California 94720-3840

Received by editor(s): June 23, 2003
Published electronically: October 26, 2004
Additional Notes: The author was supported by the National Science Foundation and the Sloan Foundation.