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Some schemes related to the commuting variety

Author(s): Allen Knutson
Journal: J. Algebraic Geom. 14 (2005), 283-294.
Posted: October 26, 2004
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Abstract | References | Additional information

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.

References:

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D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Springer GTM 150.

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E. Miller, B. Sturmfels, Combinatorial commutative algebra, in preparation. http://www.math.umn.edu/~ezra/CCA/

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R. W. Richardson, Commuting varieties of semisimple Lie algebras and algebraic groups. Compositio Math. 38 (1979), no. 3, 311-327. MR 0535074 (80c:17009)

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W. Rossmann, Equivariant multiplicities on complex varieties. Orbites unipotentes et représentations, III. Astérisque No. 173-174 (1989), 11, 313-330. MR 1021516 (91g:32042)

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Additional Information:

Allen Knutson
Affiliation: Department of Mathematics, University of California, Berkeley, 1033 Evans Hall, Berkeley, California 94720-3840
Email: allenk@math.berkeley.edu

PII: S 1056-3911(04)00389-3
Received by editor(s): June 23, 2003
Posted: October 26, 2004
Additional Notes: The author was supported by the National Science Foundation and the Sloan Foundation.

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