Some schemes related to the commuting variety

Author:
Allen Knutson

Journal:
J. Algebraic Geom. **14** (2005), 283-294

DOI:
https://doi.org/10.1090/S1056-3911-04-00389-3

Published electronically:
October 26, 2004

MathSciNet review:
2123231

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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.

[E]E D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Springer GTM 150.
- Michael Kalkbrener and Bernd Sturmfels,
*Initial complexes of prime ideals*, Adv. Math. **116** (1995), no. 2, 365–376. MR **1363769**, DOI https://doi.org/10.1006/aima.1995.1071
[M2]M2 D. Grayson, M. Stillman, Macaulay 2 – a software system for algebraic geometry and commutative algebra, available at http://www.math.uiuc.edu/Macaulay2
[MS]MS E. Miller, B. Sturmfels, Combinatorial commutative algebra, in preparation. http://www.math.umn.edu/~ezra/CCA/
- R. W. Richardson,
*Commuting varieties of semisimple Lie algebras and algebraic groups*, Compositio Math. **38** (1979), no. 3, 311–327. MR **535074**
- W. Rossmann,
*Equivariant multiplicities on complex varieties*, Astérisque **173-174** (1989), 11, 313–330. Orbites unipotentes et représentations, III. MR **1021516**
- Wolmer V. Vasconcelos,
*Arithmetic of blowup algebras*, London Mathematical Society Lecture Note Series, vol. 195, Cambridge University Press, Cambridge, 1994. MR **1275840**

[E]E D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Springer GTM 150.
[KS]KS M. Kalkbrenner, B. Sturmfels, Initial complexes of prime ideals, Adv. Math. **116** (1995), no. 2, 365–376.
[M2]M2 D. Grayson, M. Stillman, Macaulay 2 – a software system for algebraic geometry and commutative algebra, available at http://www.math.uiuc.edu/Macaulay2
[MS]MS E. Miller, B. Sturmfels, Combinatorial commutative algebra, in preparation. http://www.math.umn.edu/~ezra/CCA/
[Ri]irreducible R. W. Richardson, Commuting varieties of semisimple Lie algebras and algebraic groups. Compositio Math. 38 (1979), no. 3, 311–327.
[Ro]Ro W. Rossmann, Equivariant multiplicities on complex varieties. Orbites unipotentes et représentations, III. Astérisque No. 173-174 (1989), 11, 313–330.
[V]V W. V. Vasconcelos, Arithmetic of Blowup Algebras, London Math. Soc., Lecture Note Series 195, Cambridge University Press, 1994.

Additional Information

**Allen Knutson**

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

Email:
allenk@math.berkeley.edu

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.