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

Full-text PDF

Abstract | References | Additional Information

Abstract: The *commuting variety* is the pairs of matrices such that . We introduce the **diagonal commutator scheme**, 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 lower triangular, upper triangular, which is again a reduced complete intersection, this time with components (one for each permutation). The degrees of these components give interesting invariants of permutations.

**[E]**D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Springer GTM 150.**[KS]**Michael Kalkbrener and Bernd Sturmfels,*Initial complexes of prime ideals*, Adv. Math.**116**(1995), no. 2, 365–376. MR**1363769**, https://doi.org/10.1006/aima.1995.1071**[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]**E. Miller, B. Sturmfels, Combinatorial commutative algebra, in preparation.`http://www.math.umn.edu/~ezra/CCA/`**[Ri]**R. W. Richardson, Commuting varieties of semisimple Lie algebras and algebraic groups. Compositio Math. 38 (1979), no. 3, 311-327. MR**0535074 (80c:17009)****[Ro]**W. Rossmann,*Equivariant multiplicities on complex varieties*, Astérisque**173-174**(1989), 11, 313–330. Orbites unipotentes et représentations, III. MR**1021516****[V]**Wolmer V. Vasconcelos,*Arithmetic of blowup algebras*, London Mathematical Society Lecture Note Series, vol. 195, Cambridge University Press, Cambridge, 1994. MR**1275840**

Additional Information

**Allen Knutson**

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

Email:
allenk@math.berkeley.edu

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

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.