Resolutions of De Concini–Procesi Ideals of Hooks

Abstract

We find a minimal generating set for the defining ideal of the schematic intersection of the set of diagonal matrices with the closure of the conjugacy class of a nilpotent matrix indexed by a hook partition. The structure of this ideal allows us to compute its minimal free resolution and give an explicit description of the graded Betti numbers, and study its Hilbert series and regularity.

Key Words:

• Betti numbers
• Hilbert and Poincaré series
• Partitions
• Symmetric functions

2000 Mathematics Subject Classification:

• 13D02
• 05E10

ACKNOWLEDGMENTS

All the test examples that supported this research were run using the computer algebra program Macaulay2. We would like to thank François Bergeron, Emmanuel Briand, Tony Geramita, and the referee for useful comments, and Mark Shimozono and Jerzy Weyman for telling us about resolutions of nilpotent closures, and many helpful remarks and suggestions. M. Rosas also wishes to thank York University, for their hospitality during the completion of this project.

Notes

Communicated by I. Swanson.