Commuting Pairs in the Centralizers of 2-Regular Matrices

✍ Scribed by Michael G Neubauer; B.A Sethuraman

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 72 KB
• Volume: 214
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1998.7703

✦ Synopsis

In M k , k an algebraically closed field, we call a matrix l-regular if each n eigenspace is at most l-dimensional. We prove that the variety of commuting pairs in the centralizer of a 2-regular matrix is the direct product of various affine spaces and various determinantal varieties Z Z obtained from matrices over truncated l, m polynomial rings. We prove that these varieties Z Z are irreducible and apply this l, m to the case of the k-algebra generated by three commuting matrices: we show that if one of the three matrices is 2-regular, then the algebra has dimension at most n.

We also show that such an algebra is always contained in a commutative subalge-Ž . bra of M k of dimension exactly n. ᮊ 1999 Academic Press n * We thank Bob Guralnick for some useful discussions and for making many clarifying suggestions, particularly in Section 5. This revised version was written when the second author was visiting the University of Southern California, and he thanks the department there for its hospitality.

† The second author is supported in part by an N.S.F. grant.