A note on the variety of pairs of matrices whose product is symmetric

Majidi-Zolbanin, Mahdi; Snapp, Bart

doi:10.1090/conm/555/10995

Contemporary Mathematics

2011

DOI: 10.1090/conm/555/10995

|View full text |Cite

A note on the variety of pairs of matrices whose product is symmetric

Mahdi Majidi-Zolbanin¹,

Bart Snapp²

Abstract: We give different short proofs for a result proved by C. Mueller in [9]: Over an algebraically closed field pairs of n × n matrices whose product is symmetric form an irreducible, reduced, and complete intersection variety of dimension (3n 2 + n)/2. Our work is connected to the work of Brennan, Pinto, and Vasconcelos in [2].

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2014

Publication Types

Select...

Article1

Relationship

Self Cite0

Independent1

Authors

Journals

Cited by 1 publication

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

Commuting varieties of r-tuples over Lie algebras

Ngo¹

2014

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

Abstract. Let G be a simple algebraic group defined over an algebraically closed field k of characteristic p and let g be the Lie algebra of G. It is well known that for p large enough the spectrum of the cohomology ring for the r-th Frobenius kernel of G is homeomorphic to the commuting variety of r-tuples of elements in the nilpotent cone of g [Suslin-Friedlander-Bendel, J. Amer. Math. Soc, 10 (1997), 693-728]. In this paper, we study both geometric and algebraic properties including irreducibility, singularity, normality and Cohen-Macaulayness of the commuting varieties Cr(gl 2 ), Cr(sl2) and Cr(N ) where N is the nilpotent cone of sl2. Our calculations lead us to state a conjecture on Cohen-Macaulayness for commuting varieties of r-tuples. Furthermore, in the case when g = sl2, we obtain interesting results about commuting varieties when adding more restrictions into each tuple. In the case of sl3, we are able to verify the aforementioned properties for Cr(u). Finally, applying our calculations on the commuting variety Cr(O sub ) where O sub is the closure of the subregular orbit in sl3, we prove that the nilpotent commuting variety Cr(N ) has singularities of codimension ≥ 2.

show abstract

Commuting varieties of r-tuples over Lie algebras

Ngo¹

2014

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

hi@scite.ai

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

A note on the variety of pairs of matrices whose product is symmetric

Cited by 1 publication

References 6 publications

Commuting varieties of r-tuples over Lie algebras

Commuting varieties of r-tuples over Lie algebras

Contact Info

Product

Resources

About