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

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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].

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Commuting varieties of r-tuples over Lie algebras

Ngo¹

2014

Journal of Pure and Applied Algebra

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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.

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Commuting varieties of r-tuples over Lie algebras

Ngo¹

2014

Journal of Pure and Applied Algebra

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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

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