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- W2079836398 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, textbf{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 $C_r(mathfrak{gl}_2), C_r(fraksl_2)$ and $C_r(N)$ where $N$ is the nilpotent cone of $fraksl_2$. Our calculations lead us to state a conjecture on Cohen-Macaulayness for commuting varieties of $r$-tuples. Furthermore, in the case when $g=fraksl_2$, we obtain interesting results about commuting varieties when adding more restrictions into each tuple. In the case of $fraksl_3$, we are able to verify the aforementioned properties for $C_r(fraku)$. Finally, applying our calculations on the commuting variety $C_r(overline{calO_{sub}})$ where $overline{calO_{sub}}$ is the closure of the subregular orbit in $fraksl_3$, we prove that the nilpotent commuting variety $C_r(N)$ has singularities of codimension $ge 2$." @default.
- W2079836398 created "2016-06-24" @default.
- W2079836398 creator A5079554464 @default.
- W2079836398 date "2014-08-01" @default.
- W2079836398 modified "2023-10-17" @default.
- W2079836398 title "Commuting varieties of r-tuples over Lie algebras" @default.
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- W2079836398 doi "https://doi.org/10.1016/j.jpaa.2013.11.024" @default.
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