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Register a new userWeighted projective spaces and minimal nilpotent orbits
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Authors
C. A. Rossi
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We investigate (twisted) rings of differential operators on the resolution of singularities of a particular irreducible component of the (Zarisky) closure of the minimal orbit $bar O_{mathrm{min}}$ of $mathfrak{sp}_{2n}$, intersected with the Borel subalgebra $mathfrak n_+$ of $mathfrak{sp}_{2n}$, using toric geometry and show that they are homomorphic images of a subalgebra of the Universal Enveloping Algebra (UEA) of $mathfrak{sp}_{2n}$, which contains the maximal parabolic subalgebra $mathfrak p$ determining the minimal nilpotent orbit. Further, using Fourier transforms on Weyl algebras, we show that (twisted) rings of well-suited weighted projective spaces are obtained from the same subalgebra. Finally, investigating this subalgebra from the representation-theoretical point of view, we find new primitive ideals and rediscover old ones for the UEA of $mathfrak{sp}_{2n}$ coming from the aforementioned resolution of singularities.
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The noncommutative projective scheme $operatorname{mathsf{Proj_{nc}}} S$ of a $(pm 1)$-skew polynomial algebra $S$ in $n$ variables is considered to be a $(pm 1)$-skew projective space of dimension $n-1$. In this paper, using combinatorial methods, we give a classification theorem for $(pm 1)$-skew projective spaces. Specifically, among other equivalences, we prove that $(pm 1)$-skew projective spaces $operatorname{mathsf{Proj_{nc}}} S$ and $operatorname{mathsf{Proj_{nc}}} S$ are isomorphic if and only if certain graphs associated to $S$ and $S$ are switching (or mutation) equivalent. We also discuss invariants of $(pm 1)$-skew projective spaces from a combinatorial point of view.
We give a geometric classification of complex $n$-dimensional $2$-step nilpotent (all, commutative and anticommutative) algebras. Namely, has been found the number of irreducible components and their dimensions. As a corollary, we have a geometric classification of complex $5$-dimensional nilpotent associative algebras. In particular, it has been proven that this variety has $14$ irreducible components and $9$ rigid algebras.
We study in this paper the jet schemes of the closure of nilpotent orbits in a finite-dimensional complex reductive Lie algebra. For the nilpotent cone, which is the closure of the regular nilpotent orbit, all the jet schemes are irreducible. This was first observed by Eisenbud and Frenkel, and follows from a strong result of Mustau{t}c{a} (2001). Using induction and restriction of little nilpotent orbits in reductive Lie algebras, we show that for a large number of nilpotent orbits, the jet schemes of their closure are reducible. As a consequence, we obtain certain geometrical properties of these nilpotent orbit closures.
Projective Reed-Muller codes were introduced by Lachaud, in 1988 and their dimension and minimum distance were determined by Serre and S{o}rensen in 1991. In coding theory one is also interested in the higher Hamming weights, to study the code performance. Yet, not many values of the higher Hamming weights are known for these codes, not even the second lowest weight (also known as next-to-minimal weight) is completely determined. In this paper we determine all the values of the next-to-minimal weight for the binary projective Reed-Muller codes, which we show to be equal to the next-to-minimal weight of Reed-Muller codes in most, but not all, cases.
In this paper we present several values for the next-to-minimal weights of projective Reed-Muller codes. We work over $mathbb{F}_q$ with $q geq 3$ since in IEEE-IT 62(11) p. 6300-6303 (2016) we have determined the complete values for the next-to-minimal weights of binary projective Reed-Muller codes. As in loc. cit. here we also find examples of codewords with next-to-minimal weight whose set of zeros is not in a hyperplane arrangement.
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