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The poset of the nilpotent commutator of a nilpotent matrix  [PDF]
Leila Khatami
Mathematics , 2012,
Abstract: Let $B$ be an $n \times n$ nilpotent matrix with entries in an infinite field $\k$. Assume that $B$ is in Jordan canonical form with the associated Jordan block partition $P$. In this paper, we study a poset $\mathcal{D}_P$ associated to the nilpotent commutator of $B$ and a certain partition of $n$, denoted by $\lambda_U(P)$, defined in terms of the lengths of unions of special chains in $\mathcal{D}_P$. Polona Oblak associated to a given partition $P$ another partition $Ob(P)$ resulting from a recursive process. She conjectured that $Ob(P)$ is the same as the Jordan partition $Q(P)$ of a generic element of the nilpotent commutator of $B$. Roberta Basili, Anthony Iarrobino and the author later generalized the process introduced by Oblak. In this paper we show that all such processes result in the partition $\lambda_U(P)$.
On pairs of commuting nilpotent matrices  [PDF]
Toma? Ko?ir,Polona Oblak
Mathematics , 2007,
Abstract: Let $B$ be a nilpotent matrix and suppose that its Jordan canonical form is determined by a partition $\lambda$. Then it is known that its nilpotent commutator $N_B$ is an irreducible variety and that there is a unique partition $\mu$ such that the intersection of the orbit of nilpotent matrices corresponding to $\mu$ with $N_B$ is dense in $N_B$. We prove that map $D$ given by $D(\lambda)=\mu$ is an idempotent map. This answers a question of Basili and Iarrobino and gives a partial answer to a question of Panyushev. In the proof, we use the fact that for a generic matrix $A \in N_B$ the algebra generated by $A$ and $B$ is a Gorenstein algebra. Thus, a generic pair of commuting nilpotent matrices generates a Gorenstein algebra. We also describe $D(\lambda)$ in terms of $\lambda$ if $D(\lambda)$ has at most two parts.
On the nilpotent commutator of a nilpotent matrix  [PDF]
Polona Oblak
Mathematics , 2011,
Abstract: We study the structure of the nilpotent commutator $\nb$ of a nilpotent matrix $B$. We show that $\nb$ intersects all nilpotent orbits for conjugation if and only if $B$ is a square--zero matrix. We describe nonempty intersections of $\nb$ with nilpotent orbits in the case the $n \times n$ matrix $B$ has rank $n-2$. Moreover, we give some results on the maximal nilpotent orbit that $\nb$ intersects nontrivially.
Cayley graphs on nilpotent groups with cyclic commutator subgroup are hamiltonian  [PDF]
Ebrahim Ghaderpour,Dave Witte Morris
Mathematics , 2011,
Abstract: We show that if G is any nilpotent, finite group, and the commutator subgroup of G is cyclic, then every connected Cayley graph on G has a hamiltonian cycle.
The commutator algebra of a nilpotent matrix and an application to the theory of commutative Artinian algebras  [PDF]
Tadahito Harima,Junzo Watanabe
Mathematics , 2012,
Abstract: We show a number of properties of the commutator algebra of a nilpotent matrix over a field. In particular we determine the simple modules of the commutator algebra. Then the results are applied to prove that certain Artinian complete intersections have the strong Lefscehtz property.
Bound on the Jordan type of a generic nilpotent matrix commuting with a given matrix  [PDF]
Anthony Iarrobino,Leila Khatami
Mathematics , 2012, DOI: 10.1007/s10801-013-0433-1
Abstract: It is well-known that a nilpotent n by n matrix B is determined up to conjugacy by a partition of n formed by the sizes of the Jordan blocks of B. We call this partition the Jordan type of B. We obtain partial results on the following problem: for any partition P of n describe the type Q(P) of a generic nilpotent matrix commuting with a given nilpotent matrix of type P. A conjectural description for Q(P) was given by P. Oblak and restated by L. Khatami. In this paper we prove "half" of this conjecture by showing that this conjectural type is less than or equal to Q(P) in the dominance order on partitions.
Generic singularities of nilpotent orbit closures  [PDF]
Baohua Fu,Daniel Juteau,Paul Levy,Eric Sommers
Mathematics , 2015,
Abstract: According to a well-known theorem of Brieskorn and Slodowy, the intersection of the nilpotent cone of a simple Lie algebra with a transverse slice to the subregular nilpotent orbit is a simple surface singularity. At the opposite extremity of the nilpotent cone, the closure of the minimal nilpotent orbit is also an isolated symplectic singularity, called a minimal singularity. For classical Lie algebras, Kraft and Procesi showed that these two types of singularities suffice to describe all generic singularities of nilpotent orbit closures: specifically, any such singularity is either a simple surface singularity, a minimal singularity, or a union of two simple surface singularities of type $A_{2k-1}$. In the present paper, we complete the picture by determining the generic singularities of all nilpotent orbit closures in exceptional Lie algebras (up to normalization in a few cases). We summarize the results in some graphs at the end of the paper. In most cases, we also obtain simple surface singularities or minimal singularities, though often with more complicated branching than occurs in the classical types. There are, however, six singularities which do not occur in the classical types. Three of these are unibranch non-normal singularities: an $SL_2(\mathbb C)$-variety whose normalization is ${\mathbb A}^2$, an $Sp_4(\mathbb C)$-variety whose normalization is ${\mathbb A}^4$, and a two-dimensional variety whose normalization is the simple surface singularity $A_3$. In addition, there are three 4-dimensional isolated singularities each appearing once. We also study an intrinsic symmetry action on the singularities, in analogy with Slodowy's work for the regular nilpotent orbit.
Smallest complex nilpotent orbits with real points  [PDF]
Takayuki Okuda
Mathematics , 2014,
Abstract: Let us fix a complex simple Lie algebra and its non-compact real form. This paper focuses on non-zero adjoint nilpotent orbits in the complex simple Lie algebra meeting the real form. We show that the poset consisting of such nilpotent orbits equipped with the closure ordering has the minimum. Furthermore, we determine such the minimum orbit in terms of the Dynkin--Kostant classification even in the cases where the orbit does not coincide with the minimal nilpotent orbit in the complex simple Lie algebra. We also prove that the intersection of the orbit and the real form is the union of all minimal nilpotent orbits in the real form.
Some outer commutator multipliers and capability of nilpotent products of cyclic groups  [PDF]
Mohsen Parvizi,Behrooz Mashayekhy
Mathematics , 2012,
Abstract: In this paper, first we obtain an explicit formula for an outer commutator multiplier of nilpotent products of cyclic groups with respect to the variety $[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$, $\mathfrak{N}_{c}M(\mathbb{Z}\st{n}* \mathbb{Z}\st{n}* ... \st{n}* \mathbb{Z}\st{n}* \mathbb{Z}_{r_1}\st{n}* \mathbb{Z}_{r_2}\st{n}* ... \st{n}* \mathbb{Z}_{r_t})$ where $r_{i+1}\mid r_i \ \ (1\leq i\leq t-1)$, $c_1+c_2+1\geq n$, $2c_2-c_1>2n-2$ and $(p,r_1)=1$ for all prime less than or equal $c_1+c_2+n$, second we give a necessary condition for these groups to be $[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$-capable.
Heteroclinic cycles arising in generic unfoldings of nilpotent singularities  [PDF]
Pablo G. Barrientos,Santiago Ibá?ez,J. ángel Rodríguez
Mathematics , 2015, DOI: 10.1007/s10884-011-9230-5
Abstract: In this paper we study the existence of heteroclinic cycles in generic unfoldings of nilpotent singularities. Namely we prove that any nilpotent singularity of codimension four in $\mathbb{R}^4$ unfolds generically a bifurcation hypersurface of bifocal homoclinic orbits, that is, homoclinic orbits to equilibrium points with two pairs of complex eigenvalues. We also prove that any nilpotent singularity of codimension three in $\mathbb{R}^3$ unfolds generically a bifurcation curve of heteroclinic cycles between two saddle-focus equilibrium points with different stability indexes. Under generic assumptions these cycles imply the existence of homoclinic bifurcations. Homoclinic orbits to equilibrium points with complex eigenvalues are the simplest configurations which can explain the existence of complex dynamics as, for instance, strange attractors. The proof of the arising of these dynamics from a singularity is a very useful tool, particularly for applications.
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