Naive boundary strata and nilpotent orbits

Matt Kerr; Gregory Pearlstein

Displaying similar documents to “Naive boundary strata and nilpotent orbits”

Partial differential equations in Banach spaces involving nilpotent linear operators

Antonia Chinnì, Paolo Cubiotti (1996)

Annales Polonici Mathematici

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Let E be a Banach space. We consider a Cauchy problem of the type ⎧ $D_{t}^{k} u + \sum_{j = 0}^{k - 1} \sum_{| α | \leq m} A_{j, α} (D_{t}^{j} D_{x}^{α} u) = f$ in $ℝ^{n + 1}$ , ⎨ ⎩ $D_{t}^{j} u (0, x) = φ_{j} (x)$ in $ℝ^{n}$ , j=0,...,k-1, where each $A_{j, α}$ is a given continuous linear operator from E into itself. We prove that if the operators $A_{j, α}$ are nilpotent and pairwise commuting, then the problem is well-posed in the space of all functions $u \in C^{\infty} (ℝ^{n + 1}, E)$ whose derivatives are equi-bounded on each bounded subset of $ℝ^{n + 1}$ .

$𝒟_{n, r}$ is not potentially nilpotent for $n \geq 4 r - 2$

Yan Ling Shao, Yubin Gao, Wei Gao (2016)

Czechoslovak Mathematical Journal

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An $n \times n$ sign pattern $𝒜$ is said to be potentially nilpotent if there exists a nilpotent real matrix $B$ with the same sign pattern as $𝒜$ . Let $𝒟_{n, r}$ be an $n \times n$ sign pattern with $2 \leq r \leq n$ such that the superdiagonal and the $(n, n)$ entries are positive, the $(i, 1)$ $(i = 1, \dots, r)$ and $(i, i - r + 1)$ $(i = r + 1, \dots, n)$ entries are negative, and zeros elsewhere. We prove that for $r \geq 3$ and $n \geq 4 r - 2$ , the sign pattern $𝒟_{n, r}$ is not potentially nilpotent, and so not spectrally arbitrary.

The orbits of the Hurwitz action of the braid groups on the standard generators

Yoshiro Yaguchi (2010)

Fundamenta Mathematicae

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The Hurwitz action of the n-braid group Bₙ on the n-fold direct product $(B_{m}) ⁿ$ of the m-braid group $B_{m}$ is studied. We show that the orbit of any n- tuple of the n standard generators of $B_{n + 1}$ consists of the (n-1)th powers of n+1 elements.

The generalized Hodge and Bloch conjectures are equivalent for general complete intersections

Claire Voisin (2013)

Annales scientifiques de l'École Normale Supérieure

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We prove that Bloch’s conjecture is true for surfaces with $p_{g} = 0$ obtained as $0$ -sets $X_{σ}$ of a section $σ$ of a very ample vector bundle on a variety $X$ with “trivial” Chow groups. We get a similar result in presence of a finite group action, showing that if a projector of the group acts as $0$ on holomorphic $2$ -forms of $X_{σ}$ , then it acts as $0$ on $0$ -cycles of degree $0$ of $X_{σ}$ . In higher dimension, we also prove a similar but conditional result showing that the generalized Hodge conjecture for general $X_{σ}$ ...

One-parameter contractions of Lie-Poisson brackets

Oksana Yakimova (2014)

Journal of the European Mathematical Society

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We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra $𝒜 = 𝕂 [𝔸^{n}]$ is said to be of Kostant type, if its centre $Z (𝒜)$ is freely generated by homogeneous polynomials $F_{1}, ..., F_{r}$ such that they give Kostant’s regularity criterion on $𝔸^{n} (d_{x} F_{i}$ are linear independent if and only if the Poisson tensor has the maximal rank at $x$ ). If the initial Poisson algebra is of Kostant type and $F_{i}$ satisfy a certain degree-equality, then the contraction...

Retracts that are kernels of locally nilpotent derivations

Dayan Liu, Xiaosong Sun (2022)

Czechoslovak Mathematical Journal

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Let $k$ be a field of characteristic zero and $B$ a $k$ -domain. Let $R$ be a retract of $B$ being the kernel of a locally nilpotent derivation of $B$ . We show that if $B = R \oplus I$ for some principal ideal $I$ (in particular, if $B$ is a UFD), then $B = R^{[1]}$ , i.e., $B$ is a polynomial algebra over $R$ in one variable. It is natural to ask that, if a retract $R$ of a $k$ -UFD $B$ is the kernel of two commuting locally nilpotent derivations of $B$ , then does it follow that $B ≅ R^{[2]}$ ? We give a negative answer to this question. The interest in...

Leibniz's rule on two-step nilpotent Lie groups

Krystian Bekała (2016)

Colloquium Mathematicae

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Let be a nilpotent Lie algebra which is also regarded as a homogeneous Lie group with the Campbell-Hausdorff multiplication. This allows us to define a generalized multiplication $f g = {(f^{\lor} * g^{\lor})}^{\land}$ of two functions in the Schwartz class (*), where $^{\lor}$ and $^{\land}$ are the Abelian Fourier transforms on the Lie algebra and on the dual * and ∗ is the convolution on the group . In the operator analysis on nilpotent Lie groups an important notion is the one of symbolic calculus which can be viewed as a higher order...

Characterization of cycle domains via Kobayashi hyperbolicity

Gregor Fels, Alan Huckleberry (2005)

Bulletin de la Société Mathématique de France

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A real form $G$ of a complex semi-simple Lie group $G^{ℂ}$ has only finitely many orbits in any given $G^{ℂ}$ -flag manifold $Z = G^{ℂ} / Q$ . The complex geometry of these orbits is of interest, e.g., for the associated representation theory. The open orbits $D$ generally possess only the constant holomorphic functions, and the relevant associated geometric objects are certain positive-dimensional compact complex submanifolds of $D$ which, with very few well-understood exceptions, are parameterized by the Wolf cycle...

Hodge type decomposition

Wojciech Kozłowski (2007)

Annales Polonici Mathematici

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In the space $Λ^{p}$ of polynomial p-forms in ℝⁿ we introduce some special inner product. Let $H^{p}$ be the space of polynomial p-forms which are both closed and co-closed. We prove in a purely algebraic way that $Λ^{p}$ splits as the direct sum $d * (Λ^{p + 1}) \oplus δ * (Λ^{p - 1}) \oplus H^{p}$ , where d* (resp. δ*) denotes the adjoint operator to d (resp. δ) with respect to that inner product.

A new family of spectrally arbitrary ray patterns

Yinzhen Mei, Yubin Gao, Yan Ling Shao, Peng Wang (2016)

Czechoslovak Mathematical Journal

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An $n \times n$ ray pattern $𝒜$ is called a spectrally arbitrary ray pattern if the complex matrices in $Q (𝒜)$ give rise to all possible complex polynomials of degree $n$ . In a paper of Mei, Gao, Shao, and Wang (2014) was proved that the minimum number of nonzeros in an $n \times n$ irreducible spectrally arbitrary ray pattern is $3 n - 1$ . In this paper, we introduce a new family of spectrally arbitrary ray patterns of order $n$ with exactly $3 n - 1$ nonzeros.

A note on infinite $a S$ -groups

Reza Nikandish, Babak Miraftab (2015)

Czechoslovak Mathematical Journal

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Let $G$ be a group. If every nontrivial subgroup of $G$ has a proper supplement, then $G$ is called an $a S$ -group. We study some properties of $a S$ -groups. For instance, it is shown that a nilpotent group $G$ is an $a S$ -group if and only if $G$ is a subdirect product of cyclic groups of prime orders. We prove that if $G$ is an $a S$ -group which satisfies the descending chain condition on subgroups, then $G$ is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group...

The evolution and Poisson kernels on nilpotent meta-abelian groups

Richard Penney, Roman Urban (2013)

Studia Mathematica

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Let S be a semidirect product S = N⋊ A where N is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and A is isomorphic to $ℝ^{k}$ , k>1. We consider a class of second order left-invariant differential operators on S of the form $ℒ_{α} = L^{a} + Δ_{α}$ , where $α \in ℝ^{k}$ , and for each $a \in ℝ^{k}, L^{a}$ is left-invariant second order differential operator on N and $Δ_{α} = Δ - ⟨ α, \nabla ⟩$ , where Δ is the usual Laplacian on $ℝ^{k}$ . Using some probabilistic techniques (e.g., skew-product formulas for diffusions on S and N respectively)...

Bruhat-Tits theory from Berkovich’s point of view. I. Realizations and compactifications of buildings

Bertrand Rémy, Amaury Thuillier, Annette Werner (2010)

Annales scientifiques de l'École Normale Supérieure

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We investigate Bruhat-Tits buildings and their compactifications by means of Berkovich analytic geometry over complete non-Archimedean fields. For every reductive group $G$ over a suitable non-Archimedean field $k$ we define a map from the Bruhat-Tits building $ℬ (G, k)$ to the Berkovich analytic space $G^{an}$ associated with $G$ . Composing this map with the projection of $G^{an}$ to its flag varieties, we define a family of compactifications of $ℬ (G, k)$ . This generalizes results by Berkovich in the case of split groups. Moreover,...

Finite Groups with some $s$ -Permutably Embedded and Weakly $s$ -Permutable Subgroups

Fenfang Xie, Jinjin Wang, Jiayi Xia, Guo Zhong (2013)

Confluentes Mathematici

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Let $G$ be a finite group, $p$ the smallest prime dividing the order of $G$ and $P$ a Sylow $p$ -subgroup of $G$ with the smallest generator number $d$ . There is a set $ℳ_{d} (P) = {P_{1}, P_{2}, \dots, P_{d}}$ of maximal subgroups of $P$ such that $⋂_{i = 1}^{d} P_{i} = Φ (P)$ . In the present paper, we investigate the structure of a finite group under the assumption that every member of $ℳ_{d} (P)$ is either $s$ -permutably embedded or weakly $s$ -permutable in $G$ to give criteria for a group to be $p$ -supersolvable or $p$ -nilpotent.

On a generalization of a theorem of Burnside

Jiangtao Shi (2015)

Czechoslovak Mathematical Journal

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A theorem of Burnside asserts that a finite group $G$ is $p$ -nilpotent if for some prime $p$ a Sylow $p$ -subgroup of $G$ lies in the center of its normalizer. In this paper, let $G$ be a finite group and $p$ the smallest prime divisor of $| G |$ , the order of $G$ . Let $P \in {Syl}_{p} (G)$ . As a generalization of Burnside’s theorem, it is shown that if every non-cyclic $p$ -subgroup of $G$ is self-normalizing or normal in $G$ then $G$ is solvable. In particular, if $P ≇ 〈 a, b | a^{p^{n - 1}} = 1, b^{2} = 1, b^{- 1} a b = a^{1 + p^{n - 2}} 〉$ , where $n \geq 3$ for $p > 2$ and $n \geq 4$ for $p = 2$ , then $G$ is $p$ -nilpotent or $p$ -closed. ...

Characterization of the alternating groups by their order and one conjugacy class length

Alireza Khalili Asboei, Reza Mohammadyari (2016)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group, and let $N (G)$ be the set of conjugacy class sizes of $G$ . By Thompson’s conjecture, if $L$ is a finite non-abelian simple group, $G$ is a finite group with a trivial center, and $N (G) = N (L)$ , then $L$ and $G$ are isomorphic. Recently, Chen et al. contributed interestingly to Thompson’s conjecture under a weak condition. They only used the group order and one or two special conjugacy class sizes of simple groups and characterized successfully sporadic simple groups (see Li’s PhD dissertation)....

Iwasawa theory for symmetric powers of CM modular forms at non-ordinary primes

Robert Harron, Antonio Lei (2014)

Journal de Théorie des Nombres de Bordeaux

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Let $f$ be a cuspidal newform with complex multiplication (CM) and let $p$ be an odd prime at which $f$ is non-ordinary. We construct admissible $p$ -adic $L$ -functions for the symmetric powers of $f$ , thus verifying conjectures of Dabrowski and Panchishkin in this special case. We combine this with recent work of Benois to prove the trivial zero conjecture in this setting. We also construct “mixed” plus and minus $p$ -adic $L$ -functions and prove an analogue of Pollack’s decomposition of the admissible...