Loren Spice
The notion of a topological Jordan decomposition of a compact element of a reductive p-adic group has proven useful in many contexts. In this paper, we generalise it to groups defined over fairly general discretely-valued fields and prove the usual existence and uniqueness properties, as well as an analogue of a fixed-point result of Prasad and Yu.
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As shown by Bonnafé, a step in proving a Jordan decomposition of characters of finite special linear groups is the parametrization of unipotent characters of centralizers of semi-simple elements in projective linear groups. We show the same kind of result in the case of finite special unitary groups. The proof leads to a mild adaptation of Bonnafé's methods expounded in [B99]. The outcome is a Jordan decomposition of characters compatible with Lusztig's twisted induction.
The Bonnafé-Rouquier equivalence can be seen as a modular analogue of Lusztig's Jordan decomposition for groups of Lie type. In this paper, we show that this equivalence can be lifted to include automorphisms of the finite group of Lie type. Moreover, we prove the existence of a local version of this equivalence which satisfies similar properties.
In this article we prove that the elliptic, hyperbolic and nilpotent (or unipotent) additive (or multiplicative) Jordan components of an endomorphism $X$ (or an isomorphism $g$) of a finite dimensional vector space are given by polynomials in $X$ (or in $g$). By using this, we provide a simple proof that, for an element $X$ of a linear semisimple Lie algebra $\g$ (or $g$ of a linear semisimple connected Lie group $G$), its three Jordan components lie again in the algebra (in the group). This was previously unknown for linear Lie groups other then $\Int(\g)$. This implies that, for this class o…
Galois automorphisms and a unique Jordan decomposition in the case of connected centralizer
本文证明有限约化群特征存在Galois等变的Jordan分解，并在连通中心化子情形下给出唯一性，类比Digne--Michel的连通中心结果。
We prove a Jordan decomposition theorem for minimal connected simple groups of finite Morley rank with non-trivial Weyl group. From this, we deduce a precise structural description of Borel subgroups of this family of simple groups. Along the way we prove a Tetrachotomy theorem that classifies minimal connected simple groups. Some of the techniques that we develop help us obtain a simpler proof of a theorem of Burdges, Cherlin and Jaligot.
Let $\mathbf{G}$ be a connected reductive group with connected center defined over $\mathbb{F}_q$, with Frobenius morphism F. Given an irreducible complex character $χ$ of $\mathbf{G}^F$ with its Jordan decomposition, and a Galois automorphism $σ\in \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, we give the Jordan decomposition of the image ${^σχ}$ of $χ$ under the action of $σ$ on its character values.
If $K$ is a field of finite characteristic $p$, $G$ is a cyclic group of order $q=p^α$, $U$ and $W$ are indecomposable $KG$-modules with $\dim U=m$ and $\dim W=n$, and $λ(m,n,p)$ is a standard Jordan partition of $ m n$, we describe how to find a generator for each of the indecomposable components of the $KG$-module $U \otimes W$.
\input amssym.def \input amssym.tex Let $G$ be a connected algebraic reductive group over an algebraic closure of a prime field ${\Bbb F}_p$, defined over ${\Bbb F}_q$ thanks to a Frobenius $F$. Let $\ell$ be a prime different from $p$. Let $B$ be an $\ell$-block of the subgroup of rational points $G^F$. Under mild restrictions on $\ell$, we show the existence of an algebraic reductive group $H$ defined over ${\Bbb F}_q$ {\it via} a Frobenius $F$, and of a unipotent $\ell$-block $b$ of $H^F$ such that : the respective defect groups of $b$ and $B$ are isomorphic, the associated Brauer categorie…
The Alperin weight conjecture was reduced to simple groups by the work of Navarro, Tiep and Späth. To prove Alperin weight conjecture, it suffices to show that all finite non-abelian simple groups are BAW-good. We reduce the verification of the inductive conditons for groups of Lie type in non-defining characteristic to quasi-isolated blocks.
Let ${\mathbf G}^F$ be a finite group of Lie type, where ${\mathbf G}$ is a reductive group defined over ${\overline{\mathbb F}_q}$ and $F$ is a Frobenius root. Lusztig's Jordan decomposition parametrises the irreducible characters in a rational series${\mathcal E}({{\mathbf G}^F},(s)_{{\mathbf G}^{*F^*}})$ where $s\in{{\mathbf G}^{*F^*}}$ by the series ${\mathcal E}(C_{{\mathbf G}^*}(s)^{F^*},1)$.We conjecture that the Shintani twisting preserves the space of class functions generated by the union of the ${\mathcal E}({{\mathbf G}^F},(s')_{{\mathbf G}^{*F^*}})$ where$(s')_{{\mathbf G}^{*F^*}}…
For finite p-groups P of class 2 and exponent p the following are invariants of fully refined central decompositions of P: the number of members in the decomposition, the multiset of orders of the members, and the multiset of orders of their centers. Unlike for direct product decompositions, Aut P is not always transitive on the set of fully refined central decompositions, and the number of orbits can in fact be any positive integer. The proofs use the standard semi-simple and radical structure of Jordan algebras. These algebras also produce useful criteria for a p-group to be centrally indeco…
Let $\g$ be a semisimple Lie algebra and $G = \Int(\g)$. In this article, we relate the Jordan decomposition of $X \in \g$ (or $g \in G$) with the dynamics induced on generalized flag manifolds by the right invariant continuous-time flow generated by $X$ (or the discrete-time flow generated by $g$). We characterize the recurrent set and the finest Morse decomposition (including its stable sets) of these flows and show that its entropy always vanishes. We characterize the structurally stable ones and compute the Conley index of the attractor Morse component. When the nilpotent part of $X$ is tr…
We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If k is the minimal degree of a representation of the finite group G, then for every subset B of G with $|B| > |G| / k^{1/3}$ we have B^3 = G. We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan's theorem which implies that if k>1, then G has a proper subgroup of index at most ck^2 for some absolute constant c, hence a product-…
Jordan-Holder Theorem for profinite groups and applications
本文将合成列与合成因子概念推广到投射有限群，证明其Jordan-Hölder定理，并应用于无限投射可解扩张的Galois定理，同时探讨非强完全投射有限群中抽象与拓扑合成因子的关系。
In this paper we prove several results about the lattice of imprimitivity systems of a permutation group containing a cyclic subgroup with at most two orbits. As an application we generalize the first Ritt theorem about functional decompositions of polynomials, and some other related results. Besides, we discuss examples of rational functions, related to finite subgroups of the automorphism group of the sphere for which the first Ritt theorem fails to be true.
The purpose of this paper is to point the effectiveness of the Jordan-Chevalley decomposition, i.e. the decomposition of a square matrix $U$ with coefficients in a field $k$ containing the eigenvalues of $U$ as a sum $U=D+N,$ where $D$ is a diagonalizable matrix and $N$ a nilpotent matrix which commutes with $D.$ The most general version of this decomposition shows that every separable element $u$ of a $k$-algebra $A$ can be written in a unique way as a sum $u=d+n,$ where $d \in A$ is absolutely semi-simple and where $n\in A$ is nilpotent and commutes with $d.$ In fact an algorithm, due to C. …
Jordan correspondence and block distribution of characters
本文确定了拟单例外李型群特征标的ℓ-块分布（除Jordan分解非唯一性的微小歧义），推导了ℓ-块、e-Harish-Chandra级数与Jordan分解的相容性，并完成了Robinson特征标亏群猜想的证明。
Let $V$ be a finite-dimensional vector space over a field of characteristic two. As the main result of this paper, for every nilpotent element $e \in \mathfrak{sl}(V)$, we describe the Jordan normal form of $e$ on the $\mathfrak{sl}(V)$-modules $\wedge^2(V)$ and $S^2(V)$. In the case where $e$ is a regular nilpotent element, we are able to give a closed formula. We also consider the closely related problem of describing, for every unipotent element $u \in \operatorname{SL}(V)$, the Jordan normal form of $u$ on $\wedge^2(V)$ and $S^2(V)$. A recursive formula for the Jordan block sizes of $u$ on…
We define symmetric spaces in arbitrary dimension and over arbitrary non-discrete topological fields $\K$, and we construct manifolds and symmetric spaces associated to topological continuous quasi-inverse Jordan pairs and -triple systems. This class of spaces, called smooth generalized projective geometries, generalizes the well-known (finite or infinite-dimensional) bounded symmetric domains as well as their ``compact-like'' duals. An interpretation of such geometries as models of Quantum Mechanics is proposed, and particular attention is paid to geometries that might be considered as "stand…
A Jordan-Chevalley decomposition beyond algebraic groups
本文在 o-minimal 结构中推广了 Jordan-Chevalley 分解，证明可定义线性群可分解为无挠正规子群与半代数子群的半直积，并建立了 Levi 分解、不连通群分解及 p-群与 0-群的相关结果。
An idempotent in a Jordan algebra induces a Peirce decomposition of the algebra into subspaces whose pairwise multiplication satisfies certain fusion rules $Φ(\frac{1}{2})$. On the other hand, $3$-transposition groups $(G,D)$ can be algebraically characterised as Matsuo algebras $M_α(G,D)$ with idempotents satisfying the fusion rules $Φ(α)$ for some $α$. We classify the Jordan algebras $J$ which are isomorphic to a Matsuo algebra $M_{1/2}(G,D)$, in which case $(G,D)$ is a subgroup of the (algebraic) automorphism group of $J$; the only possibilities are $G = \operatorname{Sym}(n)$ and $G = 3^2:…
We prove that the closure of every Jordan class J in a semisimple simply connected complex group G at a point x with Jordan decomposition x = rv is smoothly equivalent to the union of closures of those Jordan classes in the centraliser of r that are contained in J and contain x in their closure. For x unipotent we also show that the closure of J around x is smoothly equivalent to the closure of a Jordan class in Lie(G) around exp^{-1}x. For G simple we apply these results in order to determine a (non-exhaustive) list of smooth sheets in G, the complete list of regular Jordan classes whose clos…
We study properties of continuous finite group actions on topological manifolds that hold true, for any finite group action, after possibly passing to a subgroup of index bounded above by a constant depending only on the manifold. These include the Jordan property, the almost fixed point property, as well as bounds on the discrete symmetry group. Most of our results apply to manifolds satisfying some restriction such as having nonzero Euler characteristic or having the integral homology of a sphere. For an arbitrary topological manifold $X$ such that $H_*(X;{\mathbf Z})$ is finitely generated,…
Étale geometry of closures of Jordan classes
本文将复数域上研究约化代数群若尔当类闭包几何性质的局部方法推广到特征$p$的代数闭域，给出了方法成立的$p$的充分限制条件。
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