Hamid Khodabandeh, Mohammad Shahryari
Using a unified method, we determine the structure of automorphisms and representations of arbitrary polyadic groups. More precisely, for a polyadic group $(G, f)=der_{θ, b}(G, \cdot)$, we obtain a complete description of automorphisms and representations of $(G,f)$ in terms of automorphisms and representations of the binary group $(G, \cdot)$.
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We prove that there is a one-one correspondence between sets of irreducible representations of a polyadic group and its Post's cover. Using this correspondence, we generalize some well-known properties of irreducible characters in finite groups to the case of polyadic groups.
In this article, we introduce the notion of representations of polyadic groups and we investigate the connection between these representations and those of retract groups and covering groups.
Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomorphisms, a 'heteromorphism' which connects polyadic systems having unequal arities, is introduced via an explicit formula, together with related definitions for multiplace representations and multiactions. Concrete examples of matrix representations for some ternary groups are then reviewed. Ternary algebras and Hopf algebras are defined, and their properties are studied. At the end some ternary generalizations …
Inadmissible representations of the tree automorphism group
研究树自同构群的不可接受表示，发现存在不可接受的不可约Banach表示，光滑向量子空间不含代数不可约分量。
Linear Representations of the Automorphism Group of a Free Group
设$F_n$是$n\ge 2$元素上的自由群，$\A(F_n)$是它的自同构群.本文给出了$\A(F_n)$的一组丰富的线性表示，它们是由它的有限指数子群在$F_n$的有限商群的关系模上的作用而产生的.我们证明（在某些条件下）我们的表示的图像是算术群。
Bounded Cohomology and Unitary Representations of Automorphism Groups of Regular Trees
本文首次计算了正则树自同构群在所有正度数下的连续有界上同调，系数来自任何不可约连续酉表示。
The faithfulness of an extension of Lawrence-Krammer representation on the group of conjugating automorphisms $C_n$ in the cases $n=3$ and $n=4$
本研究针对共轭自同构群$C_n$上Lawrence-Krammer表示的扩张$\rho$的忠实性问题，填补了$n=3$、$n=4$情形的研究空白，证明这两种情形下该表示均不忠实。
Irreducible representations of tree automorphism groups into Pontryagin spaces
研究树自同构群在有限索引连续强非退化半双线性形式上的连续不可约表示，证明索引大于1时无更多表示，完成分类。
Complex Representations of Groups and Involutions of its Automorphisms
本文建立了不可约特征标次数和与有限群自同构相关的扭曲对合数之间的关系，并开发了内自同构和对称群情形下的算法框架，进而给出了识别群具有复（非实）不可约表示的条件并探讨了结构推论。
Functor morphing and representations of automorphism groups of modules
本文提出利用函子形变研究局部环上有限模自同构群的不可约表示，证明这些群具有层级结构，可借助对称幺半范畴和不变量理论以更小的构建块对其表示进行分层。
The structure of the group of conjugating automorphisms and the linear representation of the braid groups of some manifolds
本文描述了自由群的共轭自同构群$C_n$的结构，并证明了这种结构类似于具有$n>1$弦的辫群$B_n$的结构.我们找到了群$C_n$的线性表示。证明了2-球面的辫群$B_n(S^2)$、$n$ --穿孔2-球面的映射类群M（0，n）和射影平面的辫群$B_3(P^2)$是线性的.利用J. Dyer，E. Formanek，E. Grossman和Lawrence-Krammer的忠实线性表示，构造了自同构群$B_4$的忠实线性表示.
Let $\mathfrak g_i$ be a simple complex Lie algebra, $1\leq i \leq d$, and let $G=G_1\times...\times G_d$ be the corresponding adjoint group. Consider the $G$-module $V=\oplus r_i\mathfrak g_i$ where $r_i\geq 1$ for all $i$. We say that $V$ is \emph{large} if all $r_i\geq 2$ and $r_i\geq 3$ if $G_i$ has rank 1. In [Schwarz12] we showed that when $V$ is large any algebraic automorphism $ψ$ of the quotient $Z:= V//G$ lifts to an algebraic mapping $Ψ\colon V\to V$ which sends the fiber over $z$ to the fiber over $ψ(z)$, $z\in Z$. (Most cases were already handled in [Kuttler11]). We also showed th…
Coherent state representations of the holomorphic automorphism group of the tube domain over the dual of the Vinberg cone
本文分类了 Vinberg 锥对偶上管域的全纯自同构群的所有不可约相干态表示。
Let $C_n$ be the group of conjugating automorphisms. We study the representation $ρ$ of $C_n$, an extension of Lawrence-Krammer representation of the braid group $B_n$, defined by Valerij G. Bardakov. As Bardakov proved that the representation $ρ$ is unfaithful for $n \geq 5$, the cases $n=3,4$ remain open. In our work, we make attempts towards the faithfulness of $ρ$ in the case $n=3$.
Representation theory of the group of automorphisms of a finite rooted tree
本研究构造了有限根树自同构群的常不可约表示并实现天然参数化，通过引入树合成组合学导出对应置换表示，方法适用于非球对称树且无坐标依赖。
Maximal Rigid Representations of Continuous Quivers of Type A with Automorphism
本文利用Buan和Krause对循环箭图最大刚性表示的分类结果，推导出了带自同构的A型连续箭图一类最大刚性表示的同构类数量公式。
Quiver representations respecting a quiver automorphism: a generalisation of a theorem of Kac
我们考虑了关于一个非线性自同构的非线性表示，并证明了不可分解的非线性表示的维数向量恰好是一个相关的可对称化Kac-Moody李代数的正根.而且，每个这样的李代数都是这样产生的。我们还讨论了有限域上的值箭图与类的关系。
On certain representations of automorphism groups of an algebraically closed field
设k是特征为零的代数闭域，F是其代数闭扩张，G是F的k-自同构群，G具有自然拓扑.本文的目的之一是证明G因子的任何非忠实连续表示都可以通过G的一个离散商表示。研究了由几何产生的G的表示的性质。在某些情况下，几何对象之间的态射群与相应的G-模之间的态射群是相同的，并且${\rm Ext}^1$是相关的。特别地，具有有理数张紧态射的k上的阿贝尔簇范畴可以被描述为G-模范畴。
The bar automorphism in quantum groups and geometry of quiver representations
给出了量子化包络代数正部分的条自同构的两种几何解释。第一个是在数量方面的有理点的有限域的类似物的轨道品种;第二个是在一个对偶的构造功能提供的预投射品种的箭。
Automorphisms of complex reflection groups
设$G\subset\GL(\BC^r)$为有限复反射群。我们表明，当$G$是不可约的，除了例外$G=\Sgot_6$，以及为一大类的非不可约群，任何自同构的$G$是产品的中心自同构和自同构，保持反射。我们进一步表明，保持反射的自同构是$N_{\GL(\BC^r)}(G)$的元素和“伽罗瓦”自同构的乘积：我们证明了$\Gal(K/\BQ)$，其中$K$是$G$的定义域，注入到$G$的外自同构群中，并且可以选择这种注入，使得它引起对$G$的特征的通常伽罗瓦作用，除了少数例外的字符;此外，如果需要，用2次的扩展替换$K$，注入可以提升到$\Aut(G)$，并且每个不可约表示允许关于该提升等变的模型。一路上，我们表明，$G$的基本不变量可以选择合理的。
Let $G$ be the real reductive group and let $G_0$ be the identity component. Let us assume that the unitary dual $\hat{G_0}$ is known. In this paper (in Section 5) the unitary dual $\hat{G}$ is constructed. Automorphisms of $G_0$ generated by elements of $G$ are the main ingredient of the construction. If the automorphism is outer, one has to consider the corresponding intertwining operators $S$. Operators $S$ and their properties are analyzed in Section 4. Automorphisms of $g_0$ are closely related to automorphisms of $G_0$. They are investigated in Section 3. Automorphisms of so(4,4)$ are an…
Coherent state representations of the holomorphic automorphism group of a quasi-symmetric Siegel domain of type III
本文研究了III型拟对称Siegel域全纯自同构群的泛相干态表示分类，揭示了其与Kähler代数结构理论的联系，并提出基于广义Iwasawa分解的方法。
Representations of automorphism groups of finite O-modules of rank two
设O是具有有限剩余域的完备离散赋值整环.本文描述了群Aut（M）对任意秩为2的有限O-模M的不可约表示。主要强调的是不同群体之间的相互作用和他们的代表。一个诱导方案的开发，以研究这些群体的整个家庭连贯。所得到的结果依赖于环O在一个非常弱的方式，主要是通过程度的剩余域。特别地，得到了GL（2，O/P^k）的不可约表示的统一描述，其中P是O的极大理想.
Unitary $L^{p+}$-representations of almost automorphism groups
本文研究具有 Kunze-Stein 开子群的局部紧群，证明诱导表示保持 $L^{p+}$ 性质，并据此构造树的几乎自同构群及 Neretin 群中不同 $p$ 阈值的酉表示。
The Representations of Automorphism Groups of $\mathfrak{o}$-modules of type $(\ell,1^n)$
本文针对有限剩余域的非阿基米德局部域的赋值环上的特定模，提出通过归纳计算其自同构群的表示 zeta 多项式，并证明表示维数可由有限多项式在 $q$ 处取值得到。
Given a group $G$ of automorphisms of a matroid $M$, we describe the representations of $G$ on the homology of the independence complex of the dual matroid $M^*$. These representations are related with the homology of the lattice of flats of $M$, and (when $M$ is realizable) with the top cohomology of a hyperplane arrangement. Finally we analyze in detail the case of the complete graph, which has applications to algebraic geometry.
Representations of field automorphism groups
这是一个常见的数学入门。RT/0101170，math.RT/0306333，math.RT/0506043，math.RT/0601028。与这些文献相比，得到了新的结果，包括：（i）描述了代数闭域的超越度为1的扩张的可分闭包;（ii）曲线乘积的“Künneth公式”;（iii）最高次正则型模$Ω^n_{F/k}$的半单纯性.
Consider a discrete valuation ring $R$ whose residue field is finite of cardinality at least $3$. For a finite torsion module, we consider transitive subsets $O$ under the action of the automorphism group of the module. We prove that the associated permutation representation on the complex vector space $C[O]$ is multiplicity free. This is achieved by obtaining a complete description of the transitive subsets of $O\times O$ under the diagonal action of the automorphism group.
Groups represented by incidence geometries
本研究借助关联几何框架构建群的关联几何表示理论，可同时建模群的内、外自同构，并给出多类群的相关表示示例。
Let $Q$ be a connected quiver with no oriented cycles, $k$ the field of complex numbers and $P$ a projective representation of $Q$. We study the adjoint action of the automorphism group $\Aut_{kQ} P$ on the space of radical endomorphisms $\radE_{kQ}P$. Using generic equivalence, we show that the quiver $Q$ has the property that there exists a dense open $\Aut_{kQ} P$-orbit in $\radE_{kQ} P$, for all projective representations $P$, if and only if $Q$ is a Dynkin quiver. This gives a new characterisation of Dynkin quivers.
On the automorphism groups of complex homogeneous spaces
若G是（连通）复李群，Z是G的广义旗流形，则G的（连通）实形式G0的开轨道D构成一类有趣的复齐性空间，它们在G0的表示论中起着重要作用.我们发现自同构的自同构群，即，全纯李群是一个有限维李群，除了少数的开轨道，在那里它是无限维的。在有限维的情况下，我们确定它的结构。我们的结果在表示论中有一些结果。
Representation stability and outer automorphism groups
本文通过构造推广VI-模范畴的阿贝尔范畴$\u2131\u2132$，研究外自同构群的表示族，探究了使该范畴局部诺特的有限群族$\u2131$，推导了对应表示稳定性结论，并证明有理整体同伦论的部分不变量具有表示稳定性。
Matrices for finite group representations that respect Galois automorphisms
该研究在有限群表示与伽罗瓦理论交叉领域，证明了满足特征标作用匹配条件的绝对不可约表示，其“存在对应伽罗瓦作用的矩阵形式”与“诱导表示舒尔指数为1”等价，并给出两个具体群表示实例验证。
A remark on automorphisms of tensor spaces
本文证明一大类置换群可实现为无穷维张量空间的自同构群，并据此揭示张量空间作为其自同构群表示时可能呈现病态行为。
We construct Hopf algebras whose elements are representations of combinatorial automorphism groups, by generalising a theorem of Zelevinsky on Hopf algebras of representations of wreath products. As an application we attach symmetric functions to representations of graph automorphism groups, generalising and refining Stanley's chromatic symmetric function.
In this paper, we prove the categories of lower bounded twisted modules of positive integer levels for simple vertex operator algebras associated with affine Lie algebras and general automorphisms are semisimple, using the twisted generalization of Zhu's algebra for these vertex operator algebras, constructed in \cite{HY}. We also show that the category of lower bounded twisted modules for a general automorphism is equivalent to the category of lower bounded twisted modules for the corresponding diagram automorphism.
On maximal proper subgroups of field automorphism groups
设$G$是特征为零且超越度为$n$的代数闭域的扩张$F|k$的自同构群，$1\le n\le\infty$ .本文（i）构造了极大闭非开子群$G_v$，并构造了极大闭非开子群（ii）在超越度为可数的情况下，刻画了$G$的极大开真子群;（iii）在超越度为可数的情况下，刻画了中间子域上的自同构子群（Krull的一个问题，\cite [ \S 4，问题3b）]{krull}）;（iii）在$n=\infty$的情况下，构建从$G$的光滑表示范畴到光滑表示范畴的遗忘函子的完全忠实子函子$(-)_v$（iv）使用函子$G_v$，构造$G$的光滑表示范畴上的恒等函子$Γ$，在$n=\infty$和$k=\bar{\mathbb Q}$的情况下，与构造在\cite{adm}中的$G$的光滑容许半线性表示范畴上的函子$Γ$一致（通过遗忘函子）。开子群的研究是由\cite{repr,adm}中进行的光滑表示（的稳定器）的研究激发的。函子$Γ$是光滑真代数簇上层范畴上的整体截面函子的类似物。另一个结果是“有趣的”半线性表示是“全局生成的”。
Periodic automorphisms of Takiff algebras, contractions, and $θ$-groups
设$q$是一个代数李代数，$q<m>$是一个（广义）Takiff代数. $q$的任何有限阶自同构$θ$都导出同阶的$q<m>$的自同构，记为$Θ$。我们研究了$Θ$的不动点子代数在$q<m>$中$Θ$的其他特征空间上的表示的不变性理论性质。
Hilbert spaces with generic groups of automorphisms
设G是可数群。证明了具有自同构群G的希尔伯特空间的近似理论存在一个模型伴侣。我们证明了G是顺从的当且仅当由G的正则表示的可数拷贝所诱导的结构是存在闭的。
Let $(W,S)$ be a Coxeter system and $Γ$ be a group of automorphisms of $W$ such that $γ(S)=S$ for all $γ\in Γ$. Then it is known that the group of fixed points $W^Γ$ is again a Coxeter group with a canonically defined set of generators. The usual proofs of this fact rely on the reflection representation of $W$. Here, we give a proof which only uses the combinatorics of reduced expressions in $W$. As a by-product, this shows that the length function on $W$ restricts to a weight function on $W^Γ$.
Let $Q$ be an algebraic group with $q=\Lie Q$ and $V$ a $Q$-module. The index of $V$ is the minimal codimension of the $Q$-orbits in the dual space $V^*$. There is a general inequality, due to Vinberg, relating the index of $V$ and the index of a $Q_v$-module $V/q.v$ for $v\in V$. A pair $(Q,V)$ is said to have GIB if Vinberg's inequality turns into an equality for all $v\in V$. In this article, we are interested in the GIB property of $θ$-representations, where $θ$ is a finite order automorphism of a simple Lie algebra $g$. An automorphism of order $m$ defines a $Z/mZ$-grading $g=g_0+g_1+...+…
For a local field $F$ and an Artinian local coefficient ring $Λ$ with the same positive residue characteristic $p$ we define, for any $e\in{\mathbb N}$, a category ${\mathfrak C}^{(e)}(Λ)$ of ${\rm GL}_2(F)$-equivariant coefficient systems on the Bruhat-Tits tree $X$ of ${\rm PGL}_2(F)$. There is an obvious functor from the category of ${\rm GL}_2(F)$-representations over $Λ$ to ${\mathfrak C}^{(e)}(Λ)$. If $F={\mathbb Q}_p$ then ${\mathfrak C}^{(1)}(Λ)$ is equivalent to the category of smooth ${\rm GL}_2({\mathbb Q}_p)$-representations over $Λ$ generated by their invariants under a pro-$p$-Iw…
Classification of Automorphism Groups of Smooth Cubic Threefolds and Fourfolds
本文分类了光滑三次三维簇与四维簇的自同构群，确定了156个四维簇和40个三维簇的连通族及其群作用和定义方程，方法结合表示论与格论。
We study the symplectic leaves of the subvariety of fixed points of an automorphism of a Calogero-Moser space induced by an element of finite order of the normalizer of the associated complex reflection group $W$. We give a parametrization {\it à la Harish-Chandra} of its symplectic leaves (generalizing earlier works of Bellamy and Losev). This result is inspired by the mysterious relations between the geometry of Calogero-Moser spaces and unipotent representations of finite reductive groups, which will be the theme of a forthcoming paper.
We study the subvariety of fixed points of an automorphism of a Calogero-Moser space induced by a regular element of finite order of the normalizer of the associated complex reflection group $W$. We determine some of (and conjecturally all) the ${\mathbb{C}}^\times$-fixed points of its unique irreducible component of maximal dimension in terms of the character table of $W$. This is inspired by the mysterious relations between the geometry of Calogero-Moser spaces and unipotent representations of finite reductive groups, which will be the theme of a forthcoming paper.
On twisted representations of vertex algebras
在本文中，我们开发了一个形式主义与扭曲实现的顶点和共形代数。作为例子，我们研究了共形代数的扭形式幂级数实现。我们的技术的主要应用是一个非常大的家庭表示的顶点超代数$\goth V_Λ$对应于一个整数格$Λ$的建设。对于由有穷级自同构$σ:Λ\to Λ$生成的自同构$\^σ:\goth V_Λ\to\goth V_Λ$，我们定义了$\goth V_Λ$的扭曲表示范畴$\cal O_{\^σ}$，并证明了这个范畴是半单的，它包含了许多简单对象的同构类.
The Image of Functor Morphing
本文研究了函子变形的像，通过抛物子群作用提出判别准则，并在有限域的博雷尔群上验证。
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