Fritz J. Grunewald, Natalia K. Iyudu
We give an effective solution of the conjugacy problem for two by two matrices over the polynomial ring in one variable over a finite field.
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Modules over orders, conjugacy classes of integral matrices, and abelian varieties over finite fields
本文提出算法，用于计算半单整数矩阵的共轭类代表元，以及有限域上阿贝尔簇的同构类，在特定条件下（普通或素数域无实根）有效。
In this paper we describe the set of conjugacy classes in the group SL(n,Z). We expand geometric Gauss Reduction Theory that solves the problem for SL(2,Z) to the multidimensional case. Further we find complete invariant of classes in terms of multidimensional Klein-Voronoi continued fractions, where $ς$-reduce Hessenberg matrices play the role of reduced matrices.
Drinfeld Modular Curves Subordinate to Conjugacy Classes of Nilpotent Upper-Triangular Matrices
本文引入归一化Drinfeld模曲线，参数化与幂零上三角矩阵共轭类相关的$T$-挠结构，建立矩阵分类与曲线分解的树状联系，推广塔结构，并证明$(3,2)$型曲线的几何不可约性。
We determine those k-tuples of conjugacy classes of matrices, from which it is possible to choose matrices which have no common invariant subspace and have sum zero. This is an additive version of the Deligne-Simpson problem. We deduce the result from earlier work of ours on preprojective algebras and the moment map for representations of quivers. Our answer depends on the root system for a Kac-Moody Lie algebra.
We study the possible dimension vectors of indecomposable parabolic bundles on the projective line, and use our answer to solve the problem of characterizing those collections of conjugacy classes of n by n matrices for which one can find matrices in their closures whose product is equal to the identity matrix. Both answers depend on the root system of a Kac-Moody Lie algebra. Our proofs use Ringel's theory of tubular algebras, work of Mihai on the existence of logarithmic connections, the Riemann-Hilbert correspondence and an algebraic version, due to Dettweiler and Reiter, of Katz's middle c…
We introduce a new method to study rational conjugacy of torsion units in integral group rings using integral and modular representation theory. Employing this new method, we verify the first Zassenhaus Conjecture for the group $\operatorname{PSL}(2,19)$. We also prove the Zassenhaus Conjecture for $\operatorname{PSL}(2,23)$. In a second application we show that there are no normalized units of order $6$ in the integral group rings of $M_{10}$ and $\operatorname{PGL}(2,9)$. This completes the proof of a theorem of W. Kimmerle and A. Konovalov that the Prime Graph Question has an affirmative an…
Conjugacy classes of regular integer matrices
本文研究了正则整数矩阵在 $GL_n({\mathbb Z})$ 下的共轭类理论，通过特征多项式与全格的关系，探讨了其在不同情况下的结构与性质。
A proof of a theorem of M. Hertweck presented during a seminar in January 2013 in Stuttgart is given. The proof is based on a preprint given to me by Hertweck. Let $R$ be a commutative ring, $G$ a finite group, $N$ a normal $p$-subgroup of $G$ and denote by $RG$ the group ring of $G$ over $R$. It is shown that a torsion unit $u$ in $\mathbb{Z}G$ mapping to the identity under the natural homomorphism $\mathbb{Z}G \rightarrow \mathbb{Z}G/N$ is conjugate in the unit group of $\mathbb{Z}_pG$ to an element in $N$. Here $\mathbb{Z}_p$ denotes the $p$-adic integers. The result is achieved proving a r…
Polynomials over division rings
该文研究除环上多项式的性质，得到任意除环上多项式的分解定理，证明非中心非根元素的共轭类含无穷多非根元素，还估计了四元数可除代数上几类多项式球面根的共轭类数量。
We say that two elements of a group or semigroup are $\Bbbk$-linear conjugates if their images under any linear representation over $\Bbbk$ are conjugate matrices. In this paper we characterize $\Bbbk$-linear conjugacy for finite semigroups (and, in particular, for finite groups) over an arbitrary field $\Bbbk$.
Recently we showed that Hessenberg matrices are proper to represent conjugacy classes in SL(n,Z). In this paper we focus on the reducibility properties in the set of Hessenberg matrices of SL(3,Z). We investigate the first interesting open case here: the case of matrices having one real and two complex conjugate eigenvalues.
Derivations of rings of infinite matrices
本文刻画了一般系数环上无限矩阵的若干重要结合环与李环的导子结构。
The literature about strongly clean matrices over commutative rings is quite extensive. The sharpest results are about matrices over commutative local rings, for example those by Borooah, Diesl and Dorsey. The purpose of this note is to show that, using Pierce sheaf techniques, many of the known results about matrices over commutative local rings can be extended to those over commutative clean rings in general.
Let $k = \mathbb{Q}(\sqrtα)$ be a real quadratic number field, where $α$ is a positive square-free integer. Let $\mathcal{O}_k$ be the ring of integers of $k$. In this paper, we prove that a certain set of $2 \times 2$ singular matrices with entries in $\mathcal{O}_k$ can be written as a product of a bounded number of idempotent matrices. Our main theorem can be viewed as a generalization of the recent result by Cossu and Zanardo, which studies finite generation of certain singular matrices by idempotent matrices over $\mathcal{O}_k$ instead of bounded generation of certain singular matrices b…
In this paper, the determinants of $n\times n$ matrices over commutative finite chain rings and over commutative finite principal ideal rings are studied. The number of $n\times n$ matrices over a commutative finite chain ring ${R}$ of a fixed determinant $a$ is determined for all $a\in {R}$ and positive integers $n$. Using the fact that every commutative finite principal ideal ring is a product of commutative finite chain rings, the number of $n\times n$ matrices of a fixed determinant over a commutative finite principal ideal ring is shown to be multiplicative, and hence, it can be determine…
An element of a ring $R$ is called strongly $J^{\#}$-clean provided that it can be written as the sum of an idempotent and an element in $J^{\#}(R)$ that commute. We characterize, in this article, the strongly $J^{\#}$-cleanness of matrices over projective-free rings. These extend many known results on strongly clean matrices over commutative local rings.
Lie and Jordan Isomorphisms of Algebras of Triangular Matrices over Associative Rings
研究结合环上三角矩阵代数的李同构与若尔当同构，描述其相关性质。
Let U be the group of units of an infinite twisted group algebra K_λG over a field K. We describe the maximal FC-subgroup of U and give a characterization of U with finitely conjugacy classes. In the case of group algebras we obtain the Cliff-Sehgal-Zassenhaus' theorem.
Circulant matrices over finite fields and over commutative finite chain rings have been of interest due to their nice algebraic structures and wide applications. In many cases, such matrices over rings have a closed connection with diagonal matrices over their extension rings. In this paper, the determinants of diagonal and circulant matrices over commutative finite chain rings $R$ with residue field $\mathbb{F}_q$ are studied. The number of $n\times n$ diagonal matrices over ${R}$ of determinant $a$ is determined for all elements $a$ in $ {R}$ and for all positive integers $n$. Subsequently, …
We exhibit a Cayley-Hamilton trace identity for $2\times2$ matrices with entries in a ring $R$ satisfying $[[x,y],[x,z]]=0$ and 1/2 \in R$.
An element $a$ in a ring $R$ is strongly J-clean if it is the sum of an idempotent and an element in the Jacobson radical that commutes. We characterize the strongly J-clean $2\times 2$ matrices over 2-projective-free non-commutative rings.
An element $a\in R$ is very clean provided that there exists an idempotent $e\in R$ such that $ae=ea$ and either $a-e$ or $a+e$ is invertible. A ring $R$ is very clean in case every element in $R$ is very clean. We explore the necessary and sufficient conditions under which a triangular $2\times 2$ matrix ring over local rings is very clean. The very clean $2\times 2$ matrices over commutative local rings are completely determined. Applications to matrices over power series are also obtained.
A skew polynomial ring $R=K[x;σ,δ]$ is a ring of polynomials with non-commutative multiplication. This creates a difference between left and right divisibility, and thus a concept of left and right evaluations and roots. A polynomial in such a ring may have more roots than its degree, which leads to the concepts of closures and independent sets of roots. There is also a structure of conjugacy classes on the roots. In $R=F_{q^m}[x,σ]$, this leads to matroids of right independent and left independent sets. These matroids are isomorphic via the extension of the map $ϕ:[1]\to[1]$ defined by $ϕ(a)=…
On the Waring's problems for matrices over a commutative ring, there are some trace conditions provided for matrices eligibly expressed as a sum of $k$-th powers with $k=2,3,4,5,6,7,8$ in several literatures. In this paper, we provide the similar conditions for matrices written as a sum of $k$-th powers with $k=9,10,11,12,13,14,15,16$.
An element $a\in R$ is provided that there exists an idempotent $e\in R$ such that $a-e\in U(R), ae=ea$ and $eae\in J(eRe)$. In this article, we investigate strongly rad-clean matrices over a commutative local ring. We completely determine when a $2\times 2$ matrix over a commutative local ring is strongly rad-clean.
The Deligne-Simpson problem (DSP) (resp. the weak DSP) is formulated like this: {\em give necessary and sufficient conditions for the choice of the conjugacy classes $C_j\subset GL(n,{\bf C})$ or $c_j\subset gl(n,{\bf C})$ so that there exist irreducible (resp. with trivial centralizer) $(p+1)$-tuples of matrices $M_j\in C_j$ or $A_j\in c_j$ satisfying the equality $M_1... M_{p+1}=I$ or $A_1+... +A_{p+1}=0$}. The matrices $M_j$ and $A_j$ are interpreted as monodromy operators of regular linear systems and as matrices-residua of Fuchsian ones on Riemann's sphere. For $(p+1)$-tuples of conjugacy…
Motivated by Hirano-Tominaga's work \cite{HT} on rings for which every element is a sum of two idempotents and by de Seguins Pazzis's results \cite{de} on decomposing every matrix over a field of positive characteristic as a sum of idempotent matrices, we address decomposing every matrix over a commutative ring as a sum of three idempotent matrices and, respectively, as a sum of three involutive matrices.
Group schemes over LG-rings and applications to cancellation theorems and Azumaya algebras
本文证明LG-环上约化群概形的极大环面存在性与抛物子群共轭性，推广SGA3半局部环结果，并应用于建立埃尔米特与二次型消去定理及Azumaya代数Brauer类的唯一代表与分解。
A ring R is Zhou nil-clean if every element in R is the sum of two tripotents and a nilpotent that commute. Let R be a Zhou nil-clean ring. If R is 2-primal (of bounded index), we prove that every square matrix over R is the sum of two tripotents and a nilpotent. These provides a large class of rings over which every square matrix has such decompositions by tripotent and nilpotent matrices.
Consider the Deligne-Simpson problem: {\em give necessary and sufficient conditions for the choice of the conjugacy classes $C_j\subset GL(n,{\bf C})$ (resp. $c_j\subset gl(n,{\bf C})$) so that there exist irreducible $(p+1)$-tuples of matrices $M_j\in C_j$ (resp. $A_j\in c_j$) satisfying the equality $M_1... M_{p+1}=I$ (resp. $A_1+... +A_{p+1}=0$)}. The matrices $M_j$ and $A_j$ are interpreted as monodromy operators and as matrices-residua of fuchsian systems on Riemann's sphere. We give new examples of existence of such $(p+1)$-tuples of matrices $M_j$ (resp. $A_j$) which are {\em rigid}, i.…
It is proven that every commutative semihereditary Bezout ring in which any regular element is Gelfand (adequate), is an elementary divisor ring.
In this paper, we give the complete structures of the equivalence canonical form of four matrices over an arbitrary division ring. As applications, we derive some practical necessary and sufficient conditions for the solvability to some systems of generalized Sylvester matrix equations using the ranks of their coefficient matrices. The results of this paper are new and available over the real number field, the complex number field, and the quaternion algebra.
We describe recent work on preprojective algebras and moduli spaces of their representations. We give an analogue of Kac's Theorem, characterizing the dimension types of indecomposable coherent sheaves over weighted projective lines in terms of loop algebras of Kac-Moody Lie algebras, and explain how it is proved using Hall algebras. We discuss applications to the problem of describing the possible conjugacy classes of sums and products of matrices in known conjugacy classes.
We constuct the theory of diagonalizability for matrices over Bezout rings of stable range 1 with the Kazimirsky condition. It is shown that a ring of stable range 1 with the right (left) Kazimirsky condition is an elementary divisor ring if and only if it is a duo ring.
Given a ring $T_n, n\geqslant 2$, of lower triangular $n\times n$ matrices with entries from an arbitrary field $F$, a complete description is performed of the orbits of free cyclic submodules of $^2T_n$, under the action of the general linear group $GL_2(T_n)$. It is given its total number, which is equal to the Bell number $B_n$, and there are shown their representatives.
We define the notions of finite-state and functionally recursive matrices and their growth. We also introduce two rings generated by functionally recursive matrices. The first is isomorphic to the 2-generated free ring. The second is a 2-generated monomial ring such that the multiplicative semigroup of monomials in the generators is nil of degree 5 and the ring has Gelfand Kirillov dimension 1 + log(2)/log(a) where a=1/2(1+sqrt(5)).
Let $ m , n \in \mathbb{N}$, $D$ be a division ring, and $M_{m \times n}(D)$ denote the bimodule of all $m \times n$ matrices with entries from $D$. First, we characterize one-sided submodules of $M_{m \times n}(D)$ in terms of left row reduced echelon or right column reduced echelon matrices with entries from $D$. Next, we introduce the notion of a nest module of matrices with entries from $D$. We then characterize submodules of nest modules of matrices over $D$ in terms of certain finite sequences of left row reduced echelon or right column reduced echelon matrices with entries from $D$. We …
In this paper we prove that if $R$ is a commutative refinement ring and $M$, $N$ are two $R$-modules then, $M\cong N$ if and only if for every maximal ideal $m$ of $R$, $M_m\cong N_m$. We prove if $R$ is a refinement ring, then every regular matrix over $\frac{R}{J(R)}$ admits a diagonal reduction iff every regular matrix over $R$ admits a diagonal reduction.
If A is an n \times n matrix over a ring R satisfying the polynomial identity [x,y][u,v]=0, then an invariant Cayley-Hamilton identity of the form ΣA^{i}c_{i,j}A^{j}=0 with c_{i,j}\in R and c_{n,n}=(n!)^2 holds for A.
Suppose that R is an ordered ring, G_n(R) is a subsemigroup of $GL_n(R)$, consisting of all matrices with nonnegative elements. A.V. Mikhalev and M.A. Shatalova described all automorphisms of G_n(R), where R is a linearly ordered skewfield and n>1. E.I. Bunina and A.V. Mikhalev found all automorphisms of G_n(R), if R is an arbitrary linearly ordered associative ring with 1/2, n>2. In this paper we describe automorphisms of G_n(R), if R is a commutative partially ordered ring, containing Q, n>2.
The paper is devoted to studying the orthogonality graph of the matrix ring over a commutative ring. It is proved that the orthogonality graph of the ring of matrices with size greater than 1 over a commutative ring with zero-divisors is connected and has diameter 3 or 4; a criterion for each value is obtained. It is also shown that each of its vertices has distance at most 2 from some scalar matrix.
Arithmetic of matrices over rings
本书研究交换有限生成主理想整环上矩阵环的算术，构建矩阵分解理论，并阐明其与完全线性群子群性质及单侧等价标准形的关系。
In this paper, we prove that the semigroups of invertible matrices with nonnegative elements over linearly oredered associative rings are elementarily equivalent if and only if the matrices have the same dimension and the rings are elementarily equivalent as ordered rings.
We consider the {\em Deligne-Simpson problem}: {\em Give necessary and sufficient conditions for the choice of the conjugacy classes $c_j\subset gl(n,{\bf C})$ or $C_j\subset GL(n,{\bf C})$, $j=1,..., p+1$, so that there exist irreducible $(p+1)$-tuples of matrices $A_j\in c_j$ whose sum is 0 or of matrices $M_j\in C_j$ whose product is $I$.} The matrices $A_j$ (resp. $M_j$) are interepreted as matrices-residua of Fuchsian linear systems (resp. as monodromy operators of regular systems) on Riemann's sphere. We consider the case when the sum of the dimensions of the conjugacy classes $c_j$ or $…
For an nxn matrix A over a Lie nilpotent ring R of index k, we prove that an invariant "power" Cayley-Hamilton identity of degree (n^2)2^{k-2} holds. The right coefficients are not uniquely determined by A, and the cosets lambda_i+D, with D the double commutator ideal R[[R,R],R]R of R, appear in the so-called second right characteristic polynomial of the natural image of A in the nxn matrix ring M_{n}(R/D) over the factor ring R/D.
Consider rectangular matrices over a local ring R. In the previous work we have obtained criteria for block-diagonalization of such matrices, i.e. U A V=A_1\oplus A_2, where U,V are invertible matrices over R. In this short note we extend the criteria to the decomposability of quiver representations over R.
In the present paper we prove that every 2-local inner derivation on the Lie ring of skew-symmetric matrices over a commutative ring is an inner derivation. We also apply our technique to various Lie algebras of infinite dimensional skew-adjoint matrix-valued maps on a set and prove that every 2-local spatial derivation on such algebras is a spatial derivation.
Associating to each pre-order on the indices 1,...,n the corresponding structural matrix ring, or incidence algebra, embeds the lattice of n-element pre-orders into the lattice of n x n matrix rings. Rings within the order-convex hull of the embedding, i.e. matrix rings that contain the ring of diagonal matrices, can be viewed as incidence algebras of ideal-valued, generalized pre-order relations. Certain conjugates of the upper or lower triangular matrix rings correspond to the various linear orderings of the indices, and the incidence algebras of partial orderings arise as intersections of s…
In this paper similarity classes of three by three matrices over a local principal ideal commutative ring are analyzed. When the residue field is finite, a generating function for the number of similarity classes for all finite quotients of the ring is computed explicitly.
We prove that for every trace zero matrix $A$ over a principal ideal ring $R$, there exist trace zero matrices $X,Y$ over $R$ such that $XY-YX=A$. Moreover, we show that $X$ can be taken to be regular mod every maximal ideal of $R$. This strengthens our earlier result that $A$ is a commutator of two matrices (not necessarily of trace zero), and in addition, the present proof is significantly simpler than the earlier one. Shalev has conjectured an analogous statement for group commutators in $\mathrm{SL}_{n}$ over $p$-adic integers. We prove Shalev's conjecture for $n=2$.
A commutative ring $R$ is projective free provided that every finitely generated $R$-module is free. An element in a ring is strongly clean provided that it is the sum of an idempotent and a unit that commutates. Let $R$ be a projective-free ring, and let $h\in R[t]$ be a monic polynomial of degree $n$. We prove, in this article, that every $φ\in M_n(R)$ with characteristic polynomial $h$ is strongly clean, if and only if the companion matrix $C_h$ of $h$ is strongly clean, if and only if there exists a factorization $h=h_0h_1$ such that $h_0\in {\Bbb S}_0, h_1\in {\Bbb S}_1$ and $(h_0,h_1)=1$…
Invertibility of structured perturbations of singular matrices over unital rings
本文研究单位环上奇异矩阵经结构化扰动后的可逆性，证明在核与像的自然假设下可逆性等价于扰动矩阵可逆，并给出逆的显式公式，推广了Eriksson和Nordqvist的定理。
Idempotent factorization on some matrices over quadratic integer rings
本文针对Cossu和Zanardo关于二次整数环上特定奇异矩阵幂等分解的猜想，构造了支持与反对该猜想的矩阵类，并证明存在不能分解为两个幂等矩阵乘积的此类矩阵。
Products of Traceless and Semi-Traceless Matrices over Division Rings and their Applications
研究除环上矩阵表示为无迹或半无迹矩阵乘积的问题，证明每个矩阵至多可分解为十二个无迹矩阵或四个半无迹矩阵之积，并将结果推广至有限矩阵与广义换位子乘积分解。
Representing in Low Rank I: conjugacy, topological and homological aspects
本文研究了有限群的低秩表示性质，探讨了同调良好性质、共轭类以及同余核等关键问题。
In this paper, we construct self-dual codes from a construction that involves 2x2 block circulant matrices, group rings and a reverse circulant matrix. We provide conditions whereby this construction can yield self-dual codes. We construct self-dual codes of various lengths over F2, F2 + uF2 and F4 + uF4. Using extensions, neighbours and neighbours of neighbours, we construct 32 new self-dual codes of length 68.
Mutation Of Matrices Over Group Rings
本文研究了群环上斜对称可化矩阵的突变定义及其与具有对称性的有向无环图的突变关系，并提出了新的突变规则。
Let R_{n,d} be the ring of invariants of d-tuples of n x n matrices under the simultaneous conjugation action of the general linear group. A minimal generating system and a homogeneous system of parameters for R_{3,3} are determined. Homogeneous systems of parameters for R_{3,2}, R_{4,2} are also pointed out.
The paper studies algebraic strong shift equivalence of matrices over $n$-variable polynomial rings over a principal ideal domain $D$($n\leq 2$). It is proved that in the case $n=1$, every non-zero matrix over $D[x]$ has a full rank factorization and every non-nilpotent matrix over $D[x]$ is algebraically strong shift equivalent to a nonsingular matrix. In the case $n=2$, an example of non-nilpotent matrix over $\mathbb{R}[x,y,z]=\mathbb{R}[x][y,z]$, which can not be algebraically shift equivalent to a nonsingular matrix, is given.
We consider the variety of $(p+1)$-tuples of matrices $A_j$ (resp. $M_j$) from given conjugacy classes $c_j\subset gl(n,{\bf C})$ (resp. $C_j\subset GL(n,{\bf C})$) such that $A_1+... +A_{p+1}=0$ (resp. $M_1... M_{p+1}=I$). This variety is connected with the weak {\em Deligne-Simpson problem: give necessary and sufficient conditions on the choice of the conjugacy classes $c_j\subset gl(n,{\bf C})$ (resp. $C_j\subset GL(n,{\bf C})$) so that there exist $(p+1)$-tuples with trivial centralizers of matrices $A_j\in c_j$ (resp. $M_j\in C_j$) whose sum equals 0 (resp. whose product equals $I$).} The…
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