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Let O be the ring of integers of a number field, and let n ≥ 3 . This paper studies bi-interpretability of the ring of integers Z with the special linear group SL n ( O ) , the general linear group GL n ( O ) and the subgroup T n ( O ) of GL n ( O ) consisting of all the uppertriangular matrices. For each of these groups we provide a complete characterization of arbitrary models of their complete first-order theories.

Final steps are done in proving that the groups SL ⁡ ( n , Z ) \operatorname {SL}(n,\mathbb {Z}) , GL ⁡ ( n , Z ) \operatorname {GL}(n,\mathbb {Z}) and PGL ⁡ ( n , Z ) \operatorname {PGL}(n,\mathbb {Z}) are ( 2 , 3 ) (2,3) -generated if and only if n ≥ 5 n\ge 5 , and PSL ⁡ ( n , Z ) \operatorname {PSL}(n,\mathbb {Z}) is ( 2 , 3 ) (2,3) -generated if and only if n = 2 n=2 or n ≥ 5 n\ge 5 . In particular, the results cover the remaining cases of n = 8 n=8 , …, 12 12 , and 14 14 .

In this paper, we present a uniform formula for the integration of polynomials over the unitary, orthogonal, and symplectic groups using Weingarten calculus. From this description, we further simplify the integration formulas and give several optimizations for various cases. We implemented the optimized Weingarten calculus in Maple in a package called IntHaar for all three compact groups. Here we will discuss its functions, provide instructions for the package, and produce some examples of computed integrals., Comment: 21 pages, 4 figures, software available at: http://www.math.rutgers.edu/~ag930/

A graph \(X\) is said to be \(G\)-semisymmetric if it is regular and there exists a subgroup \(G\) of \(A := \operatorname{Aut}(X)\) acting transitively on its edge set but not on its vertex set. In the case of \(G = A\), we call \(X\) a semisymmetric graph. Finding elementary abelian covering projections can be grasped combinatorially via a linear representation of automorphisms acting on the first homology group of the graph. The method essentially reduces to finding invariant subspaces of matrix groups over prime fields. In this study, by applying concept linear algebra, we classify the connected semisymmetric \(z_{p}\)-covers of the \(C20\) graph.

Birkhoff's theorem tells that any doubly stochastic matrix can be decomposed as a weighted sum of permutation matrices. A similar theorem reveals that any unitary matrix can be decomposed as a weighted sum of complex permutation matrices. Unitary matrices of dimension equal to a power of 2 (say 2 w ) deserve special attention, as they represent quantum qubit circuits. We investigate which subgroup of the signed permutation matrices suffices to decompose an arbitrary such matrix. It turns out to be a matrix group isomorphic to the extraspecial group E 2 2 w + 1 + of order 2 2 w + 1 . An associated projective group of order 2 2 w equally suffices.

In this paper, W Z factorization is optimized with a proposed Cramer’s rule and compared with classical Cramer’s rule to solve the linear systems of the factorization technique. The matrix norms and performance time of WZ factorization together with LU factorization are analyzed using sparse matrices on MATLAB via AMD and Intel processor to deduce that the optimized Cramer’s rule in the factorization algorithm yields accurate results than LU factorization and conventional W Z factorization. In all, the matrix group and Schur complement for every Zsystem (2×2 block triangular matrices from Z-matrix) are established.

We investigate the structure of the multiplicative semigroup generated by the set of matrices that are unitarily equivalent to a given invertible matrix A. In particular, we give necessary and sufficient conditions for such a semigroup to be the special linear group or the general linear group or other classical matrix groups. Furthermore, we use the above to determine the subgroups of the general linear group that are normalized by the unitary group.

A complete procedure is described for constructing the irreducible KG-modules and their Brauer characters, where K is a finite field of characteristic p and G is a finite permutation or matrix group. The central idea is to construct a sequence { S 1 , … , S n } of KG-modules, each having relatively small dimension, such that each S i has one or more irreducible constituents that are not constituents of S 1 , … , S i − 1 . The Meataxe, used in conjunction with condensation, is used to extract the new irreducibles from each S i . The algorithm has been implemented in Magma and is capable of constructing irreducibles of dimension over 200 000 .

This paper concerns matrix decompositions in which the factors are restricted to lie in a closed subvariety of a matrix group. Such decompositions are of relevance in control theory: given a target matrix in the group, can it be decomposed as a product of elements in the subvarieties, in a given order? And if so, what can be said about the solution set to this problem? Can an irreducible curve of target matrices be lifted to an irreducible curve of factorisations? We show that under certain conditions, for a sufficiently long and complicated such sequence, the solution set is always irreducible, and we show that every connected matrix group has a sequence of one-parameter subgroups that satisfies these conditions, where the sequence has length less than 1.5 times the dimension of the group., 11 pages

We determine the minimal degree permutation representations of all finite groups with trivial soluble radical, and describe applications to structural computations in large finite matrix groups that use the output of the CompositionTree algorithm. We also describe how this output can be used to help find an effective base and strong generating set for such groups. We have implemented the resulting algorithms in Magma , and we report on their performance.

We call a function of k variables reducing for a multiplicative semigroup S ⊆ M n ( C ) , if the semigroup is reducible whenever f sends S to zero. For continuous reducing functions on groups of unitary matrices we prove that such a group is reducible whenever f is uniformly small enough on the group. Extensions of this result for certain types of semigroups are also given. Note that our results contain and extend many known results of the kind. In particular, it is shown that a group of complex unitary matrices is reducible if, for fixed indices i and j, its ( i , j ) entries are uniformly small.

We develop an efficient algorithm for sampling the eigenvalues of random matrices distributed according to the Haar measure over the orthogonal or unitary group. Our technique samples directly a factorization of the Hessenberg form of such matrices, and then computes their eigenvalues with a tailored core-chasing algorithm. This approach requires a number of floating-point operations that is quadratic in the order of the matrix being sampled, and can be adapted to other matrix groups. In particular, we explain how it can be used to sample the Haar measure over the special orthogonal and unitary groups and the conditional probability distribution obtained by requiring the determinant of the sampled matrix be a given complex number on the complex unit circle.

We determine the minimal number of separating invariants for the invariant ring of a matrix group $G < \mathrm{GL}_n(\mathbb{F}_q)$ over the finite field $\mathbb{F}_q$. We show that this minimal number can be obtained with invariants of degree at most $|G|n(q-1)$. In the non-modular case this construction can be improved to give invariants of degree at most $n(q-1)$. As examples we study separating invariants over the field $\mathbb{F}_2$ for two important representations of the symmetric group, 18 pages

If every element of a matrix group is similar to a permutation matrix, then it is called a permutation-like matrix group. References [3] , [4] and [5] showed that, if a permutation-like matrix group contains a maximal cycle such that the maximal cycle generates a normal subgroup and the length of the maximal cycle equals a prime, or a square of a prime, or a power of an odd prime, then the permutation-like matrix group is similar to a permutation matrix group. In this paper, we prove that if a permutation-like matrix group contains a maximal cycle such that the maximal cycle generates a normal subgroup and the length of the maximal cycle equals any power of 2, then it is similar to a permutation matrix group.

We study properties of the Hirzebruch class of quotient singularities $\mathbb{C}^n/G$, where $G$ is a finite matrix group. The main result states that the Hirzebruch class coincides with the Molien series of $G$ under suitable substitution of variables. The Hirzebruch class of a crepant resolution can be described specializing the orbifold elliptic genus constructed by Borisov and Libgober. It is equal to the combination of Molien series of centralizers of elements of $G$. This is an incarnation of the McKay correspondence. The results are illustrated with several examples, in particular of 4-dimensional symplectic quotient singularities., v2: 29 pages; introduction extended, several minor changes

We get a formula for all finite Riordan involutions. Using the natural approximation induced by the inverse limit construction, we also describe all infinite Riordan matrices which are involutions in the Riordan group. Finally, we get some consequences.

For x an element of a field other than 0 or 1, we compute the order n Massey products 〈 ( 1 − x ) − 1 , x − 1 , … , x − 1 , ( 1 − x ) − 1 〉 of n − 2 factors of x − 1 and two factors of ( 1 − x ) − 1 by embedding P 1 − { 0 , 1 , ∞ } into its Picard variety and constructing Gal ( k s / k ) equivariant maps from π 1 et applied to this embedding to unipotent matrix groups. This method produces obstructions to π 1 -sections of P 1 − { 0 , 1 , ∞ } , partial computations of obstructions of Jordan Ellenberg, and also computes the Massey products 〈 x − 1 , ( − x ) − 1 , … , ( − x ) − 1 , x − 1 〉 .

We consider what some authors call “parabolic Mobius subgroups” of matrices over Z , Q , and R and focus on the membership problem in these subgroups and complexity of relevant algorithms.

In the present article, the authors continue the line of inquiry started by Cigler and Jerman, who studied the separation of eigenvalues of a matrix under an action of a matrix group. The authors consider groups \Fam{G} of matrices of the form $\left[\small{\begin{smallmatrix} G & 0\\ 0& z \end{smallmatrix}}\right]$, where $z$ is a complex number, and the matrices $G$ form an irreducible subgroup of $\GL(\C)$. When \Fam{G} is not essentially finite, the authors prove that for each invertible $A$ the set $\Fam{G}A$ contains a matrix with more than one eigenvalue. The authors also consider groups $\Fam{G}$ of matrices of the form $\left[\small{\begin{smallmatrix} G & x\\ 0& 1 \end{smallmatrix}}\right]$, where the matrices $G$ comprise a bounded irreducible subgroup of $\GL(\C)$. When \Fam{G} is not finite, the authors prove that for each invertible $A$ the set $\Fam{G}A$ contains a matrix with more than one eigenvalue.

The still-unsolved problem of determining the set of eigenvalues realized by $n$-by-$n$ doubly stochastic matrices, those matrices with row sums and column sums equal to $1$, has attracted much attention in the last century. This problem is somewhat algebraic in nature, due to a result of Birkhoff demonstrating that the set of doubly stochastic matrices is the convex hull of the permutation matrices. Here we are interested in a general matrix group $G \subseteq GL_n(\mathbb{C})$ and the hull spectrum $\text{HS}(G)$ of eigenvalues realized by convex combinations of elements of $G$. We show that hull spectra of matrix groups share many nice properties. Moreover, we give bounds on the hull spectra of matrix groups, determine $\text{HS}(G)$ exactly for important classes of matrix groups, and study the hull spectra of representations of abstract groups., Comment: 19 pages. This work was completed at the 2019 Matrix Analysis REU at the College of William & Mary

Let S be a semigroup of invertible matrices. It is shown that if P is an idempotent matrix of rank and co-rank at least two such that the rank of ( 1 − P ) S P is never more than one for S in S (the range of the kind of P is said to be near-invariant), then S has an invariant subspace within one dimension of the range of P (the kind of range is said to be nearly invariant).

We use Nagao’s Theorem to construct matrix subgroups of dimension 2 over skew polynomial rings acting sharply transitively on the projective line of the quotient skew field. This way we get new examples of nearfields. Under additional conditions we prove that these nearfields are wild nearfields, by which we mean that they are not Dickson nearfields. To our knowledge these are the first known examples of nearfields of dimension greater than 2 over their kernel for which it could be certainly proven that they are wild.

Separability is one of the basic topological properties. Most classical topological groups and Banach spaces are separable, as examples we mention compact metric groups, matrix groups, connected (finite-dimensional) Lie groups, and the Banach spaces C ( K ) for metrizable compact spaces K, and ℓ p , for p &ge, 1 . This survey focuses on the wealth of results that have appeared in recent years about separable topological groups. In this paper, the property of separability of topological groups is examined in the context of taking subgroups, finite or infinite products, and quotient homomorphisms. The open problem of Banach and Mazur, known as the Separable Quotient Problem for Banach spaces, asks whether every Banach space has a quotient space which is a separable Banach space. This paper records substantial results on the analogous problem for topological groups. Twenty open problems are included in the survey.

In \cite{rump_goml}, Rump defined and characterized noncommutative universal groups $G(X)$ for generalized orthomodular lattices $X$. We give an explicit description of $G(X)$ in terms of \emph{paraunitary} matrix groups, whenever $X$ is the orthomodular lattice of subspaces of a finite-dimensional $k$-vector space $V$ that is equipped with an anisotropic, symmetric $k$-bilinear form., Comment: 20 pages; Comments are welcome!

We study the joint distribution of the solutions to the equation $gh=x$ in $G(\mathbb{F}_p)$ as $p\to\infty$, for any fixed $x\in G(\mathbb{Z})$, where $G=\operatorname{GL}_n$, $\operatorname{SL}_n$, $\operatorname{Sp}_{2n}$ or $\operatorname{SO}_{n}^\pm$. In the special linear case, this answers in particular a question raised by S. Hu and Y. Li, and improves their error terms. Similar results are derived in certain subgroups, and when the entries of $g,h$ lie in fixed intervals. The latter shows for example the existence of $g\in\operatorname{GL}_n(\mathbb{F}_p)$ such that $g,g^{-1}$ have all entries in $[0, c_np^{1-1/(2n^2+2)+\varepsilon}]$ for some absolute constant $c_n>0$. The key for these results is to use Deligne's extension of the Weil conjectures on a sheaf on $G$, along with the stratification theorem of Fouvry, Katz and Laumon, instead of reducing to bounds on classical Kloosterman sums., 16 pages; incorporating the referees' comments

Let be the ring of all matrices over a field . It is proved that a map is additive if and only if for all , the set of all invertible matrices over .

If every element of a matrix group is similar to a permutation matrix, then it is called a permutation-like matrix group. Refs. [3] and [4] showed that, if a permutation-like matrix group contains a maximal cycle of length equal to a prime or a square of a prime and the maximal cycle generates a normal subgroup, then it is similar to a permutation matrix group. In this paper, we prove that if a permutation-like matrix group contains a maximal cycle of length equal to any power of any odd prime and the maximal cycle generates a normal subgroup, then it is similar to a permutation matrix group.

We describe an algorithm to compute a composition tree for a matrix group defined over a finite field, and show how to use the associated structure to carry out computations with such groups; these include finding composition and chief series, the soluble radical, and Sylow subgroups.

We investigate the commutators of elements of the group of infinite unitriangular matrices over an associative ring with and a commutative group of invertible elements. We prove that every unitriangular matrix of a specified form is a commutator of two other unitriangular matrices. As a direct consequence, we give a complete characterization of the lower central series of the group including the width of its terms with respect to basic commutators and Engel words. With an additional restriction on the ring , we show that the derived subgroup of coincides with the group . These results generalize the results obtained for triangular groups over a field.

Let R denote the Galois ring of characteristic p s and cardinality p s h . In this paper, we determine the Smith normal forms of matrices over R , compute the number of the orbits of m × n matrices under the group GL m ( R ) × GL n ( R ) and the length of each orbit. Moreover, we discuss their applications to association schemes, systems of linear equations, generalized invertible matrices, idempotent matrices and involutory matrices.

A dominance rule for group invertible matrices using proper splitting is proposed, and used this notion to show that a matrix is group monotone. Then some possible applications are discussed.

A matrix group is said to be permutation-like if any matrix of the group is similar to a permutation matrix. G. Cigler proved that, if a permutation-like matrix group contains a normal cyclic subgroup which is generated by a maximal cycle and the matrix dimension is a prime, then the group is similar to a permutation matrix group. This paper extends the result to the case where the matrix dimension is a square of a prime.

We study products of matrices of fixed orders. We show that if g is an upper triangular matrix, finite or infinite, over a field of q elements, then g can be expressed as a product of at most four triangular matrices whose orders are divisors of q − 1 . This result can be applied to the general linear and to the Vershik–Kerov group. We also present some facts about conjugacy of elements of orders dividing q − 1 .

The purpose in this paper is to complete the classification of primitive sharp permutation groups of type ({0, k}, n) by proving that no such group can be almost simple.

We determine explicitly the Gauss sum τ l ( χ , e ) = ∑ X ∈ GL 2 ( Z / p l Z ) χ ( X ) e ( Tr X ) on the general linear group GL 2 ( Z / p l Z ) for every irreducible character χ of GL 2 ( Z / p l Z ) and a nontrivial additive character e of Z / p l Z , where p is an odd prime and l is an integer ⩾2. While there are several studies of the Gauss sums on finite algebraic groups defined over a finite field, this paper seems to be the first one which determines the Gauss sums on a matrix group over a finite ring.

Classical group representation theory is used to determine which finite groups have finite negative K-theory. There follows a computation of the K −1 of integral group rings ZG for all finite non-abelian subgroups of the group SU(2) of unit quaternions. In principle, the method applies to any finite group.

This article describes a setup that – given a composition tree – provides functionality for calculation in finite matrix groups using the Trivial-Fitting approach that has been used successfully for permutation groups. It treats the composition tree as a black-box object. It thus is applicable to other classes of groups for which a composition tree can be obtained. As an example, we consider an effective algorithm for determining conjugacy class representatives.

In this article, the invertible matrices over commutative semirings are studied. Some properties and equivalent descriptions of the invertible matrices are given and the inverse matrix of an invertible matrix is presented by analogues of the classic adjoint matrix. Also, Cramer's rule over a commutative semiring is established. The main results obtained in this article generalize the corresponding results for matrices over commutative rings, for lattice matrices, for incline matrices, for matrices over zerosumfree semirings and for matrices over additively regular semirings.

Many key invariants in the representation theory of classical groups (symmetric groups $S_n$, matrix groups $GL_n$, $O_n$, $Sp_{2n}$) are polynomials in $n$ (e.g., dimensions of irreducible representations). This allowed Deligne to extend the representation theory of these groups to complex values of the rank $n$. Namely, Deligne defined generically semisimple families of tensor categories parametrized by $n\in \mathbb{C}$, which at positive integer $n$ specialize to the classical representation categories. Using Deligne's work, Etingof proposed a similar extrapolation for many non-semisimple representation categories built on representation categories of classical groups, e.g., degenerate affine Hecke algebras (dAHA). It is expected that for generic $n\in \mathbb{C}$ such extrapolations behave as they do for large integer $n$ ("stabilization"). The goal of our work is to provide a technique to prove such statements. Namely, we develop an algebro-geometric framework to study categories indexed by a parameter $n$, in which the set of values of $n$ for which the category has a given property is constructible. This implies that if a property holds for integer $n$, it then holds for generic complex $n$. We use this to give a new proof that Deligne's categories are generically semisimple. We also apply this method to Etingof's extrapolations of dAHA, and prove that when $n$ is transcendental, "finite-dimensional" simple objects are quotients of certain standard induced objects, extrapolating Zelevinsky's classification of simple dAHA-modules for $n\in \mathbb{N}$. Finally, we obtain similar results for the extrapolations of categories associated to wreath products of the symmetric group with associative algebras., 26 pages; Comments welcome

We study the real elements in triangular matrix groups. We describe some classes of elements that are real in T n (K) – the groups of upper triangular matrices over a commutative field K. From the obtained results there follow some applications for finding real elements in general linear groups – GL n (K).

Firstly, we give conditions for the existence of the group inverse of the 2 × 2 matrixM=acbdover an arbitrary ring R with unity 1. Our second purpose is to derive a representation for the group inverse of M in the case when either the entry a or d has group inverse in R. We illustrate our results with some numerical examples in the setting of 2 × 2 complex block matrices.

In this article we consider the elements in upper triangular finite and infinite dimensional matrix groups over fields, whose order is equal to (). For the case when the characteristic of does not divide, we give a description of such elements. Next using this criterion, we show how to find the number of these elements in finite dimensional groups when is a finite field.

We provide upper and lower bounds for the number of completely reducible homomorphisms from a finite group Γ to general linear and unitary groups over arbitrary finite fields, and to orthogonal and symplectic groups over finite fields of odd characteristic.

This is the first in a series of four papers on fixed point ratios in actions of finite classical groups. Our main result states that if G is a finite almost simple classical group and Ω is a faithful transitive non-subspace G-set then either fpr(x)&lesssim;|xG|−1/2 for all elements x∈G of prime order, or (G, Ω) is one of a small number of known exceptions. Here fpr(x) denotes the proportion of points in Ω which are fixed by x. In this introductory note we present our results and describe an application to the study of minimal bases for primitive permutation groups. A further application concerning monodromy groups of covers of Riemann surfaces is also outlined.

This paper investigates the critical group of a faithful representation of a finite group. It computes the order of the critical group in terms of the character values, and gives some restrictions on its subgroup structure. It also computes the precise structure of the critical group both for the regular representation of any finite group, and for the reflection representation of the symmetric group., 8 pages

In this note, we give explicit expressions of Gauss sums for general (resp. special) linear groups over finite fields, which involves Gauss sums (resp. Kloosterman sums). The key ingredient is averaging such sums over Borel subgroups. As applications, we count the number of invertible matrices of zero-trace over finite fields and we also improve two bounds by Ferguson, Hoffman, Luca, Ostafe and Shparlinski in [ Some additive combinatorics problems in matrix rings, Rev. Mat. Complut. (23) 2010, 501--513 ]., 9 pages

Almost periodic homogeneous linear difference systems are considered. It is supposed that the coefficient matrices belong to a group. The aim was to find such groups that the systems having no non-trivial almost periodic solution form a dense subset of the set of all considered systems. A closer examination of the used methods reveals that the problem can be treated in such a generality that the entries of coefficient matrices are allowed to belong to any complete metric field. The concepts of transformable and strongly transformable groups of matrices are introduced, and these concepts enable us to derive efficient conditions for determining what matrix groups have the required property.

The sign pattern of a real matrix M is the (0, 1, −1)-matrix obtained from M by replacing each entry by its sign. Let N ∈ ℝ n×n be a group invertible matrix. Let Q(N) be the set of real matrices with the same sign pattern as N. For any , if is group invertible and the group inverses of N and have the same sign pattern, then N is called an S2GI matrix. In this article, we present the existence and the representations for the group inverse of some block matrices with one or two full rank sub-blocks, and give a family of block matrices which are S2GI matrices. Applying these results, we can partially determine the sign pattern of the solution of singular linear system with index one.

Let G and H be groups of complex n × n matrices. We say that G is an H -like group if every matrix in G is similar to a matrix from H . For several groups H we consider two questions: (A) Is every H -like group (simultaneously) similar to a subgroup of H ? (B) Is H the only H -like group containing H ? Among other results we prove that the symmetric group S n is the only S n -like group containing S n .