Group Of Upper Triangular Matrices Research Articles

On groups in which irreducible systems of elements form a matroid

Matroid is defined as a pair $(X,\mathcal{I})$, where $X$ is a non-empty finite set, and $\mathcal{I}$ is a non-empty set of subsets of $X$ that satisfies the hereditary axiom and the augmentation axiom. The paper investigates for which groups (primarily finite) $G$, the pair $(\widehat{G}, \mathcal{I})$ will be a matroid. The obtained criteria of matroidality for finite and infinite abelian groups, for finite nilpotent, finite symmetric, and finite dihedral groups, as well as for certain classes of finite matrix groups, are presented. Additionally, the non-matroidality of a whole range of finite groups has been proven, including Hamiltonian groups, groups of diagonal matrices, general and special linear groups, groups of upper triangular matrices with determinant $1$, and others.

On Galois representations with large image

For every prime number p ≥ 3 p\geq 3 and every integer m ≥ 1 m\geq 1 , we prove the existence of a continuous Galois representation ρ : G Q → G l m ( Z p ) \rho : G_\mathbb {Q} \rightarrow Gl_m(\mathbb {Z}_p) which has open image and is unramified outside { p , ∞ } \{p,\infty \} if p ≡ 3 p\equiv 3 mod 4 4 and is unramified outside { 2 , p , ∞ } \{2,p,\infty \} if p ≡ 1 p \equiv 1 mod 4 4 . We also revisit the question of the lifting of residual Galois representations in terms of embedding problems; that allows us to produce Galois representations with open image in the group of upper triangular matrices with diagonal entries equal to 1 1 , unramified outside { p , ∞ } \{p,\infty \} , for m m “small” comparing to p p .

Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories

In this paper, we use a geometric technique developed by Gonz\'alez-Prieto, Logares, Mu\~noz, and Newstead to study the $G$-representation variety of surface groups $\mathfrak{X}_G(\Sigma_g)$ of arbitrary genus for $G$ being the group of upper triangular matrices of fixed rank. Explicitly, we compute the virtual classes in the Grothendieck ring of varieties of the $G$-representation variety and the moduli space of $G$-representations of surface groups for $G$ being the group of complex upper triangular matrices of rank $2$, $3$, and $4$ via constructing a topological quantum field theory. Furthermore, we show that in the case of upper triangular matrices the character map from the moduli space of $G$-representations to the $G$-character variety is not an isomorphism.

Conjugacy classes of centralizers in the group of upper triangular matrices

Let [Formula: see text] be a group. Two elements [Formula: see text] are said to be in the same [Formula: see text]-class if their centralizers in [Formula: see text] are conjugate within [Formula: see text]. In this paper, we prove that the number of [Formula: see text]-classes in the group of upper triangular matrices is infinite provided that the field is infinite and size of the matrices is at least [Formula: see text], and finite otherwise.

Menon-type identities derived from actions of subgroups of general linear groups

In this article, we consider the actions of subgroups of the general linear group GLr(Zn) on Znr, including groups of upper triangular matrices in GLr(Zn), unipotent groups, Heisenberg groups and extended Heisenberg groups. By applying the Cauchy–Frobenius–Burnside lemma, we obtain several generalizations of the well known Menon's identity.

Closed expressions for averages of set partition statistics

Abstract In studying the enumerative theory of super characters of the group of upper triangular matrices over a finite field, we found that the moments (mean, variance, and higher moments) of novel statistics on set partitions of [n]={1,2,⋯,n} have simple closed expressions as linear combinations of shifted bell numbers. It is shown here that families of other statistics have similar moments. The coefficients in the linear combinations are polynomials in n. This allows exact enumeration of the moments for small n to determine exact formulae for all n.

Real elements in T n (K)

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).

Group measure space decomposition of II1 factors and W*-superrigidity

We prove a "unique crossed product decomposition" result for group measure space II_1 factors arising from arbitrary free ergodic probability measure preserving (p.m.p.) actions of groups \Gamma in a fairly large family G, which contains all free products of a Kazhdan group and a non-trivial group, as well as certain amalgamated free products over an amenable subgroup. We deduce that if T_n denotes the group of upper triangular matrices in PSL(n,Z), then any free, mixing p.m.p. action of the amalgamated free product of PSL(n,Z) with itself over T_n, is W*-superrigid, i.e. any isomorphism between L^\infty(X) \rtimes \Gamma and an arbitrary group measure space factor L^\infty(Y) \rtimes \Lambda, comes from a conjugacy of the actions. We also prove that for many groups \Gamma in the family G, the Bernoulli actions of \Gamma are W*-superrigid.

Biorthogonal Laurent Polynomials, Toplitz Determinants, Minimal Toda Orbits and Isomonodromic Tau Functions

We consider the class of biorthogonal polynomials that are used to solve the inverse spectral problem associated to elementary co-adjoint orbits of the Borel group of upper triangular matrices; these orbits are the phase space of generalized integrable lattices of Toda type. Such polynomials naturally interpolate between the theory of orthogonal polynomials on the line and orthogonal polynomials on the unit circle and tie together the theory of Toda, relativistic Toda, Ablowitz-Ladik and Volterra lattices. We establish corresponding Christoffel-Darboux formulae. For all these classes of polynomials a 2 × 2 system of Differential-Difference-Deformation equations is analyzed in the most general setting of pseudo-measures with arbitrary rational logarithmic derivative. They provide particular classes of isomonodromic deformations of rational connections on the Riemann sphere. The corresponding isomonodromic tau function is explicitly related to the shifted Toplitz determinants of the moments of the pseudo-measure. In particular, the results imply that any (shifted) Toplitz (Hankel) determinant of a symbol (measure) with arbitrary rational logarithmic derivative is an isomonodromic tau function.

ACTION OF THE BOREL GROUP ON MONOMIAL IDEALS

ABSTRACT Let be the Borel group of upper triangular matrices. In this paper we want to study the action of on the set of monomial ideals of ( a field of characteristic zero) from a computational point of view. More specifically, we show that the stabilizer of a monomial ideal in is a purely combinatorial object and we give an algorithm for computing it. Then we characterize the subgroups of that are stabilizers of monomial ideals, we give an algorithm which finds if a given ideal is in the orbit of a monomial ideal under the action of and in the affirmative case, finds the matrices such that . We show that the entries of can be directly obtained from the coefficients of the generators of , so in particular no solutions of polynomial equations are required.

The Ring Generated by the Elements of Degree 2 in H*(Un([formula omitted]p), [formula omitted

We compute all the relations in cohomology satisfied by the elements of degree two of H*(Un(Fp),Z) where p≥n and Un(Fp) is the group of upper triangular matrices of GLn(Fp) with 1 on the main diagonal.

A Presentation for the Unipotent Group over Rings with Identity

For a ring R with identity, define Unipn(R) to be the group of upper-triangular matrices over R all of whose diagonal entries are 1. For i=1,2,…,n−1, let Si denote the matrix whose only nonzero off-diagonal entry is a 1 in the ith row and (i+1)st column. Then for any integer m (including m=0), it is easy to see that the Si generate Unipn(Z/mZ). Reiner gave relations among the Si which he conjectured gave a presentation for Unipn(Z/2Z). This conjecture was proven by Biss [Comm. Algebra26 (1998), 2971–2975] and an analogous conjecture was made for Unipn(Z/mZ) in general. We prove this conjecture, as well as a generalization of the conjecture to unipotent groups over arbitrary rings.

Good representations and solvable groups.

We give a characterization of connected solvable groups in terms of the existence of representations with certain geometric properties. The existence of such representations for the group of upper triangular matrices played an important role in the proof of the authors' equivariant Riemann-Roch theorem.

Two Random Walks on Upper Triangular Matrices

We study two random walks on a group of upper triangular matrices. In each case, we give upper bound on the mixing time by using a stopping time technique.

Inversion quasi-invariant Gaussian measures on the group of upper-triangular matrices of infinite order

Lemma 2. p L~ ,,~ #b Vt E B~ r '7 Sk,~(b) = ~'~,~=,~+1 bk,,~b,, n-1 < ~ Vk < n. We shMt prove the following theorem. ~L (b~b-1 Theorem 1. /f E(b) = ~ ~k<n~,kn~ / k n < CO, then #~ ~ ,b- In this case the right and left regular representations T ~'~* and T L''b of B~ are well defined, and the commutation theorem holds. Moreover, the operator 4*b given by (Jubf)(x) = (dttb(x-l)/d#b(X ))W2f(x -1) is an intertwining operator: T L't'b = J, bT~'"bJl,b, t e B~.

Scale Functions on Linear Groups Over Local Skew Fields

LetGbe a general or special linear group over a local skew field. ThenGis a totally disconnected, locally compact group, to which G. Willis (Math. Ann.300, 1994, 341–363) associates its scale functions:G→N. We computeson the subset of diagonalizable matrices. We also consider the projective situation, and we discuss scale functions on groups of upper triangular matrices. The latter can be computed completely, provided that the underlying field is commutative. The data computed suffice to determine the modular functions on the groups considered, and they facilitate an easy proof of the fact that general or special linear groups over local skew fields with distinct modules cannot be isomorphic as topological groups.

Conjugacy in Groups of Upper Triangular Matrices

We show that in the groupUn(F2), a Sylow 2-subgroup ofGLn(F2), there are elements not conjugate to their inverses ifn≥13. It follows that there are irreducible characters of this group that are not real-valued, and that a conjecture of Kirillov is not correct as stated. We give a partial explanation of why these group elements do not exist ifn≤12, and an example to show that analogous phenomena can occur for odd primes.

Quantum Heisenberg group and reflection equation. I

The quantum group of upper triangular matrices and the Hopf algebra associated with the reflection equation are constructed for the R-matrix corresponding to the quantum Heisenberg group. Bibliography: 12 titles.

Eigenvalues of Random Walks on Groups

In this paper we discuss and apply a novel method for bounding the eigenvalues of a random walk on a group $G$ (or equivalently on its Cayley graph). This method works by looking at the action of an Abelian normal subgroup $H$ of $G$ on $G$. We may then choose eigenvectors which fall into representations of $H$. One is then left with a large number (one for each representation of $H$) of easier problems to analyze. This analysis is carried out by new geometric methods. This method allows us to give bounds on the second largest eigenvalue of random walks on nilpotent groups with low class number. The method also lets us treat certain very easy solvable groups and to give better bounds for certain nice nilpotent groups with large class number. For example, we will give sharp bounds for two natural random walks on groups of upper triangular matrices.

Random Walks on the Groups of Upper Triangular Matrices

This paper gives sharp bounds on the eigenvalues of a natural random walk on the group of upper triangular $n \times n$ matrices over the field of characteristic $p$, an odd prime, with 1's on the diagonal. In particular, this includes the finite Heisenberg groups as a special case. As a consequence we get bounds on the time required to achieve randomness for these walks. Some of the steps are done using the geometric bounds on the eigenvalues of Diaconis and Stroock. However, the crucial step is done using more subtle and idiosyncratic techniques. We bound the eigenvalues inductively over a sequence of subspaces.