Ring of integers

About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.

Accessing the Farrell-Tate cohomology of discrete groups

Abstract: We introduce a method to explicitly determine the Farrell-Tate cohomology of discrete groups. We apply this method to the Coxeter triangle and tetrahedral groups as well as to the Bianchi groups, i.e. PSL_2 over the ring of integers in an imaginary quadratic number field. We show that the Farrell-Tate cohomology of the Bianchi groups is completely determined by the numbers of conjugacy classes of finite subgroups. In fact, our access to Farrell-Tate cohomology allows us to detach the information about it from geometric models for the Bianchi groups and to express it only with the group structure. Formulae for the numbers of conjugacy classes of finite subgroups in the Bianchi groups have been determined in a thesis of Kramer, in terms of elementary number-theoretic information on the ring of integers. An evaluation of these formulae for a large number of Bianchi groups is provided numerically in the appendix. Our new insights about the homological torsion allow us to give a conceptual description of the cohomology ring structure of the Bianchi groups.

3D Coprime Arrays in Sparse Sensing

Abstract: Coprime arrays are a class of sensor arrays that play a crucial role in various signal processing tasks because of their desirable properties such as sparsity and increased degrees of freedom (DOF) of coarrays. In this contribution, a new class of three-dimensional (3D) arrays is constructed from pure cubic fields. By studying the properties of cubic integers, we convert the problem of finding two coprime 3-by-3 integer matrices to that of two coprime integers in the ring of integers of a cubic field, which significantly reduces the design complexity and expands the design space of these matrices. The proposed construction offers naturally commutative matrices and includes generalized circulant matrices as a special case (under certain restriction of a parameter). The surged DOF is guaranteed by the generalized Chinese Remainder Theorem (CRT) for rings and ideals.

Involutions and free pairs of bass cyclic units in integral group rings

Abstract: Let ℤG be the integral group ring of the finite nonabelian group G over the ring of integers ℤ, and let * be an involution of ℤG that extends one of G. If x and y are elements of G, we investigate when pairs of the form (uk, m(x), uk, m(x*)) or (uk, m(x), uk, m(y)), formed respectively by Bass cyclic and *-symmetric Bass cyclic units, generate a free noncyclic subgroup of the unit group of ℤG.

On the Quasi-cyclicity of the Gray Map Image of a Class of Codes over Galois Rings

Abstract: Results on the quasi-cyclicity of the Gray map image of a class of codes defined over the Galois ring GR(p2,m) are given. These results generalize some appearing in [8] for codes over the ring of integers modulo p2(pa prime). The ring of (truncated) Witt vectors is a useful tool in proving the main results.

Representation Growth of Special Compact Linear Groups of Order Two

Abstract: Let $\mathfrak{o}$ be the ring of integers of a non-Archimedean local field $F$ with finite residue field of even characteristic and maximal ideal $\mathfrak{p}.$ Let $\mathrm{e}(\mathfrak{o})$ denotes the ramification index of $\mathfrak{o}$ in case $\mathfrak{o}$ has characteristic zero. We prove that the abscissa of convergence of representation zeta function of Special Linear group $\mathrm{SL}_2(\mathfrak{o})$ is $1.$ We prove that for any $\mathfrak{o}$ of characteristic zero with the residue field of cardinality $q$ such that $2 \mid q$ the group algebras $\mathbb C[\mathrm{SL}_2( \mathfrak{o}/\mathfrak{p}^{2 r})]$ and $\mathbb C[\mathrm{SL}_2(\mathbb F_q[t]/(t^{2r}))]$ are not isomorphic for any $r > \mathrm{e}(\mathfrak{o}).$ Further we give a construction of all primitive irreducible representations of groups $\mathrm{SL}_2\left (\mathbb F_q[t]/(t^{2r}) \right) $ for all $r \geq 1$ and of groups $\mathrm{SL}_2( \mathfrak{o}/\mathfrak{p}^{2 r}),$ where $\mathfrak{o}$ has characteristic zero and $r \geq 2\mathrm{e}(\mathfrak{o}).$