Related papers: Presentations of matrix rings
In this paper, by using matix representation for quaternions andoctonions, we provide a procedure to obtain some example of k potent matrices of order 4 or 8, over the real field or over the field ZP, with p a prime number.
We use Fox calculus to assign a marked polytope to a `nice' group presentation with two generators and one relator. Relating the marked vertices to Novikov-Sikorav homology we show that they determine the Bieri-Neumann-Strebel invariant of…
Let $R$ be a commutative ring with identity and a fixed invertible element $q^{\frac{1}{2}}$, and suppose $q+q^{-1}$ is invertible in $R$. For each planar surface $\Sigma_{0,n+1}$, we present its Kauffman bracket skein algebra over $R$ by…
A compact classification of the projective lines defined over (commutative) rings (with unity) of all orders up to thirty-one is given. There are altogether sixty-five different types of them. For each type we introduce the total number of…
We show that lower bounds on the border rank of matrix multiplication can be used to non-trivially derandomize polynomial identity testing for small algebraic circuits. Letting $\underline{R}(n)$ denote the border rank of $n \times n \times…
We significantly strengthen results on the structure of matrix rings over finite fields and apply them to describe the structure of the so-called weakly $n$-torsion clean rings. Specifically, we establish that, for any field $F$ with either…
We make a list of finite simple groups whose group rings over a given field are serial.
We show that the process matrix in the basis of tensor products of Pauli operators or SU(N) generators representing low rank and sparse dynamical maps will have only a few distinct entries which goes as $O(r^2)$ ($r$ is the rank).
We give a generators and relations presentation for the full monoidal subcategory of representations of the quantum orthogonal group generated by the quantum exterior powers of the defining representation.
We suggest a two-matrix model depending on three (infinite) sets of parameters which interpolates between all the models proposed in arXiv:2206.13038, and defined there through $W$-representations. We also discuss further generalizations of…
In Grayson's combinatorial description of higher K-groups, the generators are bounded acyclic binary multi-complexes of arbitrary size. Generalising work by Kasprowski, Winges and the author, we show in this paper that multi-complexes of…
Two new matrix classes are introduced; inverse cyclic matrices and bi-diagonal south-west matrices. An interesting relation is established between these classes. Applications to two classes of inverse $Z$-matrices are provided.
A cyclic presentation of a group is a presentation with an equal number of generators and relators that admits a particular cyclic symmetry. We characterise the orientable, non-orientable, and redundant cyclic presentations and obtain…
We define recurrence matrices and study a few properties (links with automatic sequences, branch groups etc.) of them.
In 2003, Cohn and Umans proposed a group-theoretic approach to bounding the exponent of matrix multiplication. Previous work within this approach ruled out certain families of groups as a route to obtaining $\omega = 2$, while other…
In this article we generalize results of Clozel and Ray (for $SL_2$ and $SL_n$ respectively) to give explicit ring-theoretic presentation in terms of a complete set of generators and relations of the Iwasawa algebra of the pro-$p$ Iwahori…
For a connected graph L, let G(L) be a group with generators the vertex set of L, subject only to the relations that the ends of each edge commute. Now let H(L) be the kernel of the homomorphism from G(L) to the integers that takes each…
We find an explicit presentation of relative odd unitary Steinberg groups constructed by odd form rings and of relative doubly laced Steinberg groups over commutative rings, i.e. the Steinberg groups associated with the Chevalley group…
We briefly review the connection between the fuzzy field theories and matrix models and describe the main features of the models that appear. We summarize the different approaches to their analysis, some of the recent results and the…
It is known that a presentation of the knot group of a branched twist spin is obtained from a Wirtinger presentation of the original 1-knot group by adding a generator corresponding to a regular orbit of the circle action and a certain…