Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2608.27414

Very good gradings on structural matrix rings

Patrik Lundström·Johan Öinert·Laura Orozco·Héctor Pinedo
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $R$ be a nonzero associative unital ring, let $G$ be a group, and let $\rho$ be a preorder on $\{1,\ldots,n\}$. A $G$-grading on $\rho$ induces a very good $G$-grading on the structural matrix ring $M_n(\rho,R)$. We show that, for each of the properties trivial, symmetric, epsilon-strong and strong, the grading on $\rho$ has the property if and only if the induced ring grading does. The epsilon-crossed product and crossed product properties pass from $\rho$ to the ring, but the converses fail in general. We also give a concrete criterion for epsilon-strongness and show that a very good $G$-grading on $M_n(\rho,R)$ that is strong satisfies $|G|\leq n$. When $\rho$ is an equivalence relation and the neutral component is diagonal, very good gradings correspond bijectively to free partial actions of $G$ on $\{1,\ldots,n\}$ with orbit relation $\rho$. These gradings are epsilon-crossed products, and over a field the correspondence gives a classification up to graded algebra isomorphism.

Explore related subjects

math.RA

Keep this discovery

BibTeXRIS

Patrik Lundström, Johan Öinert, Laura Orozco, Héctor Pinedo. 2026-08-27. Very good gradings on structural matrix rings. https://arxiv.org/abs/2608.27414

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Invariants of Nilpotent Lie Algebras via Geometry and Algebra with a Focus on Computation

We consider the problem of computing rational invariants of nilpotent Lie algebras. We compare two methods that are commonly used for this task: the method of integral curves and the Dixmier map. Given a derivation of a rational function field with polynomial coefficients, we formulate a condition under which the kernel can be recovered from a family of rational integral curves, and we show that triangular derivations satisfy this hypothesis. This yields an explicit description of the kernel as a purely transcendental extension and produces algebraically independent generators. We also show that, in the triangular case, the resulting generators agree with those obtained from the Dixmier map via a local slice. A careful analysis of the generating set obtained from this method leads to an algorithm for computing generators of the rational invariant field of a nilpotent Lie algebra. An implementation of the methods is available in the SageMath system.

math.RA↗
arXiv2026

Quasilinear multiplication in the real Cayley--Dickson tower

Direct evaluation of the defining product in the real Cayley--Dickson algebra $A_n$, of dimension $N=2^n$, has quadratic arithmetic complexity. This paper gives a uniform algorithm for multiplication using $O(N\log N)$ real arithmetic operations and $O(N)$ auxiliary storage. The algorithm reduces multiplication to the alternating product on the imaginary subspace, then evaluates that product by a two-call recursion over one fixed quadratic coefficient extension. For $n\ge1$, the resulting bilinear algorithm uses at most $(9n-15)2^{n-1}+10$ input-dependent real multiplications, and for $n\ge3$, the specified arithmetic schedule uses $(34n-83)2^{n-1}+50$ real operations in total. Under this counting convention, the quasilinear schedule uses fewer operations than direct multiplication for $N\ge16$ and than the uniform Cariow--Cariowa method for $N\ge32$. The algorithm is implemented in the MIT-licensed C11 library fastCD, with a NumPy-backed Python interface, and its results are checked against an independent implementation of the defining recursion. In single-core benchmarks against direct multiplication and the uniform Cariow--Cariowa method, the quasilinear implementation had the lowest mean time of the three at every tested dimension $N\ge32$, for both single and batched products, and was roughly $16$ times faster than direct multiplication at $N=1024$.

math.RA↗
arXiv2026

Graded classification of Leavitt path algebras in terms of strong shift equivalence

Given two finite essential adjacency matrices $A$ and $B$, Hazrat's graded classification conjectures posit that an order preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism of $K_0$ groups implies graded Morita equivalence of the Leavitt path algebras of $A$ and $B$, while the pointed version predicts a graded isomorphism of the Leavitt path algebras when the $K_0$ group isomorphism additionally preserves the class of the regular module. For any field $k$, we show that the Leavitt path algebras over $k$ of $A$ and $B$ are graded Morita equivalent if and only if $A$ and $B$ are strong shift equivalent. By appealing to counterexamples of Kim and Roush from symbolic dynamics, this shows that Hazrat's graded classification conjectures are false.

math.RA↗