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Presentations of categories are a well-known algebraic tool to provide descriptions of categories by means of generators, for objects and morphisms, and relations on morphisms. We generalize here this notion, in order to consider situations…

Building on work of Saxl, we classify the multiplicity-free permutation characters of all symmetric groups of degree 66 or more. A corollary is a complete list of the irreducible characters of symmetric groups (again of degree 66 or more)…

The left patience sorting (lPS) monoid, also known in the literature as the Bell monoid, and the right patient sorting (rPS) monoid are introduced by defining certain congruences on words. Such congruences are constructed using insertion…

We characterize the words that can be mapped to arbitrarily high powers by injective morphisms. For all other words, we prove a linear upper bound for the highest power that they can be mapped to, and this bound is optimal up to a constant…

The computational cost of simulating quantum many-body systems can often be reduced by taking advantage of physical symmetries. While methods exist for specific symmetry classes, a general algorithm to find the full permutation symmetry…

We show that in the free group of rank 3, given an arbitrary number of automorphisms, the intersection of their fixed subgroups is equal to the fixed subgroup of some other single automorphism.

We study the representability problem for torsion-free arithmetic matroids. By using a new operation called "reduction" and a "signed Hermite normal form", we provide and implement an algorithm to compute all the representations, up to…

The PC algorithm uses conditional independence tests for model selection in graphical modeling with acyclic directed graphs. In Gaussian models, tests of conditional independence are typically based on Pearson correlations, and…

For a natural number $c$, a $c$-arrangement is an arrangement of dimension $c$ subspaces satisfying the following condition: the sum of any subset of the subspaces has dimension a multiple of $c$. Matroids arising as normalized rank…

A symmetric matrix $C$ is completely positive (CP) if there exists an entrywise nonnegative matrix $B$ such that $C=BB^T$. The CP-completion problem is to study whether we can assign values to the missing entries of a partial matrix (i.e.,…

Graph independence (also known as $\epsilon$-independence or $\lambda$-independence) is a mixture of classical independence and free independence corresponding to graph products or groups and operator algebras. Using conjugation by certain…

We describe the endomorphisms of the direct product of two free groups of finite rank and obtain conditions for which the subgroup of fixed points is finitely generated and we do the same for periodic points. We also describe the…

The Modular Isomorphism Problem asks, if an isomorphism between modular group algebras of finite $p$-groups over a field $F$ implies an isomorphism of the group bases. We explore the differences of knowledge on the problem when $F$ is…

We study the complexity of computation in finitely generated free left, right and two-sided adequate semigroups and monoids. We present polynomial time (quadratic in the RAM model of computation) algorithms to solve the word problem and…

Previously, the authors proved that the presentation complex of a one-relator group $G$ satisfies a geometric condition called negative immersions if every two-generator, one-relator subgroup of $G$ is free. Here, we prove that one-relator…

We investigate a version of the Green correspondence for categories of complexes, including homotopy categories and derived categories. The correspondence is an equivalence between a category defined over a finite group $G$ and the same for…

Let u be a word over the positive integers. Motivated in part by a question from representation theory, we study the centralizer set of u which is C(u) = {w | uw is Knuth-equivalent to wu}. In particular, we give various necessary…

Let $F$ be a finitely generated free group. We present an algorithm such that, given a subgroup $H\leqslant F$, decides whether $H$ is the fixed subgroup of some family of automorphisms, or family of endomorphisms of $F$ and, in the…

Machine learning and pattern recognition techniques have been successfully applied to algorithmic problems in free groups. In this paper, we seek to extend these techniques to finitely presented non-free groups, with a particular emphasis…

We consider the group isomorphism problem: given two finite groups G and H specified by their multiplication tables, decide if G and H are isomorphic. The n^(log n) barrier for group isomorphism has withstood all attacks --- even for the…