Related papers: Infinite Permutation Groups
These twenty-two lectures, with exercises, comprise the extent of what was meant to be a full-year graduate-level course on the strong interactions and QCD, given at Caltech in 1987-88. The course was cut short by the illness that led to…
Notes used for a course held in 2016 in the School of Advances in Group Theory and Applications, for some lectures given in 2018 for the students of the Master in Mathematics of the Vrije Universiteit Brussels, a course for master and Ph.D.…
Forty-five years ago, a young researcher in finite permutation group theory encountered a paper by Robert Woodrow. The homogeneous triangle-free graph Woodrow described there seemed to be an infinite analogue of the Higman--Sims graph which…
Permutation polynomials over finite fields play important roles in finite fields theory. They also have wide applications in many areas of science and engineering such as coding theory, cryptography, combinatorial design, communication…
Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, two conjectures on permutation polynomials proposed recently by Wu and Li [19] are…
We offer a perspective on some recent results obtained in the context of the group field theory approach to quantum gravity, on top of reviewing them briefly. These concern a natural mechanism for the emergence of non-commutative field…
This is a presentation of recent work on quantum permutation groups. Contains: a short introduction to operator algebras and Hopf algebras; quantum permutation groups, and their basic properties; diagrams, integration formulae, asymptotic…
We study asymptotics of traces of (noncommutative) monomials formed by images of certain elements of the universal enveloping algebra of the infinite-dimensional unitary group in its Plancherel representations. We prove that they converge…
Let X be a finite set with at least two elements, and let k be any commutative field. We prove that the inversion height of the embedding k<X> ---> D, where D denotes the universal (skew) field of fractions of the free algebra k<X>, is…
We introduce a new product for permutation groups. It takes as input two permutation groups, M and N, and produces an infinite group M [X] N which carries many of the permutational properties of M. Under mild conditions on M and N the group…
This expository article revolves around the question to find short presentations of finite simple groups. This subject is one of the most active research areas of group theory in recent times. We bring together several known results on…
The ring of fluxions (real sequential germs at infinity) provides a rigorous approach to infinitesimals, different from the better-known approach of Abraham Robinson. The basic idea was first espoused in a paper by Curt Schmieden and Detlof…
In a preprint from 1982, John Milnor formulated various fundamental questions concerning infinite-dimensional Lie groups. In this note, we describe some of the answers (and partial answers) obtained in the preceding years.
This is an expanded version of the talk I gave at Oberwolfach, MIT and several other universities.
Permutations are usually enumerated by size, but new results can be found by enumerating them by inversions instead, in which case one must restrict one's attention to indecomposable permutations. In the style of the seminal paper by Simion…
In the infinite dimensional Heisenberg group, we construct a left invariant weak Riemannian metric that gives a degenerate geodesic distance. The same construction yields a degenerate sub-Riemannian distance. We show how the standard notion…
We introduce a notion of permutation presentations of modules over finite groups, and completely determine finite groups over which every module has a permutation presentation. To get this result, we prove that every coflasque module over a…
In 1965, I published a paper, exhibiting a hitherto unknown limit of the Lorentz group, which I christened Carroll group because of its seemingly paradoxical physical contents. Since I saw it as more curious than relevant, I published it in…
The goal of this paper is threefold. First, we describe the notion of dissociation for closed subgroups of the group of permutations on a countably infinite set and explain its numerous consequences on unitary representations…
The impossibility of theories with hidden variables as an alternative and replacement for quantum mechanics was discussed by J. von Neumann in 1932. His proof was criticized as being logically circular, by Grete Hermann soon after, and as…