Related papers: Virtual permutations and polymorhisms

We show that any smooth permutation $\sigma\in S_n$ is characterized by the set ${\mathbf{C}}(\sigma)$ of transpositions and $3$-cycles in the Bruhat interval $(S_n)_{\leq\sigma}$, and that $\sigma$ is the product (in a certain order) of…

It is shown that some convolution semigroups of infinitely divisible measures are invariant under the random integral mappings $I^{h,r}_{(a,b]}$ defined in $(\star)$ below. The converse implication is specified for the semigroups of…

This paper constructs and studies the Gromov-Witten invariants and their properties for noncompact geometrically bounded symplectic manifolds. Two localization formulas for GW-invariants are also proposed and proved. As applications we get…

Let $\mathbb{N}$ be a set of the natural numbers. Symmetric inverse semigroup $R_\infty$ is the semigroup of all infinite 0-1 matrices $[g_{ij}]$ with at most one 1 in each row and each column such that $g_{ii}=1$ on the complement of a…

We derive a multiplication law for free non-hermitian random matrices allowing for an easy reconstruction of the two-dimensional eigenvalue distribution of the product ensemble from the characteristics of the individual ensembles. We define…

Starting with an integral domain $D$ of characteristic $0$, we consider a class of iterated wreath product $W_n$ of $n$ copies of $D$. In order that $W_n$ be transfinite hypercentral, it is necessary to restrict to the case of wreath…

Let $\mu$ be a probability measure (or corresponding random variable) such that all moments $\mu_n$ exist. Knowledge of the moments is not sufficient to determine infinite divisibility of the measure; we show also that infinitely divisible,…

We discuss permutation representations which are obtained by the natural action of $S_n \times S_n$ on some special sets of invertible matrices, defined by simple combinatorial attributes. We decompose these representations into…

Let $M$ be a compact Riemannian manifold endowed with an isometric action of a compact Lie group. The method of the Witten deformation is used to compute the virtual representation-valued equivariant index of a transversally elliptic, first…

The Mallows measure on the symmetric group $S_n$ is the probability measure such that each permutation has probability proportional to $q$ raised to the power of the number of inversions, where $q$ is a positive parameter and the number of…

Let VB$_n$ be the virtual braid group on $n$ strands and let $\mathfrak{S}_n$ be the symmetric group on $n$ letters. Let $n,m \in \mathbb{N}$ such that $n \ge 5$, $m \ge 2$ and $n \ge m$. We determine all possible homomorphisms from VB$_n$…

A polynomial ensemble is a probability density function for the position of $n$ real particles of the form $\frac{1}{Z_n} \, \prod_{j<k} (x_k-x_j) \, \det \left[ f_k (x_j) \right]_{j,k=1}^n$, for certain functions $f_1, \ldots, f_n$. Such…

Motivated by proving the loss of ergodicity in expanding systems of piecewise affine coupled maps with arbitrary number of units, all-to-all coupling and inversion symmetry, we provide ad-hoc substitutes - namely inversion-symmetric maps of…

We show that if the complexity difference function p(n+1)-p(n) of a infinite minimal shift is bounded, then the the automorphism group of the one-sided shift is finite, and the automorphism group of the corresponding two-sided shift "modulo…

We consider the probability $p(S_n)$ that a pair of random permutations generates either the alternating group $A_n$ or the symmetric group $S_n$. Dixon (1969) proved that $p(S_n)$ approaches $1$ as $n\to\infty$ and conjectured that…

Let $\mathcal{Q}$ be a vertex subset problem on graphs. In a reconfiguration variant of $\mathcal{Q}$ we are given a graph $G$ and two feasible solutions $S_s, S_t\subseteq V(G)$ of $\mathcal{Q}$ with $|S_s|=|S_t|=k$. The problem is to…

The Fortuin-Kasteleyn mapping between the Ising model and the site-bond correlated percolation model is shown to be only one of an infinite class of exact mappings. These new cluster representations are a result of "renormalized"…

Group-invariant probability distributions appear in many data-generative models in machine learning, such as graphs, point clouds, and images. In practice, one often needs to estimate divergences between such distributions. In this work, we…

We give an efficient algorithm for the enumeration up to isomorphism of the inverse semigroups of order n, and we count the number S(n) of inverse semigroups of order n<=15. This improves considerably on the previous highest-known value…

In earlier papers, we showed a decomposition of the arcs of 2-diregular digraphs (2-dds) and used it to prove some conditions for these graphs to be non-Hamiltonian; we then extended this decomposition to a larger class of digraphs and used…