Related papers: Minimum transformation representations of diagram …
We study the representations of the commutator subgroup of the braid group with n strands in the symmetric group of degree r. Motivated by some experimental results, we conjecture that for n>r, every such representation is trivial.
We increase the scope of previous work on change of basis between finite bases of polynomials by defining ascending and descending bases and introducing three techniques for defining them from known ones. The minimum degrees of polynomials…
We call \emph{Alphabet model} a generalization to N types of particles of the classic ABC model. We have particles of different types stochastically evolving on a one dimensional lattice with an exchange dynamics. The rates of exchange are…
We define non-pivotal analogs of the Temperley-Lieb, Motzkin, and planar rook monoids, and compute bounds for the sizes of their nontrivial simple representations. From this, we assess the two types of monoids in their relative suitability…
We investigate the real Lie algebra of first-order differential operators with polynomial coefficients, which is subject to the following requirements. (1) The Lie algebra should admit a basis of differential operators with homogeneous…
A binary modified de Bruijn sequence is an infinite and periodic binary sequence derived by removing a zero from the longest run of zeros in a binary de Bruijn sequence. The minimal polynomial of the modified sequence is its unique…
The Betti tables of a multigraded module encode the grades at which there is an algebraic change in the module. Multigraded modules show up in many areas of pure and applied mathematics, and in particular in topological data analysis, where…
The Yangian of the Lie algebra $gl_N$ has a distinguished family of irreducible finite-dimensional representations, called elementary representations. They are parametrized by pairs, consisting of a skew Young diagram and a complex number.…
The representation theory of the symmetric group has been intensively studied for over 100 years and is one of the gems of modern mathematics. The full transformation monoid $\mathfrak T_n$ (the monoid of all self-maps of an $n$-element…
A spline is an assignment of polynomials to the vertices of a graph, where the difference of two polynomials along an edge must belong to the ideal labeling that edge. We consider a ring of splines $\mathcal{M}_{H}$ constructed on a graph…
The question of computing the reductions modulo $p$ of two-dimensional crystalline $p$-adic Galois representations has been studied extensively, and partial progress has been made for representations that have small weights, very small…
Let $G$ be a graph of order $n$. The maximum and minimum degree of $G$ are denoted by $\Delta$ and $\delta$ respectively. The \emph{path partition number} $\mu (G)$ of a graph $G$ is the minimum number of paths needed to partition the…
We show that the poset of non-trivial partitions of 1,2,...,n has a fundamental homology class with coefficients in a Lie superalgebra. Homological duality then rapidly yields a range of known results concerning the integral representations…
We define the notion of a minimal affinization of an irreducible representation of $U_q(g)$. We prove that minimal affinizations exist and establish their uniqueness in the rank 2 case.
We derive a recurrence relation for the number of simple vertex-labelled bipartite graphs with given degrees of the vertices and use this result to obtain a new method for computing the growth function of the Artin monoid of type $A_{n-1}$…
Let p be a prime, and let f : Z/pZ -> R be a function with average value 0 and ||f||_A <= 1, where ||f||_A denotes the algebra norm (L^1 norm of the Fourier transform). Then f(x) is small for some x, specifically min_x |f(x)| is no more…
Let each point of a homogeneous Poisson process in R^d independently be equipped with a random number of stubs (half-edges) according to a given probability distribution mu on the positive integers. We consider translation-invariant schemes…
We study the obstruction degrees of translates of sub-tori of multiplicative tori and we show how they are connected to lower bounds for the essential minimum of these varieties. In particular, we combine our computations with results of A.…
Basis partitions are minimal partitions corresponding to successive rank vectors. We show combinatorially how basis partitions can be generated from primary partitions which are equivalent to the Rogers-Ramanujan partitions. This leads to…
Paraparticles beyond bosons and fermions can be exchanged via either the braid group (anyons, existing up to $D=2$ space dimensions) or the permutation group; in the latter case the space dimensions are not limited. Besides being predicted,…