Related papers: Extended partial order and applications to tensor …

The set of triangulations of a cyclic polytope possesses two a priori different partial orders, known as the higher Stasheff-Tamari orders. The first of these orders was introduced by Kapranov and Voevodsky, while the second order was…

Let $p$ be an odd prime and let $\widehat{\sigma}$ be an order $p$ automorphism of $V_{\sqrt{2}A_{p-1}}$ which is a lift of a $p$-cycle in the Weyl group ${\rm Weyl}(A_{p-1})\cong {\mathfrak S}_p$. We study a certain extension $V$ of a…

We demonstrate how exact structures can be placed on the additive category of right operator modules over an operator algebra in order to discuss global dimension for operator algebras. The properties of the Haagerup tensor product play a…

A previous study of exactly solvable rationally-extended radial oscillator potentials and corresponding Laguerre exceptional orthogonal polynomials carried out in second-order supersymmetric quantum mechanics is extended to $k$th-order one.…

The authors continue a series of articles studying certain unitary representations of the Richard Thompson groups $F,T,V$ called Pythagorean. They all extend to the Cuntz algebra $\mathcal{O}$ and conversely all representations of…

We discuss extensions of an inner product from a vector space to its full antidual. None of these extensions is weakly continuous, but partial extensions recapture some familiar structure including the Hilbert space completion and the…

We show that the ordinary cohomology functor from the category of augmented $k$-algebras to itself exchanges coproducts and products, and that Hochschild cohomology is close to sending coproducts to products if the factors are…

Let $k$ be a field of characteristic $0$, let $\mathsf{C}$ be a finite split category, let $\alpha$ be a 2-cocycle of $\mathsf{C}$ with values in the multiplicative group of $k$, and consider the resulting twisted category algebra…

We develop a theory of ordered *-vector spaces with an order unit. We prove fundamental results concerning positive linear functionals and states, and we show that the order (semi)norm on the space of self-adjoint elements admits multiple…

Let $\mathbf{G}$ be either a simple linear algebraic group over an algebraically closed field of characteristic $\ell>0$ or a quantum group at an $\ell$-th root of unity. We define a tensor ideal of singular $\mathbf{G}$-modules in the…

Originally motivated by questions of P. Etingof related to growth rates of tensor powers in symmetric tensor categories, we obtain general bounds on the order of finite subgroups of ${\rm GL}(n,\mathbb{C})$ with restricted composition…

We consider the space of tensor densities on the n-dimensional sphere with degree lambda (or, equivalently, of conformal densities with degree lambda). This space is a module over the group of diffeomorphisms, and consequently over the Lie…

Octupolar order is described in two space dimensions in terms of the maxima (and conjugated minima) of the probability density associated with a third-rank, fully symmetric and traceless tensor. Such a representation is shown to be…

We study extremal problems for tuples of integers chosen from sets $A_i \subset [X_i,2X_i]$ for $1\le i\le k$, under large GCD and small LCM conditions. For the GCD problem, we extend the work of Green and Walker to higher dimensions.…

We present a unified and completely general formulation of extended geometry, characterised by a Kac-Moody algebra and a highest weight coordinate module. Generalised diffeomorphisms are constructed, as well as solutions to the section…

We consider the tensor power $V=(C^N)^{\otimes n}$ of the vector representation of $gl_N$ and its weight decomposition $V=\oplus_{\lambda=(\lambda_1,...,\lambda_N)}V[\lambda]$. For $\lambda = (\lambda_1 \geq ... \geq \lambda_N)$, the…

Let $F$ be a finite extension of $\mathbb{Q}_p$, let $\Omega_F$ be Drinfeld's upper half-plane over $F$ and let $G^0$ the subgroup of $GL_2(F)$ consisting of elements whose determinant has norm $1$. By working locally on $\Omega_F$, we…

Using symmetric algebras we simplify (and slightly strengthen) the Bruns-Eisenbud-Evans "generalized principal ideal theorem" on the height of order ideals of non-minimal generators in a module. We also obtain a simple proof and an…

We present an approach to cohomological dimension theory based on infinite symmetric products and on the general theory of dimension called the extension dimension. The notion of the extension dimension $\ExD(X)$ was introduced by…

Let $A$ be a finite dimensional $k$-algebra standardly stratified for a partial order $\leqslant$ and $\Delta$ be the direct sum of all standard modules. In this paper we study the extension algebra $E= \text{Ext}_A^{\ast} (\Delta, \Delta)$…