Related papers: On the Combinatorial Inverse Monoid IO3

In this paper, we present a new connection between representation theory of noncommutative hypersurfaces and combinatorics. Let $S$ be a graded ($\pm 1$)-skew polynomial algebra in $n$ variables of degree $1$ and $f =x_1^2 + \cdots +x_n^2…

Given a surface F, we are interested in Z/2 valued invariants of immersions of F into R^3, which are constant on each connected component of the complement of the quadruple point discriminant in Imm(F,R^3). Such invariants will be called…

Suppose given a Frobenius category E, i.e. an exact category with a big enough subcategory B of bijectives. Let_E_ := E/B denote its classical stable category. For example, we may take E to be the category of complexes C(A) with entries in…

This is the third part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. In this paper (Part III), we introduce and study…

We present a construction for the holomorph of an inverse semigroup, derived from the cartesian closed structure of the category of ordered groupoids. We compare the holomorph with the monoid of mappings that preserve the ternary heap…

In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2-categories. In particular, we show that every Frobenius object in a monoidal category M arises from an ambijunction…

Using arbitrary bases for the finite field $\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}$, we obtain the generalized M\"obius transformations (GMTs), which are a class of bijections between the projective geometry $PG(n-1,q)$ and the set of roots…

The stable category of modules over the algebra of a finite group with coefficients in a field is a compactly generated tensor triangulated category, that has been studied extensively in representation theory. In this paper, we provide a…

Let \(E\) be a finite-dimensional real vector space. We study invertible objects in the monoidal category of constructible sheaves on \(E\), endowed with the convolution product \(\star\). We show that the inverse of an invertible…

Let $\mathbb{T}$ be the unit circle and $\Gamma\setminus G$ the 3-dimensional Heisenberg nilmanifold. We prove that the M\"obius function is linearly disjoint from a class of distal skew products on $\mathbb{T}\times\Gamma\setminus G$.…

We curry the elementary arithmetic operations of addition and multiplication to give monotone injections on N, and describe & study the inverse monoids that arise from also considering their generalised inverses. This leads to well-known…

We study some combinatorial properties of partial wreath $k$-th power of the semigroup $\mathcal{IS}_d$. In particular, we calculate its order, the number of idempotents and the number of D-classes.

In this paper we completely describe the unital compressed commuting graph of the ring $\mathcal{M}_3(\mathrm{GF}(p))$ of $3 \times 3$ matrices over the finite prime field $\mathrm{GF}(p)$. To achieve this we combine methods from linear…

We consider $\mathbb{R}^3$ as a homogeneous manifold for the action of the motion group given by rotations and translations. For an arbitrary $\tau\in \widehat{SO(3)}$, let $E_\tau$ be the homogeneous vector bundle over $\mathbb{R}^3$…

This is the abstruct of the revised paper. We study the equivariant analytic torsion for K3 surfaces with an anti-symplectic involution with the invariant lattice M (such a surface is called a 2-elementary K3 surface of type M in this…

For a commutative ring $A$, we have the category of (bounded-below) chain complexes of $A$-modules $Ch_{+}(A\mymod)$, a closed symmetric monoidal category with a compatible stable Quillen model structure. The associated homotopy category is…

We compute the group of Morita auto-equivalences of the even parts of the $ADE$ subfactors, and Galois conjugates. To achieve this we study the braided auto-equivalences of the Drinfeld centres of these categories. We give planar algebra…

Let R be a compact oriented surface of genus g with one boundary component. Homology cylinders over R form a monoid IC into which the Torelli group I of R embeds by the mapping cylinder construction. Two homology cylinders M and M' are said…

Let $\mathcal{C}$ be a finite tensor category with simple unit object, let $\mathcal{Z}(\mathcal{C})$ denote its monoidal center, and let $L$ and $R$ be a left adjoint and a right adjoint of the forgetful functor $U:…

The theory of Gauss diagrams and Gauss diagram formulas provides convenient ways to compute knot invariants, such as coefficients of the HOMFLYPT polynomial. In \cite{4,5}, the author uses Gauss diagram formulas to find combinatorial…