Related papers: Families of zero cycles and divided powers: I. Rep…

In this paper we study the group $A_0(X)$ of zero dimensional cycles of degree 0 modulo rational equivalence on a projective homogeneous algebraic variety $X$. To do this we translate rational equivalence of 0-cycles on a projective variety…

Let k be a field of characteristic zero. We consider graded subalgebras A of k[x_1,...,x_m]/(x_1^2,...,x_m^2) generated by d linearly independant linear forms. Representations of matroids over k provide a natural description of the…

Given an ideal $I$ in a commutative ring $A$, a divided power structure on $I$ is a collection of maps $\{\gamma_n \colon I \to A\}_{n \in \mathbb{N}}$, subject to axioms that imply that it behaves like the family $\{x \mapsto…

Let $X$ be a quasi-compact and quasi-separated scheme. There are two fundamental and pervasive facts about the unbounded derived category of $X$: (1) $\mathsf{D}_{\mathrm{qc}}(X)$ is compactly generated by perfect complexes and (2) if $X$…

A cycle is algebraically trivial if it can be exhibited as the difference of two fibers in a family of cycles parameterized by a smooth scheme. Over an algebraically closed field, it is a result of Weil that it suffices to consider families…

We show that the $n-$th symmetric product of an affine scheme $X=\mathrm{Spec} A$ over a characteristic zero field is isomorphic as a scheme to the quotient by the general linear group of the scheme parameterizing $n-$dimensional linear…

It is known that semi-magic square matrices form a 2-graded algebra or superalgebra with the even and odd subspaces under centre-point reflection symmetry as the two components. We show that other symmetries which have been studied for…

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field and $A$ a standard graded $S$-algebra. In terms of the Gr\"obner basis of the defining ideal $J$ of $A$ we give a condition, called the x-condition, which implies that all graded…

In quantum mechanics, associative algebras play an important role in understanding symmetries and operator algebras, providing new algebraic frameworks for describing physical systems. This work classifies associative algebras over a field…

We give a necessary and sufficient smoothness condition for the scheme parameterizing the n-dimensional representations of a finitely generated associative algebra over an algebraically closed field of characteristic zero. In particular,…

We determine the derived representation types of algebras with radical square zero and give a description of the indecomposable objects in their bounded derived categories.

The algebra of symmetric functions contains several interesting families of symmetric functions indexed by integer partitions or skew partitions. Given a sequence $\{u_n\}$ of symmetric functions taken from one of these families such that…

Functional digraphs are unlabelled finite digraphs where each vertex has exactly one out-neighbor. They are isomorphic classes of finite discrete-time dynamical systems. Endowed with the direct sum and product, functional digraphs form a…

We consider a class of C*-algebras C(X) associated with quantum spaces such as spheres, projective spaces, and lens spaces. We introduce a non-self-adjoint operator algebra A together with an explicit functor from the category of…

Let $m>1$ and $\mathfrak{d} \neq 0$ be integers such that $v_{p}(\mathfrak{d}) \neq m$ for any prime $p$. We construct a matrix $A(\mathfrak{d})$ of size $(m-1) \times (m-1)$ depending on only of $\mathfrak{d}$ with the following property:…

The purpose of this paper is to give a characterisation of divided power algebras over a reduced operad. Such a characterisation is given in terms of polynomial operations, following the classical example of divided power algebras. We…

For a graph X, let f_0(X) denote its number of vertices, d(X) its minimum degree and Z_1(X;Z/2) its cycle space in the standard graph-theoretical sense (i.e. 1-dimensional cycle group in the sense of simplicial homology theory with…

We study the space of irreducible representations of a crossed product C*-algebra AxG, where G is a finite group. We construct a space $\Gamma$ which consists of pairs of irreducible representations of A and irreducible projective…

Using the Polya Enumeration Theorem, we count with particular attention to C^3/Gamma up to C^6/Gamma, abelian orbifolds in various dimensions which are invariant under cycles of the permutation group S_D. This produces a collection of…

In the paper we prove for every finite algebra A that either it has the polynomially generated powers (PGP) property, or it has the exponentially generated powers (EGP) property. For idempotent algebras we give a simple criteria for the…