Related papers: Algebra of overpartitions

There is a classical connection between the representation theory of the symmetric group and the general linear group called Schur-Weyl duality. Variations on this principle yield analogous connections between the symmetric group and other…

Let $f: \mathbb{Z}_+\rightarrow \mathbb{Z}_+$ be a polynomial with the property that corresponding to every prime $p$ there exists an integer $\ell$ such that $p\nmid f(\ell)$. In this paper, we establish some equidistributed results…

In this paper we study the structure of the monoid $\mathbf{I}\mathbb{N}_{\infty}^n$ of cofinite partial isometries of the $n$-th power of the set of positive integers $\mathbb{N}$ with the usual metric for a positive integer $n\geqslant…

The notion of multiplier Hopf monoid in any braided monoidal category is introduced as a multiplier bimonoid whose constituent fusion morphisms are isomorphisms. In the category of vector spaces over the complex numbers, Van Daele's…

In this paper we show that for a given set of pairwise comaximal ideals $\{X_i\}_{i\in I}$ in a ring $R$ with unity and any right $R$-module $M$ with generating set $Y$ and $C(X_i)=\sum\limits_{k\in\mathbb{N}}\underline{\ell}_M(X_i^{k})$,…

We define and study a family of partitions of the wonderful compactification \bar{G} of a semi-simple algebraic group G of adjoint type. The partitions are obtained from subgroups of G \times G associated to triples (A_1, A_2, a), where A_1…

For $k\geq i\geq 1$, let $B_{k,i}(n)$ denote the number of partitions of $n$ such that part 1 appears at most $i-1$ times, two consecutive integers l and $l+1$ appear at most $k-1$ times and if l and $l+1$ appear exactly $k-1$ times then…

The following problem has been known since the 80s. Let $\Gamma$ be an Abelian group of order $m$ (denoted $|\Gamma|=m$), and let $t$ and $\{m_i\}_{i=1}^{t}$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when…

We prove several congruences satisfied by the generalized cubic and generalized overcubic partition functions, recently introduced by Amdeberhan, Sellers, and Singh. We also prove infinite families of congruences modulo powers of $2$ and…

Let $k$ be a field containing an algebraically closed field of characteristic zero. If $G$ is a finite group and $D$ is a division algebra over $k$, finite dimensional over its center, we can associate to a faithful $G$-grading on $D$ a…

For a partition $\underline{\lambda} = (\lambda_{1}^{\rho _1}>\lambda_{2}^{\rho _2}>\lambda_{3}^{\rho _3}>\ldots>\lambda_{k}^{\rho _k})$ and its associated finite $\mathcal{R}$-module…

We study the number of $k \times r$ plane partitions, weighted on the sum of the first row. Using Erhart reciprocity, we prove an identity for the generating function. For the special case $k=1$ this result follows from the classical theory…

We introduce a family of univariate polynomials indexed by integer partitions. At prime powers, they count the number of subspaces in a finite vector space that transform under a regular diagonal matrix in a specified manner. This…

We introduce the inverse monoid of inner partial automorphisms of a semigroup -- a tool that associates to every semigroup an inverse semigroup. When the semigroup is a group, this inverse semigroup is isomorphic to the group of inner…

Let $A$ be an algebra over any field. We do not assume that $A$ has an identity. The \emph{multiplier algebra} $M(A)$ is a unital algebra associated to $A$. If we require the product in $A$ to be non-degenerate (as a bilinear form), the…

In this Ph.D. dissertation (2018, Emory University) we prove theorems at the intersection of the additive and multiplicative branches of number theory, bringing together ideas from partition theory, $q$-series, algebra, modular forms and…

Inspired by Andrews' and Bachraoui's work on partitions with repeated smallest part, we extend the concept to overpartitions. We study overpartitions with the restriction that the smallest non-overlined part appears exactly $k$ times and…

A family of quantum cluster algebras is introduced and studied. In general, these algebras are new, but subclasses have been studied previously by other authors. The algebras are indexed by double partitions or double flag varieties.…

The paper is devoted to homology groups of cubical sets with coefficients in contravariant systems of Abelian groups. The study is based on the proof of the assertion that the homology groups of the category of cubes with coefficients in…

We call a poset factorable if its characteristic polynomial has all positive integer roots. Inspired by inductive and divisional freeness of a central hyperplane arrangement, we introduce and study the notion of inductive posets and their…