Related papers: A branching rule for partition complexes

We study the restrictions, the strict fixed points, and the strict quotients of the partition complex $|\Pi_n|$, which is the $\Sigma_n$-space attached to the poset of proper nontrivial partitions of the set $\{1,\ldots,n\}$. We express the…

A partition of a positive integer $n$ is defined as a non-increasing sequence $P = [y_0, y_1, ..., y_m]$ of positive integers which sum to $n$, where the $y_i$ are called the $parts$ of the partition. A Young diagram is a visual…

Using a new presentation for partition algebras (J. Algebraic Combin. 37(3):401-454, 2013), we derive explicit combinatorial formulae for the seminormal representations of the partition algebras. These results generalise to the partition…

Given a variety $Y$ with a rectangular Lefschetz decomposition of its derived category, we consider a degree $n$ cyclic cover $X \to Y$ ramified over a divisor $Z \subset Y$. We construct semiorthogonal decompositions of $\mathrm{D^b}(X)$…

Assume $\mathsf{M}_n$ is the $n$-dimensional permutation module for the symmetric group $\mathsf{S}_n$, and let $\mathsf{M}_n^{\otimes k}$ be its $k$-fold tensor power. The partition algebra $\mathsf{P}_k(n)$ maps surjectively onto the…

We consider a certain quotient of a polynomial ring categorified by both the isomorphic Green rings of the symmetric groups and Schur algebras generated by the signed Young permutation modules and mixed powers respectively. They have bases…

In recent work, M. Schneider and the first author studied a curious class of integer partitions called "sequentially congruent" partitions: the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to…

We derive group branching laws for formal characters of subgroups $H_\pi$ of GL(n) leaving invariant an arbitrary tensor $T^\pi$ of Young symmetry type $\pi$ where $\pi$ is an integer partition. The branchings $GL(n)\downarrow GL(n-1)$,…

The paper contains an exposition of part of topology using partitions of unity. The main idea is to create variants of the Tietze Extension Theorem and use them to derive classical theorems. This idea leads to a new result generalizing…

New exact modular branching rules are obtained for modules over the symmetric groups that are close to completely splittable modules. These results are based on some upper bounds for the Ext^1-spaces between simple modules.

We start with $n$-torsions in the Jacobian of an $m$-gonal curve and produce $n$-torsions in the class group of certain number field $K$.

For $ k \in \mathbb{N}$ we introduce an idempotent subalgebra, the spherical partition algebra ${\mathcal{SP} }_{k}$, of the partition algebra ${\mathcal{P} }_{k}$, that we define using an embedding associated with the trivial…

For stacked simplicial complexes, (special subclasses of such are: trees, triangulations of polygons, stacked polytopes), we give an explicit bijection between partitions of facets (for trees: edges), and partitions of vertices into…

In this paper, we further develop the theory of circles of partition by introducing the notion of complex circles of partition. This work generalizes the classical framework, extending from subsets of the natural numbers as base sets to…

We show that many theorems which assert that two kinds of partitions of the same integer $n$ are equinumerous are actually special cases of a much stronger form of equality. We show that in fact there correspond partition statistics $X$ and…

In this note we consider the question how the set of inversions of a permutation $\pi \in S_n$ can be partitioned into two subset, such that those are itself inversion sets of permutations. This is archived by exploiting a connection to a…

We explore partitions that lie in the intersection of several sets of classical interest: partitions with parts indivisible by $m$, appearing fewer than $m$ times, or differing by less than $m$. We find results on their behavior and…

The partition algebras are algebras of diagrams (which contain the group algebra of the symmetric group and the Brauer algebra) such that the multiplication is given by a combinatorial rule and such that the structure constants of the…

This article investigates structural connections between unrefinable partitions into distinct parts and numerical semigroups. By analysing the hooksets of Young diagrams associated with numerical sets, new criteria for recognising…

This paper introduces new structural decompositions for almost symmetric numerical semigroups through the combinatorial lens of Young diagrams. To do that, we use the foundational correspondence between numerical sets and Young diagrams,…