Algebraic aspects of increasing subsequences

Homological aspects of oriented hypergraphs

We define an algebraic setup of homology for hypergraphs, which defaults to simplicial homology in the case of graphs, and study its basic properties. As part of our study we define algebraic spanning trees of hypergraphs, along with fundamental cuts and cycles playing their usual roles.

Continuous tensor categories from quantum groups I: algebraic aspects

We describe the algebraic ingredients of a proof of the conjecture of Frenkel and Ip that the category of positive representations $\mathcal{P}_λ$ of the quantum group $U_q(\mathfrak{sl}_{n+1})$ is closed under tensor products. Our results generalize those of Ponsot and Teschner in the rank 1 case of $U_q(\mathfrak{sl}_2)$. In higher rank, many nontrivial features appear, the most important of these being a surprising connection to the quantum integrability of the open Coxeter-Toda lattice. We show that the closure under tensor products follows from the orthogonality and completeness of the To…

无条件格点多面体的代数性质

Algebraic aspects of unconditional lattice polytopes

本文研究了反阻多格点多面体与其关联的无条件格点多面体之间的代数关系，证明了反阻多面体的 торic 圎环正常当且仅当关联的无条件多面体的 toric 圎环正常，并且反阻多面体的 toric 理想由二次二项式生成当且仅当关联的无条件多面体的 toric 理想也由二次二项式生成。

多项式Littlewood-Offord问题的代数方面

Algebraic aspects of the polynomial Littlewood-Offord problem

本文研究了多项式Littlewood-Offord问题，证明了除非多项式接近具有特殊代数结构的多项式，否则Meka-Nguyen-Vu界可被显著改进，并修正了Costello关于多重线性形式的猜想。

Algorithmic and algebraic aspects of unshuffling permutations

A permutation is said to be a square if it can be obtained by shuffling two order-isomorphic patterns. The definition is intended to be the natural counterpart to the ordinary shuffle of words and languages. In this paper, we tackle the problem of recognizing square permutations from both the point of view of algebra and algorithms. On the one hand, we present some algebraic and combinatorial properties of the shuffle product of permutations. We follow an unusual line consisting in defining the shuffle of permutations by means of an unshuffling operator, known as a coproduct. This strategy all…

Algebraic and combinatorial aspects of sandpile monoids on directed graphs

The sandpile group of a graph is a well-studied object that combines ideas from algebraic graph theory, group theory, dynamical systems, and statistical physics. A graph's sandpile group is part of a larger algebraic structure on the graph, known as its sandpile monoid. Most of the work on sandpiles so far has focused on the sandpile group rather than the sandpile monoid of a graph, and has also assumed the underlying graph to be undirected. A notable exception is the recent work of Babai and Toumpakari, which builds up the theory of sandpile monoids on directed graphs from scratch and provide…

Some algebraic aspects of mesoprimary decomposition

Recent results of Kahle and Miller give a method of constructing primary decompositions of binomial ideals by first constructing "mesoprimary decompositions" determined by their underlying monoid congruences. Monoid congruences (and therefore, binomial ideals) can present many subtle behaviors that must be carefully accounted for in order to produce general results, and this makes the theory complicated. In this paper, we examine their results in the presence of a positive $A$-grading, where certain pathologies are avoided and the theory becomes more accessible. Our approach is algebraic: whil…

Combinatorial aspects of dynamical Yang-Baxter maps and dynamical braces

In this article we propose an algebraic system, which is an abelian group $(A,+)$ with a family of non-associative and non-(left)distributive multiplications $\{\cdot_λ\}_{λ\in H}$. We call this algebraic system dynamical brace. The dynamical brace corresponds to a certain dynamical Yang-Baxter map (which is left nondegenerate and satisfy the unitary condition). Combinatorial aspects of the dynamical brace give us a correspondence between the dynamical brace and a certain family of subsets of $A\rtimes Aut(A)$. From this viewpoint we give an interpretation and examples of the dynamical brace.

Analytic aspects of the shuffle product

There exist very lucid explanations of the combinatorial origins of rational and algebraic functions, in particular with respect to regular and context free languages. In the search to understand how to extend these natural correspondences, we find that the shuffle product models many key aspects of D-finite generating functions, a class which contains algebraic. We consider several different takes on the shuffle product, shuffle closure, and shuffle grammars, and give explicit generating function consequences. In the process, we define a grammar class that models D-finite generating functions…

组合矩阵的代数方面

Algebraic Aspects of combined matrices

本文研究了数域$K$上非奇异矩阵$A$的组合矩阵$\mathcal{C}(A)$的代数性质，分析了$A$属于$\mathrm{GL}_n(K)$代数子群的情况，给出了$n=2$和$n=3$时$\mathcal{C}(A)$的显式对角化结果。

配分函数与Kurepa分解I：代数计算及若干物理应用

Partition Functions and Kurepa Decomposition I: Algebraic computation and some physical Applications

本文通过Dobinski数、Bell数和矩生成函数分解多项式，建立与Kurepa阶乘的等价关系以研究Kurepa猜想，并探讨相关物理分布。研究提供代数计算工具并连接组合与物理现象。

Combinatorial aspects of the quantized universal enveloping algebra of $\mathfrak{sl}_{n+1}(\mathbb{C})$

Quasi-triangular Hopf algebras were introduced by Drinfel'd in his construction of solutions to the Yang--Baxter Equation. This algebra is built upon $\mathcal{U}_h(\mathfrak{sl}_2)$, the quantized universal enveloping algebra of the Lie algebra $\mathfrak{sl}_2$. In this paper, combinatorial structure in $\mathcal{U}_h(\mathfrak{sl}_2)$ is elicited, and used to assist in highly intricate calculations in this algebra. To this end, a combinatorial methodology is formulated for straightening algebraic expressions to a canonical form in the case $n=1$. We apply this formalism to the quasi-triangu…