J. -C. Novelli, J. -Y. Thibon
We realize several combinatorial Hopf algebras based on set compositions, plane trees and segmented compositions in terms of noncommutative polynomials in infinitely many variables. For each of them, we describe a trialgebra structure, an internal product, and several bases.
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We realize the free dendriform trialgebra on one generator, as well as several other examples of dendriform trialgebras, as sub-trialgebras of an algebra of noncommutative polynomials in infinitely many variables.
We construct explicit polynomial realizations of some combinatorial Hopf algebras based on various kind of trees or forests, and some more general classes of graphs, ranging from the Connes-Kreimer algebra to an algebra of labelled forests isomorphic to the Hopf algebra of parking functions, and to a new noncommutative algebra based on endofunctions admitting many interesting subalgebras and quotients.
The INIEP: Irreducible and Positive Realizations
本文研究非负可实现谱（或特征多项式）的不可约非负矩阵实现，通过图论方法刻画了迹为零情况下的不可约非负可实现性，并讨论了正实现问题。
Polynomial realizations of natural Hopf algebras of nonsymmetric operads
本文为自然Hopf代数引入新基，通过变量字母表构造多项式实现，并揭示其与多个已知Hopf代数（如词拟对称函数、Connes-Kreimer装饰版本等）的联系。
Bounds on the realizations of zero-nonzero patterns and sign conditions of polynomials restricted to varieties and applications
本文研究了多项式在代数集上零非零模式和符号条件的实现界限，并提出了相对秩的概念，应用于代数集的复杂性分析及量子计算领域。
A Geometric Realization of Partially-Symmetric Macdonald Polynomials
本文提出猜想将部分对称Macdonald多项式与抛物线旗Hilbert概形联系，扩展了Haiman的几何实现，并证明其与代数$\mathbb{A}_{t,q}$元素作用相容作为证据。
We study the nonsymmetric Macdonald polynomials specialized at infinity from various points of view. First, we define a family of modules of the Iwahori algebra whose characters are equal to the nonsymmetric Macdonald polynomials specialized at infinity. Second, we show that these modules are isomorphic to the dual spaces of sections of certain sheaves on the semi-infinite Schubert varieties. Third, we prove that the global versions of these modules are homologically dual to the level one affine Demazure modules.
Polynomial realizations of Hopf algebras built from nonsymmetric operads
本文利用带一元与二元关系的非交换变量字母表，为非对称 operad 的自然 Hopf 代数构造多项式实现，并将其与多种经典 Hopf 代数联系起来。
S-meandric Permutations and Tangency Polynomials
本文研究允许切触时 meander 配置与排列的非唯一对应，给出可实现性判据，证明实现构成仿射空间，引入切触多项式并证明其分解性质，推导中心极限定理与渐近公式。
We give a new interpretation of the chromatic polynomial of a simple graph G in terms of the Kac-Moody Lie algebra with Dynkin diagram G. We show that the chromatic polynomial is essentially the q-Kostant partition function of this Lie algebra evaluated on the sum of the simple roots. Applying the Peterson recurrence formula for root multiplicities, we obtain a new realization of the chromatic polynomial as a weighted sum of paths in the bond lattice of G.
Intersection cohomology without spaces
本文探讨了三种无需实际空间的交集上同调构造，涵盖Schubert簇、toric簇和arrangement Schubert簇，通过组合sheaves和双模理论实现，并应用于KLS多项式。
Dense, irregular, yet always graphic $3$-uniform hypergraph degree sequences
本文针对3-均匀超图度序列问题，证明了当所有度落在$\ rac{2n^2}{63}+O(n)$与$\ rac{5n^2}{63}-O(n)$之间、顶点数至少45且度和可被3整除时，度序列必可实现，并首次给出构造高度不规则稠密度序列的多项式时间算法。
Q-Kostka polynomials and spin Green polynomials
本文利用顶点算子实现研究Q-Kostka多项式并建立其稳定性，同时引入自旋Green多项式作为Green多项式的类似物和自旋不可约特征标的形变，给出迭代公式及相关性质。
Orbit portraits were introduced by Milnor as a combinatorial tool to describe the patterns of all periodic dynamical rays landing on a periodic cycle of a quadratic polynomial. This encodes information about the dynamics and the parameter spaces of these maps. We carry out a similar analysis for unicritical anti-polynomials and give an explicit description of the orbit portraits that can occur for such maps in terms of their characteristic angles, which turns out to be rather restricted when compared with the holomorphic case. Finally, we prove a realization theorem for these combinatorial obj…
For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Young tableaux occur, and do so with weight 0. Standard Young tableaux also parametrize the vertices of dual equivalence graphs. Motivated by the underlying representation theory, in this paper, we explainthis connection by giving a combinatorial manifestation of Schur-Weyl duality. In particular, we put a dual equivalence graph structure on the 0-weight space of certain crystal graphs, producing edges combi…
On Two Algebraic Realizations of Schubert Calculus
研究复格拉斯曼流形上舒伯特演算的两种代数实现，即通过作用于舒尔多项式的微分算子和作用于外代数的顶点算子，利用玻色子 - 费米子对应建立统一框架，关联舒伯特演算与多种数学理论。
The main result of the paper is a construction of a five-parameter family of new bases in the algebra of symmetric functions. These bases are inhomogeneous and share many properties of systems of orthogonal polynomials on an interval of the real line. This means, in particular, that the algebra of symmetric functions is embedded into the algebra of continuous functions on a certain compact space Omega, and under this realization, our bases turn into orthogonal bases of weighted Hilbert spaces corresponding to certain probability measures on Omega. These measures are of independent interest ---…
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