Grothendieck bialgebras, Partition lattices and symmetric functions in noncommutative variables

Antipode formulas for some combinatorial Hopf algebras

Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras that can be thought of as K-theoretic analogues of the Hopf algebras of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, and of the Malvenuto-Reutenauer Hopf algebra of permutations. They described the bialgebra structure in all cases that were not yet known but left open the question of finding explicit formulas for the antipode maps. We give combinatorial formulas for the antipode map for the K-theoretic…

Noncommutative Symmetric Systems over Associative Algebras

This paper is the first of a sequence papers ([Z4]--[Z7]) on the {\it ${\mathcal N}$CS $(\text{noncommutative symmetric})$ systems} over differential operator algebras in commutative or noncommutative variables ([Z4]); the ${\mathcal N}$CS systems over the Grossman-Larson Hopf algebras ([GL],[F]) of labeled rooted trees ([Z6]); as well as their connections and applications to the inversion problem ([BCW],[E4]) and specializations of NCSFs ([Z5],[Z7]). In this paper, inspired by the seminal work [GKLLRT] on NCSFs (noncommutative symmetric functions), we first formulate the notion {\it ${\mathca…

The peak algebra and the Hecke-Clifford algebras at $q=0$

Using the formalism of noncommutative symmetric functions, we derive the basic theory of the peak algebra of symmetric groups and of its graded Hopf dual. Our main result is to provide a representation theoretical interpretation of the peak algebra and its graded dual as Grothendieck rings of the tower of Hecke-Clifford algebras at $q=0$.

非交换变量中的额外基

The extra basis in noncommuting variables

本文利用 Hopf 幺半群理论和 Fock 函子，给出了非交换变量对称函数 Hopf 代数 NCSym 中 x-基余积的组合描述，并发现了对称函数代数的一个新的乘法基。

混合 Grothendieck 多项式

Hybrid Grothendieck polynomials

本文定义混合 Grothendieck 多项式作为集值反向平面分割的权重生成函数，证明其对称性，通过晶体结构得到 Schur 展开，并推导饱和 Newton 多面体及 omega 对合下的组合公式。

Hopf Algebras in Combinatorics

These notes -- originating from a one-semester class by their second author at the University of Minnesota -- survey some of the most important Hopf algebras appearing in combinatorics. After introducing coalgebras, bialgebras and Hopf algebras in general, we study the Hopf algebra of symmetric functions, including Zelevinsky's axiomatic characterization of it as a "positive self-adjoint Hopf algebra" and its application to the representation theory of symmetric and (briefly) finite general linear groups. The notes then continue with the quasisymmetric and the noncommutative symmetric function…

Representation theories of some towers of algebras related to the symmetric groups and their Hecke algebras

We study the representation theory of three towers of algebras which are related to the symmetric groups and their Hecke algebras. The first one is constructed as the algebras generated simultaneously by the elementary transpositions and the elementary sorting operators acting on permutations. The two others are the monoid algebras of nondecreasing functions and nondecreasing parking functions. For these three towers, we describe the structure of simple and indecomposable projective modules, together with the Cartan map. The Grothendieck algebras and coalgebras given respectively by the induct…