EL-labelings, Supersolvability and 0-Hecke Algebra Actions on Posets

正则Schur标记斜形状偏序集及其0-Hecke模

Regular Schur labeled skew shape posets and their 0-Hecke modules

该研究在Stanley的$P$-分拆猜想假设下，刻画了正则Schur标记斜形状偏序集的性质，分类其线性扩张集合与相关0-Hecke模，并构造了模关于Schur基的特异滤过。

Poset edge densities, nearly reduced words, and barely set-valued tableaux

In certain finite posets, the expected down-degree of their elements is the same whether computed with respect to either the uniform distribution or the distribution weighting an element by the number of maximal chains passing through it. We show that this coincidence of expectations holds for Cartesian products of chains, connected minuscule posets, weak Bruhat orders on finite Coxeter groups, certain lower intervals in Young's lattice, and certain lower intervals in the weak Bruhat order below dominant permutations. Our tools involve formulas for counting nearly reduced factorizations in 0-H…

0-Hecke代数的偏序集模与相关拟对称幂和展开

Poset modules of the $0$-Hecke algebras and related quasisymmetric power sum expansions

本文研究0-Hecke代数的偏序集模，证明相关Grothendieck群的直和具有拟对称函数Hopf代数结构，还分析Schur函数拟类似物的拟对称幂和展开，给出新算法并通过边界条表表示对偶无污函数等的展开系数。

The biHecke monoid of a finite Coxeter group and its representations

For any finite Coxeter group W, we introduce two new objects: its cutting poset and its biHecke monoid. The cutting poset, constructed using a generalization of the notion of blocks in permutation matrices, almost forms a lattice on W. The construction of the biHecke monoid relies on the usual combinatorial model for the 0-Hecke algebra H_0(W), that is, for the symmetric group, the algebra (or monoid) generated by the elementary bubble sort operators. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort o…

The Whitney Duals of a Graded Poset

We introduce the notion of a \emph{Whitney dual} of a graded poset. Two posets are Whitney duals to each other if (the absolute value of) their Whitney numbers of the first and second kind are interchanged between the two posets. We define new types of edge and chain-edge labelings which we call \emph{Whitney labelings}. We prove that every graded poset with a Whitney labeling has a Whitney dual. Moreover, we show how to explicitly construct a Whitney dual using a technique involving quotient posets. As applications of our main theorem, we show that geometric lattices, the lattice of noncrossi…

Young walls and graded dimension formulas for finite quiver Hecke algebras of type $A^{(2)}_{2\ell}$ and $D^{(2)}_{\ell+1}$

We study graded dimension formulas for finite quiver Hecke algebras $R^{Λ_0}(β)$ of type $A^{(2)}_{2\ell}$ and $D^{(2)}_{\ell+1}$ using combinatorics of Young walls. We introduce the notion of standard tableaux for proper Young walls and show that the standard tableaux form a graded poset with lattice structure. We next investigate Laurent polynomials associated with proper Young walls and their standard tableaux arising from the Fock space representations consisting of proper Young walls. Then we prove the graded dimension formulas described in terms of the Laurent polynomials. When evaluatin…

The Hecke group algebra of a Coxeter group and its representation theory

Let W be a finite Coxeter group. We define its Hecke-group algebra by gluing together appropriately its group algebra and its 0-Hecke algebra. We describe in detail this algebra (dimension, several bases, conjectural presentation, combinatorial construction of simple and indecomposable projective modules, Cartan map) and give several alternative equivalent definitions (as symmetry preserving operator algebra, as poset algebra, as commutant algebra, ...). In type A, the Hecke-group algebra can be described as the algebra generated simultaneously by the elementary transpositions and the elementa…

On some actions of the 0-Hecke monoids of affine symmetric groups

There are left and right actions of the 0-Hecke monoid of the affine symmetric group $\tilde{S}_n$ on involutions whose cycles are labeled periodically by nonnegative integers. Using these actions we construct two bijections, which are length-preserving in an appropriate sense, from the set of involutions in $\tilde{S}_n$ to the set of $\mathbb{N}$-weighted matchings in the $n$-element cycle graph. As an application, we compute a formula for the bivariate generating function counting the involutions in $\tilde S_n$ by length and absolute length. The 0-Hecke monoid of $\tilde{S}_n$ also acts on…

On the representation theory of finite J-trivial monoids

In 1979, Norton showed that the representation theory of the 0-Hecke algebra admits a rich combinatorial description. Her constructions rely heavily on some triangularity property of the product, but do not use explicitly that the 0-Hecke algebra is a monoid algebra. The thesis of this paper is that considering the general setting of monoids admitting such a triangularity, namely J-trivial monoids, sheds further light on the topic. This is a step to use representation theory to automatically extract combinatorial structures from (monoid) algebras, often in the form of posets and lattices, both…