关于S6抛物商的熵偏序

Noncrossing Arc Diagrams, Tamari Lattices, and Parabolic Quotients of the Symmetric Group

Ordering permutations by containment of inversion sets yields a fascinating partial order on the symmetric group: the weak order. This partial order is, among other things, a semidistributive lattice. As a consequence, every permutation has a canonical representation as a join of other permutations. Combinatorially, these canonical join representations can be modeled in terms of arc diagrams. Moreover, these arc diagrams also serve as a model to understand quotient lattices of the weak order. A particularly well-behaved quotient lattice of the weak order is the well-known Tamari lattice, which…

Bijective Projections on Parabolic Quotients of Affine Weyl Groups

Affine Weyl groups and their parabolic quotients are used extensively as indexing sets for objects in combinatorics, representation theory, algebraic geometry, and number theory. Moreover, in the classical Lie types we can conveniently realize the elements of these quotients via intuitive geometric and combinatorial models such as abaci, alcoves, coroot lattice points, core partitions, and bounded partitions. Berg, Jones, and Vazirani described a bijection between n-cores with first part equal to k and (n-1)-cores with first part less than or equal to k, and they interpret this bijection in te…

$J$-上升集在Coxeter群的抛物商中

$J$-ascent sets in parabolic quotients of Coxeter groups

本文针对Coxeter群抛物商中的$J$-上升集，建立局部转移公式并证明其基数在右乘简单反射下的变化受限于$\max\{1,\deg(s)-1\}$。

仿射对称群核分拆上的双射与抛物商

A bijection on core partitions and a parabolic quotient of the affine symmetric group

设$\ell,k$为固定的正整数。在早期的工作中，第一和第三作者建立了$\ell$ -第一部分等于$k$的核和$(\ell-1)$ -第一部分小于或等于$k$的核之间的双射。本文给出了这种双射的几种新的解释。$\ell$ -核心索引表示$\widetilde{S_{\ell}} / S_{\ell}$的最小长度陪集，其中$\widetilde{S_{\ell}}$表示仿射对称群，$S_{\ell}$表示有限对称群.在这种情况下，双射在$A_{\ell-1}$类型的根格上有一个漂亮的几何解释。我们还表明，双射有一个自然的描述，由于Lapointe和莫尔斯的另一个对应。

线性B型抛物塔马里格

Parabolic Tamari Lattices in Linear Type B

本研究聚焦B型Coxeter群的抛物对齐元，给出其模式避免的组合模型，定义了对应等价关系并证明其为弱序同余，得到了A型抛物Tamari格的B型类似物，填补了相关文献空白。

抛物型Kazhdan-Lusztig多项式的不变性猜想之间的等价性

Equivalence between invariance conjectures for parabolic Kazhdan-Lusztig polynomials

本研究证明Mario Marietti提出的抛物型Kazhdan-Lusztig多项式组合不变性猜想等价于其在极大商上的限制，且该等价性与Barkley、Gaetz近期证明的Kazhdan-Lusztig多项式不变性猜想相关等价性处于相反极端。

Classifying quotients on Coxeter groups by isomorphism in Bruhat order

We classify all quotients $W/W_J$ up to isomorphism in Bruhat order, with $(W,S)$ a Coxeter system and $W_J$ a parabolic subgroup of $W$. In particular, the non-trivial isomorphisms fall into a small number of cases which are highly restricted; all have $W$ finite and $W_J$ a maximal parabolic. This has the immediate application of classifying dominant and antidominant blocks of category $\mathcal O$ for Kac-Moody algebras up to equivalence.

Covering relations of k-Grassmannian permutations of type B

The main result of this work is the characterization of the covering relations of the Bruhat order of the maximal parabolic quotients of type B. Our approach is mainly combinatorial and is based in the pattern of the corresponding permutations also called signed $k$-Grassmannians permutations. We obtain that a covering relation can be classified in four different pairs of permutations. This answers a question raised by Ikeda and Matsumura providing a nice combinatorial model for maximal parabolic quotients of type B.

On torus quotients of Schubert varieties in orthogonal Grassmannian-II

Let $G=SO(8n+4,\mathbb{C})$ ($n\ge 1$). Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $P (\supset B)$ denote the maximal parabolic subgroup of $G$ corresponding to the simple root $α_{4n+2}$. In this article, we prove projective normality of the GIT quotients of certain Schubert varieties in the orthogonal Grassmannian $G/P$ with respect to the descent of a suitable $T$-linearized very ample line bundle.

线性扩张与剥壳序

Linear extensions and shelling orders

该研究证明拟阵Bruhat序的线性扩张是剥壳序、拟阵重心重分为Coxeter拟阵，类似结论适用于对称群极小商的序理想，还给出了剥壳序的提升与排空操作。

The Sperner property for $132$-avoiding intervals in the weak order

A well-known result of Stanley from 1980 implies that the weak order on a maximal parabolic quotient of the symmetric group $S_n$ has the Sperner property; this same property was recently established for the weak order on all of $S_n$ by Gaetz and Gao, resolving a long-open problem. In this paper we interpolate between these results by showing that the weak order on any parabolic quotient of $S_n$ (and more generally on any $132$-avoiding interval) has the Sperner property. This result is proven by exhibiting an action of $\mathfrak{sl}_2$ respecting the weak order on these intervals. As a cor…

Parabolic double cosets in Coxeter groups

Parabolic subgroups $W_I$ of Coxeter systems $(W,S)$, as well as their ordinary and double quotients $W / W_I$ and $W_I \backslash W / W_J$, appear in many contexts in combinatorics and Lie theory, including the geometry and topology of generalized flag varieties and the symmetry groups of regular polytopes. The set of ordinary cosets $w W_I$, for $I \subseteq S$, forms the Coxeter complex of $W$, and is well-studied. In this article we look at a less studied object: the set of all double cosets $W_I w W_J$ for $I, J \subseteq S$. Double coset are not uniquely presented by triples $(I,w,J)$. W…

The intermediate orders of a Coxeter group

We define a class of partial orders on a Coxeter group associated with sets of reflections. In special cases, these lie between the left weak order and the Bruhat order. We prove that these posets are graded by the length function and that the projections on the right parabolic quotients are always order preserving. We also introduce the notion of $k$-Bruhat graph, $k$-absolute length and $k$-absolute order, proposing some related conjectures and problems.

On torus quotients of Schubert varieties in Orthogonal Grassmannian

Let $G=Spin(8n, \mathbb{C})(n\ge 1)$ and $T_{G}$ be a maximal torus of $G.$ Let $P^{α_{4n}}(\supset T_{G})$ be the maximal parabolic subgroup of $G$ corresponding to the simple root $α_{4n}.$ Let $X$ be a Schubert variety in $G/P^{α_{4n}}$ admitting semi-stable point with respect to the $T$-linearized very ample line bundle $\mathcal{L}(2ω_{4n}).$ Let $R=\bigoplus_{k \in \mathbb{Z}_{\geq 0}}R_k,$ where $R_k=H^{0}(X, \mathcal{L}^{\otimes k}(2ω_{4n}))^{T_{G}}.$ In this article, we prove that for $n=1$ and $X=G/P^{α_4},$ the graded $\mathbb{C}$-algebra $R$ is generated by $R_1.$ As a consequence,…