避免S_k的极大抛物子群

从射影空间到最小副群旗的态射

Morphisms from projective spaces to flags of minimal parabolic subgroups

本文研究了从射影空间到最小副群旗的非常数态射分类，通过上同调和Schubert交集恒等式证明了某些情况下不存在非常数态射，且修正了早期错误。

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…

Higher generating subgroups and Cohen-Macaulay complexes

We show how to find higher generating families of subgroups, in the sense of Abels and Holz, for groups acting on Cohen-Macaulay complexes. We apply this to groups with a BN-pair to prove higher generation by parabolic and Levi-subgroups and describe higher generating families of parabolic subgroups in Aut(F_n).

群中的避和集

Sum-avoiding sets in groups

令$A$是任意加法群$G$的有限子集，并且令$ϕ(A)$表示$A$中的最大子集$B$在$A$中是避免和的基数（也就是说，对于所有不同的$b_1,b_2 \in B$，$b_1+b_2 \not \in A$）。当$G$无扭转时，根据$ϕ(A)$控制$A$尺寸的问题由Erdens和Moser提出。当$G$有挠时，由于子群的存在，对于固定的$ϕ(A)$，$A$可以任意大。尽管如此，我们提供了一个定性的答案，在这种设置类似的Erdens-Moser问题，通过建立一个结构定理，粗略地说，断言$A$是有效地覆盖的$ϕ(A)$有限子群的$G$，或少于$ϕ(A)$有限子群的$G$连同一个剩余集的有界基数。为了避免大量的嵌套归纳论证，我们的证明使用了非标准分析的语言。我们还否定地回答了Erdens关于有限加性群$G$的大子集$A$与$ϕ(A)$有界的问题，但给出了一个积极的结果时，$|G|$是不可分的小素数。

关于抛物型Coxeter元分解的传递Hurwitz作用的注记

A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements

本文给出了一个简短而完备的证明：n股上的辫群传递作用于有限秩n的Coxeter群中Coxeter元的约化分解为反射积的集合.此外，我们使用相同的参数也表明，所有的分解的一个元素在一个抛物子群的W谎言，以及在这个抛物子群。

Counting nearest faraway flats for Coxeter chambers

In a finite Coxeter group $W$ and with two given conjugacy classes of parabolic subgroups $[X]$ and $[Y]$, we count those parabolic subgroups of $W$ in $[Y]$ that are full support, while simultaneously being simple extensions (i.e., extensions by a single reflection) of some standard parabolic subgroup of $W$ in $[X]$. The enumeration is given by a product formula that depends only on the two parabolic types. Our derivation is case-free and combines a geometric interpretation of the "full support" property with a double counting argument involving Crapo's beta invariant. As a corollary, this a…

Parabolic orbits of $2$-nilpotent elements for classical groups

We consider the conjugation-action of the Borel subgroup of the symplectic or the orthogonal group on the variety of nilpotent complex elements of nilpotency degree $2$ in its Lie algebra. We translate the setup to a representation-theoretic context in the language of a symmetric quiver algebra. This makes it possible to provide a parametrization of the orbits via a combinatorial tool that we call symplectic/orthogonal oriented link patterns. We deduce information about numerology. We then generalize these classifications to standard parabolic subgroups for all classical groups. Finally, our r…

抛物型Mirković-Vilonen交与Levi子群分支的Alcove游走模型

Alcove walk models for parabolic Mirković-Vilonen intersections and branching to Levi subgroups

本文通过建立由正折叠alcove游走索引的显式胞腔分解，给出了分裂约化群仿射格拉斯曼流形中Schubert簇与半无限轨道的抛物型Mirković-Vilonen交的模型，并由此推导出不可约最高权表示到Levi子群分支的新组合描述及特征标计算算法。

有限Coxeter群中元素中心化子的注记

A Note on Element Centralizers in Finite Coxeter Groups

有限Coxeter群$W$的标准抛物子群$W_J$的正规化子$N_W(W_J)$在由$W$中的$W_J$的某些最小长度陪集表示组成的补$N_J$的抛物子群上分裂。在本说明中，我们表明（除了少数情况下产生的情况下，Coxeter群的类型$D_n$）的中心化子$C_W(w)$的一个元素$w \in W$是在一个类似的方式半直积的中心化子$w$在一个适当的小抛物子群$W_J$与补充同构的正规化补充$N_J$。

A Lattice Isomorphism Theorem for Cluster Groups of Mutation-Dynkin Type $A_{n}$

Each quiver appearing in a seed of a cluster algebra determines a corresponding group, which we call a cluster group, which is defined via a presentation. Grant and Marsh showed that, for quivers appearing in seeds of cluster algebras of finite type, the associated cluster groups are isomorphic to finite reflection groups. As for finite Coxeter groups, we can consider parabolic subgroups of cluster groups. We prove that, in the type $A_{n}$ case, there exists an isomorphism between the lattice of subsets of the defining generators of the cluster group and the lattice of its parabolic subgroups…

一般线性群的抛物共轭

Parabolic conjugacy in general linear groups

设q是素数的幂，n是正整数.设P（q）是有限一般线性群GL（n，q）的抛物子群.我们证明了GL（n，q）中P（q）-共轭类的个数是q中一个整系数多项式的函数。这回答了J. Alperin的一个问题。

The B-orbits on a Hermitian symmetric variety in characteristic 2

Let $G$ be a reductive linear algebraic group over an algebraically closed field $\mathbb{K}$ of characteristic $2$. Fix a parabolic subgroup $P$ such that the corresponding parabolic subgroup over $\mathbb{C}$ has abelian unipotent radical and fix a Levi subgroup $L\subseteq P$. We parametrize the orbits of a Borel $B\subseteq P$ over the Hermitian symmetric variety $G/L$ supposing the root system $Φ$ is irreducible. For $Φ$ simply laced we prove a combinatorial characterization of the Bruhat order over these orbits. We also prove a formula to compute the dimension of the orbits from combinat…

On the Induction of p-Cells

We study cells with respect to the $p$-canonical basis of the Hecke algebra of a crystallographic Coxeter system (see arXiv:1510.01556, arXiv:1901.02323) and their compatibility with standard parabolic subgroups. We show that after induction to the surrounding bigger Coxeter group the cell module of a right $p$-cell in a standard parabolic subgroup decomposes as a direct sum of cell modules. Along the way, we state some new positivity properties of the $p$-canonical basis.