A Unified Construction of Coxeter Group Representations (I)

Conjectures P1-P15 for hyperbolic Coxeter groups of rank 3

We prove Lusztig's conjectures P1-P15 for hyperbolic Coxeter groups of rank 3. Our proof enables us to give a description of the a-function and Kazhdan-Lusztig cells for these Coxeter groups.

Coxeter群及其忠实表示的真联合谱

Coxeter groups and the proper joint spectrums of their faithful representations

本文分析了二面体群的忠实表示，证明了Coxeter群可由其忠实表示的真联合谱唯一确定。

The boundness of weighted Coxeter groups of rank 3

We prove that a weighted Coxeter group (W,S,L) is bounded in the sense of G.Lusztig if the rank of W is 3.

Involutions in Coxeter groups

This paper constructs a representation of a Hecke algebra on a vector space spanned by the involutions in a Coxeter group.

Determinants of Representations of Coxeter Groups

In [APS], the authors characterize the partitions of $n$ whose corresponding representations of $S_n$ have nontrivial determinant. The present paper extends this work to all irreducible finite Coxeter groups $W$. Namely, given a nontrivial multiplicative character $ω$ of $W$, we give a closed formula for the number of irreducible representations of $W$ with determinant $ω$. For Coxeter groups of type $B_n$ and $D_n$, this is accomplished by characterizing the bipartitions associated to such representations.

Lusztig's $a$ function for Coxeter groups of rank 3

We show that Lusztig's $a$-function of a Coxeter group is bounded if the rank of the Coxeter group is 3.

Diagrammatics for Coxeter groups and their braid groups

We give a monoidal presentation of Coxeter and braid 2-groups, in terms of decorated planar graphs. This presentation extends the Coxeter presentation. We deduce a simple criterion for a Coxeter group or braid group to act on a category.

Minimal length elements in some double cosets of Coxeter groups

We study the minimal length elements in some double cosets of Coxeter groups and use them to study Lusztig's $G$-stable pieces and the generalization of $G$-stable pieces introduced by Lu and Yakimov. We also use them to study the minimal length elements in a conjugacy class of a finite Coxeter group and prove a conjecture in \cite{GKP}.

晶体学Coxeter群的标准表示

Standard Representations of Crystallographic Coxeter Groups

该论文研究将特定Coxeter群表示为Kac-Moody代数Weyl群的Cartan表示，证明了相关同构条件、有限特征域上的部分结果，明确了晶体学Coxeter群的含义并给出对应Weyl群的条件，还提供了示例。

Coxeter orbits and modular representations

We study the modular representations of finite groups of Lie type arising in the cohomology of certain quotients of Deligne-Lusztig varieties associated with Coxeter elements. These quotients are related to Gelfand-Graev representations and we present a conjecture on the Deligne-Lusztig restriction of Gelfand-Graev representations. We prove the conjecture for restriction to a Coxeter torus. We deduce a proof of Broué's conjecture on equivalences of derived categories arising from Deligne-Lusztig varieties, for a split group of type $A\_n$ and a Coxeter element. Our study is based on Lusztig's …

Some power series involving involutions in Coxeter groups

Let W be a Coxeter group. We show that a certain power series involving a sum over all involutions in W can be expressed in terms of the Poincare series of W. (The case where W is finite is already known,)

Minimal length elements of finite Coxeter groups

We give a geometric proof that minimal length elements in a (twisted) conjugacy class of a finite Coxeter group $W$ have remarkable properties with respect to conjugation, taking powers in the associated Braid group and taking centralizer in $W$.

Rigid reflections of rank 3 Coxeter groups and reduced roots of rank 2 Kac--Moody algebras

In a recent paper by K.-H. Lee and K. Lee, rigid reflections are defined for any Coxeter group via non-self-intersecting curves on a Riemann surface with labeled curves. When the Coxeter group arises from an acyclic quiver, the rigid reflections are related to the rigid representations of the quiver. For a family of rank $3$ Coxeter groups, it was conjectured in the same paper that there is a natural bijection from the set of reduced positive roots of a symmetric rank $2$ Kac--Moody algebra onto the set of rigid reflections of the corresponding rank $3$ Coxeter group. In this paper, we prove t…

A remark on descent for Coxeter groups

Let $Γ$ be a finite Coxeter group with reflection representation $R$. We show that a $Γ$-equivariant quasicoherent sheaf on $R$ descends to the quotient space $R//Γ$ if it descends to the quotient space $R//\langle s_i\rangle$ for every simple reflection $s_i\in Γ$.

Conjectures P1-P15 for Coxeter groups with complete graph

We prove Lusztig's conjectures P1-P15 for Coxeter groups with complete graph, using deceasing induction on $ \mathbf{a} $-values and a kind of decomposition formula of Kazhdan-Lusztig basis elements. As a byproduct, we give a description of the left, right, and two-sided cells. In the appendix, we prove P1-P15 for right-angled Coxeter groups by the same methods.

Coxeter群的Lusztig a函数值1的表示

Representations of Coxeter groups of Lusztig's a-function value 1

本文研究Coxeter群中Lusztig a函数值1的表示，确定了某些单理Coxeter群的不可约表示。

Elements with finite Coxeter part in an affine Weyl group

Let $W_a$ be an affine Weyl group and $η:W_a\longrightarrow W_0$ be the natural projection to the corresponding finite Weyl group. We say that $w\in W_a$ has finite Coxeter part if $η(w)$ is conjugate to a Coxeter element of $W_0$. The elements with finite Coxeter part is a union of conjugacy classes of $W_a$. We show that for each conjugacy class $\mathcal{O}$ of $W_a$ with finite Coxeter part there exits a unique maximal proper parabolic subgroup $W_J$ of $W_a$, such that the set of minimal length elements in $\mathcal{O}$ is exactly the set of Coxeter elements in $W_J$. Similar results hold…

保正交体的Coxeter群

Coxeter groups preserving orthants

本文证明保正交体的Coxeter群必为对称群乘积，并给出仿射类比，由此揭示量子几何Satake等价两版本的存在性是纯A型现象。

Higher signs for Coxeter groups

We define and study cocycles on a Coxeter group in each degree generalizing the sign function. When the Coxeter group is a Weyl group, we explain how the degree three cocycle arises naturally from geometry representation theory.

On paired root systems of Coxeter groups

This paper examines a systematic method to construct a pair of (inter-related) root systems for arbitrary Coxeter groups from a class of non-standard geometric representations. This method can be employed to construct generalizations of root systems for a large family of groups generated only by involutions. We then give a characterization of Coxeter groups, among these groups, in terms of such paired root systems. Furthermore, we use this method to construct and study the paired root systems for reflection subgroups of Coxeter groups.

Preprojective roots of Coxeter groups

Certain results on representations of quivers have analogs in the structure theory of general Coxeter groups. A fixed Coxeter element turns the Coxeter graph into an acyclic quiver, allowing for the definition of a preprojective root. A positive root is an analog of an indecomposable representation of the quiver. The Coxeter group is finite if and only if every positive root is preprojective, which is analogous to the well-known result that a quiver is of finite representation type if and only if every indecomposable representation is preprojective. Combinatorics of orientation-admissible word…

Birman-Murakami-Wenzl algebras for general Coxeter groups

We introduce a BMW type algebra for every Coxeter group. These new algebras are introduced as deformations of the Brauer type algebras introduced by the author, they have the corresponding Hecke algebras as quotients.

The strong Lefschetz property of the coinvariant ring of the Coxeter group of type H4

We prove that the coinvariant ring of the irreducible Coxeter group of type H4 has the strong Lefschetz property.

Vanishing ranges for the mod $p$ cohomology of alternating subgroups of Coxeter groups

We obtain vanishing ranges for the mod $p$ cohomology of alternating subgroups of finite $p$-free Coxeter groups. Here a Coxeter group $W$ is $p$-free if the order of the product $st$ is prime to $p$ for every pair of Coxeter generators $s,t$ of $W$. Our result generalizes those for alternating groups formerly proved by Kleshchev-Nakano and Burichenko. As a byproduct, we obtain vanishing ranges for the twisted cohomology of finite $p$-free Coxeter groups with coefficients in the sign representations. In addition, a weak version of the main result is proved for a certain class of infinite Coxet…

Relatively compact sets in the reduced $C^{\ast}$-algebras of Coxeter groups

We characterize relatively norm compact sets in the regular $C^*$-algebra of finitely generated Coxeter groups using a geometrically defined positive semigroup acting on the algebra.

On convex hulls of orbits of Coxeter groups and Weyl groups

The notion of a linear Coxeter system introduced by Vinberg generalizes the geometric representation of a Coxeter group. Our main theorem asserts that if $v$ is an element of the Tits cone of a linear Coxeter system and $\cW$ is the corresponding Coxeter group, then $\cW v \subeq v - C_v,$ where $C_v$ is the convex cone generated by the coroots $\check α$, for which $α(v) > 0$. This implies that the convex hull of $\cW v$ is completely determined by the image of $v$ under the reflections in $\cW$. We also apply an analogous result for convex hulls of $\cW$-orbits in the dual space, although th…