Related papers: Coxeter and Dynkin Diagrams

This paper is a sequel to work of Dynkin on subroot lattices of root lattices and to work of Carter on presentations of Weyl group elements as products of reflections. The quotients $L/L_1$ are calculated for all irreducible root lattices…

The author is mainly interest in the Gr\"{o}bner-Shirshov bases of finite Coxeter groups. It is known that the finite Coxeter groups are classified in terms of Coxeter-Dynkin diagrams. Under the fixed order, it is worth mention that the…

We introduce the concept of hyperreflection groups, which are a generalization of Coxeter groups. We prove the Deletion and Exchange Conditions for hyperreflection groups, and we discuss special subgroups and fundamental sectors of…

We give a criterion for a finitely generated odd-angled Coxeter group to have a proper finite index subgroup generated by reflections. The answer is given in terms of the least prime divisors of the exponents of the Coxeter relations.

Let $W$ be a finite Coxeter group. We classify the reflection subgroups of $W$ up to conjugacy and give necessary and sufficient conditions for the map that assigns to a reflection subgroup $R$ of $W$ the conjugacy class of its Coxeter…

We study associative multiplications in semi-simple associative algebras over C compatible with the usual one. An interesting class of such multiplications is related to the affine Dynkin diagrams of A, D, E-type. In this paper we…

We construct the fcc (face centered cubic), bcc (body centered cubic) and sc (simple cubic) lattices as the root and the weight lattices of the affine Coxeter groups W(D3) and W(B3)=Aut(D3). The rank-3 Coxeter-Weyl groups describing the…

We discuss the classification of reflection subgroups of finite and affine Weyl groups from the point of view of their root systems. A short case free proof is given of the well known classification of the isomorphism classes of reflection…

Given any Coxeter group, we define rigid reflections and rigid roots using non-self-intersecting curves on a Riemann surface with labeled curves. When the Coxeter group arises from an acyclic quiver, they are related to the rigid…

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…

I show that the new topological field theories recently associated by Dubrovin with each Coxeter group may be all obtained in a simple way by a ``restriction'' of the standard ADE solutions. I then study the Chebichev specializations of…

In present work, we find a class of Lie algebras, which are defined from the symmetrizable generalized intersection matrices. However, such algebras are different from generalized intersection matrix algebras and intersection matrix…

We study diagrams associated with a finite simplicial complex K, in various algebraic and topological categories. We relate their colimits to familiar structures in algebra, combinatorics, geometry and topology. These include: right-angled…

We continue the Coxeter spectral analysis of finite connected posets $I$ that are non-negative in the sense that their symmetric Gram matrix $G_I:=\frac{1}{2}(C_I + C_I^{tr})\in\mathbb{M}_{m}(\mathbb{Q})$ is positive semi-definite of rank…

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,…

Affine Coxeter groups are fundamental objects in mathematics and in crystallography. If two group elements are conjugate, then they have very similar algebraic and geometric properties. Using recent structural results of Mili\'cevi\'c,…

We extend Carter's notion of admissible diagrams and attach a "Dynkin-like" diagram to each reduced reflection factorization of an element in a finite Weyl group. We give a complete classification for the diagrams attached to reduced…

All quasi-affine connected Generalized Dynkin Diagram with rank $= 3$ and $2$ are found. All quasi-affine Nichols (Lie braided) algebras with rank $ 3$ and $2$ are also found.

In a recent paper we claimed that both the group algebra of a finite Coxeter group $W$ as well as the Orlik-Solomon algebra of $W$ can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each…

For simply-laced quivers, we consider the fixed-point subalgebra of the quiver Hecke algebra under the homogeneous sign map. This leads to a new family of algebras we call alternating quiver Hecke algebras. We give a basis theorem and a…