Related papers: Introduction to finite Coxeter groups and their re…
In [5], Elnitsky constructed three elegant bijections between classes of reduced words for Type $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{D}$ families of Coxeter groups and certain tilings of polygons. This paper offers a particular…
The article covers developments in the representation theory of finite group schemes over the last fifteen years. We start with the finite generation of cohomology of a finite group scheme and proceed to discuss various consequences and…
In this short note we discuss the interplay between finite Coxeter groups and construction of wavelet sets, generalized multiresolution analysis and sampling.
We lay the foundations of the first-order model theory of Coxeter groups. Firstly, with the exception of the $2$-spherical non-affine case (which we leave open), we characterize the superstable Coxeter groups of finite rank, which we show…
In this paper we use families of finite subgroups to study Grothendieck rings associated to certain discrete groups, such as the arithmetic ones.
To every group of $I$-type, we associate a finite quotient group that plays the role that Coxeter groups play for Artin-Tits groups. Since groups of I-type are examples of Garside groups, this answers a question of D. Bessis in the…
In this expository note, I showcase the relevance of Coxeter groups to quiver representations. I discuss (1) real and imaginary roots, (2) reflection functors, and (3) torsion free classes and c-sortable elements. The first two topics are…
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…
In this article, I study some classes of finitely presented groups with the aim of finding out whether the maximal metabelian quotients of the members of these classes admit finite presentations. The considered classes include those of…
We give formulas for the second and third integral homology of an arbitrary finitely generated Coxeter group, solely in terms of the corresponding Coxeter diagram. The first of these calculations refines a theorem of Howlett, while the…
Let $d$ be a positive integer. We study the proportion of irreducible characters of infinite families of irreducible Coxeter groups whose values evaluated on a fixed element $g$ are divisible by $d$. For Coxeter groups of types $A_n, B_n$…
In this paper, we give a characterization of Coxeter group representations of Lusztig's a-function value 1, then determine all the irreducible such representations for certain simply laced Coxeter groups.
These are the notes of some lectures given by the author for a workshop held at TIFR, Mumbai in December, 2011, giving an exposition of the Deligne-Lusztig theory.
In this paper, we summarize the work on the characterization of finite simple groups and the study on finite groups with the set of element orders and two orders (the order of group and the set of element orders). Some related topics, and…
Let W be an irreducible finitely generated Coxeter group. The geometric representation of W in GL(V) provides a discrete embedding in the orthogonal group of the Tits form (the associated bilinear form of the Coxeter group). If the Tits…
Coxeter groups are a special class of groups generated by involutions. They play important roles in the various areas of mathematics. This survey particularly focuses on how one uses Coxeter groups to construct interesting examples of…
Coxeter and Dynkin diagrams classify a wide variety of structures, most notably finite reflection groups, lattices having such groups as symmetries, compact simple Lie groups and complex simple Lie algebras. The simply laced or "ADE" Dynkin…
Computer based techniques for recognizing finitely presented groups are quite powerful. Tools available for this purpose are outlined. They are available both in stand-alone programs and in more comprehensive systems. A general…
We construct the first explicit finite presentations for a family of K\"ahler groups with arbitrary finiteness properties, answering a question of Suciu.
In this article we introduce the notion of a \textit{regular partition} of a Coxeter group. We develop the theory of these partitions, and show that the class of regular partitions is essentially equivalent to the class of automata (not…