A generalization of Coxeter groups, root systems, and Matsumoto’s theorem

We begin this paper with a short survey on finite reflection groups. First we establish what a reflection in Euclidean space is. Then we introduce a root system, which is then partitioned into two sets: one of positive roots and one with negative roots. Th is articulates our understanding of groups generated by simple reflections. Furthermore, we develop our insight to Weyl groups and crystallographic groups before exploring crystallographic root systems. The section section of this paper examines the twisted group algebra along with the Demazure element Xi and the Demazure-Lusztig element Ti. Lastly, the third section of this paper computes such Xi and Ti in the case of S3 where S3 is the symmetric group Sn, n=3.

We generalize the notion of a root system by relaxing the conditions that ensure that it is invariant under reflections and study the resulting structures, which we call generalized root systems (GRSs). Since both Kostant root systems and root systems of Lie superalgebras are examples of GRSs, studying GRSs provides a uniform axiomatic approach to studying both of them. GRSs inherit many of the properties of root systems. In particular, every GRS defines a crystallographic hyperplane arrangement. We believe that GRSs provide an intrinsic counterpart to finite Weyl groupoids and crystallographic hyperplane arrangements, extending the relationship between finite Weyl groupoids and crystallographic hyperplane arrangements established by Cuntz. An important difference between GRSs and root systems is that GRSs may lack a (large enough) Weyl group. In order to compensate for this, we introduce the notion of a virtual reflection, building on a construction of Penkov and Serganova in the context of root systems of Lie superalgebras. The most significant new feature of GRSs is that, along with subsystems, one can define quotient GRSs. Both Kostant root systems and root systems of Lie superalgebras are equivalent to quotients of root systems and all root systems are isomorphic to quotients of simply-laced root systems. We classify all rank 2 GRSs and show that they are equivalent to quotients of root systems. Finally, we discuss in detail quotients of root systems. In particular we provide all isomorphisms and equivalences among them. Our results on quotient of root systems provide a different point of view on flag manifolds, reproving results of Alekseevsky and Graev.

Given a root system Φ of rank r and a global field F containing the n-th roots of unity, it is possible to define a Weyl group multiple Dirichlet series whose coefficients are n-th order Gauss sums. It is a function of r complex variables, and it has meromorphic continuation to all of C, with functional equations forming a group isomorphic to the Weyl group of Φ. Weyl group multiple Dirichlet series and their residues unify many examples that have been studied previously in a case-bycase basis, often with applications to analytic number theory. (Examples may be found in the final section of the paper.) We believe these Weyl group multiple Dirichlet series are fundamental objects. The goal of this paper is to define these series for any such Φ and F , and to indicate how to study them. We will note the following points. • The coefficients of the Weyl group multiple Dirichlet series are multiplicative, but the multiplicativity is twisted , so the Dirichlet series is not an Euler product. • Due to the multiplicativity, description of the coefficients reduces to the case where the parameters are powers of a single prime p. There are only finitely many such coefficients (for given p). • In the “stable case” where n is sufficiently large (depending on Φ), the number of nonzero coefficients in the p-part is equal to the order of the Weyl group. Indeed, these nonzero coefficients are parametrized in a natural way by the Weyl group elements. • The p-part coefficient parametrized by a Weyl group element w is a product of l(w) Gauss sums, where l is the length function on the Weyl group. We note a curious similarity between this description and the coefficients of the generalized theta series on the n-fold cover of GL(n) and GL(n − 1); these coefficients are determined in Kazhdan and Patterson [17] and discussed further in Patterson [21]. See Bump and Hoffstein [11] or Hoffstein [15] for a “classical” description of these coefficients. The noted similarity means that the complete Mellin transform of the theta function would be a multiple Dirichlet series resembling our An+1 multiple Dirichlet series. There is no a priori reason that we are

Let \(\mathfrak{g}_{0}\) be a real semisimple Lie algebra. Let \(\mathfrak{g}_{0} = \mathfrak{k}_{0} \oplus \mathfrak{p}_{0}\) be the corresponding Cartan decomposition and \(\mathfrak{h}_{0} = \mathfrak{t}_{0} \oplus \mathfrak{a}_{0}\) be a maximally compact Cartan subalgebra of \(\mathfrak{g}_{0}\). Let \(\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}\) and \(\mathfrak{h} = \mathfrak{t} \oplus \mathfrak{a}\) be the corresponding complexifications. The set \(\Delta {\left( {\mathfrak{g},\mathfrak{t}} \right)}\) consists of all the linear forms on \(\mathfrak{t}\) which are the restriction to \(\mathfrak{t}\) of the roots in the root system \(\Delta {\left( {\mathfrak{g},\mathfrak{h}} \right)}\) of \(\mathfrak{g}\) with respect to \(\mathfrak{h}\). The main result of the paper is to prove that \(\Delta {\left( {\mathfrak{g},\mathfrak{t}} \right)}\) is also a (maybe non-reduced) root system and its Weyl group can be identified with a subgroup of the Weyl group of \(\Delta {\left( {\mathfrak{g},\mathfrak{h}} \right)}\). Let \(Spin\,\nu :\mathfrak{k} \to End\,S\) be the composition of the isotropy representation \(\nu :\mathfrak{k} \to \mathfrak{s}\mathfrak{o}{\left( \mathfrak{p} \right)}\) with the spin representation \(Spin:\mathfrak{s}\mathfrak{o}{\left( \mathfrak{p} \right)} \to End\,S\). Finally as an application, we give a nice description of the \(\mathfrak{k}\)-module structure on S in terms of the restricted root system \(\Delta {\left( {\mathfrak{g},\mathfrak{t}} \right)}\) and its Weyl group.

We introduce inversions for classical Weyl group elements and relate them, by counting, to the length function, root systems and Schubert cells in flag manifolds. Special inversions are those that only change signs in the Weyl groups of types Bn, Cnand Dn. Their counting is related to the (only) generator of the Weyl group that changes signs, to the corresponding roots, and to a special subvariety in the flag manifold fixed by a finite group.

It is well known that the Weyl groups play a key role in the theory of the structure of Lie algebras and their linear representations, as well as in the theory of Chevalley groups. Suppose g is a complex semisimple Lie algebra, h its Cartan subalgebra, Σ and Π denote its root system and the basis of the root system, respectively, and w α i indicates the reflection determined by αi ∈ Π. The so-called Weyl group, denoted by W,refers to the finite group generated by all w α i ( αi ∈ Π). There have been many works published on the structure and properties of W [6, 7]. The following Molien identity related to the global structure of W is well known [3]: $$\underset{w\in W}{\mathop{\Sigma}}\,\frac{1}{\det\left( 1-tw \right)}=\frac{1}{\prod{{{\left( 1-{{t}^{{{d}_{i}}}} \right)}^{,}}}}$$ (8.1) where d i signify the Poincare indices of g. In 1956, by topological methods based on Morse’s theory, Bott proved another identity [2]: $$\underset{w\in W}{\mathop \sum }\,{{t}^{l\left(w \right)}}=\prod{\frac{1-{{t}^{{{d}_{i}}}}}{1-{{t}^{,}}}}$$ (8.2) where l(w) refers to the Π-length of w;i.e.,wmay be expressed by the number of minimal products of w αi ( αi ∈ Π). This important formula may shed new light on our understanding the structure of W. In 1966, Solomon offered a purely algebraic proof of (8.2), thus solving the order problem of the Chevalley simple groups [5, 3].

To each reduced root system Φ of rank r, and each sufficiently large integer n, we define a family of multiple Dirichlet series in r complex variables, whose group of functional equations is isomorphic to the Weyl group of Φ. The coefficients in these Dirichlet series exhibit a multiplicativity that reduces the specification of the coefficients to those that are powers of a single prime p. For each p, the number of nonzero such coefficients is equal to the order of the Weyl group, and each nonzero coefficient is a product of n-th order Gauss sums. The root system plays a basic role in the combinatorics underlying the proof of the functional equations.

For connected reductive groups together with a Frobenius root $F$, we show that the cohomology of the structure sheaf and respectively the canonical sheaf for compactified Deligne–Lusztig varieties associated to an element in the free monoid generated by the simple reflections is isomorphic to that of a minimal length element in an $F$-conjugacy class in the Weyl group.

Classification of crystallographic groups associated with Coxeter groups

A Weyl group multiple Dirichlet seriesis a Dirichlet series in several complex variables attached to a root system Φ. The number of variables equals the rank rof the root system, and the series satisfies a group of functional equations isomorphic to the Weyl group Wof Φ. In this paper we construct a Weyl group multiple Dirichlet series over the rational function field using n th order Gauss sums attached to the root system of type A 2. The basic technique is that of [11, 10]; namely, we construct a rational function in rvariables invariant under a certain action of W, and use this to build a “local factor” of the global series.

Let $\Phi$ be an irreducible crystallographic root system with Weyl group $W$, coroot lattice $\check{Q}$ and Coxeter number $h$. Recently the second named author defined a uniform $W$-isomorphism $\zeta$ between the finite torus $\check{Q}/(mh+1)\check{Q}$ and the set of non-nesting parking functions $\operatorname{Park}^{(m)}(\Phi)$. If $\Phi$ is of type $A_{n-1}$ and $m=1$ this map is equivalent to a map defined on labelled Dyck paths that arises in the study of the Hilbert series of the space of diagonal harmonics. In this paper we investigate the case $m=1$ for the other infinite families of root systems ($B_n$, $C_n$ and $D_n$). In each type we define models for the finite torus and for the set of non-nesting parking functions in terms of labelled lattice paths. The map $\zeta$ can then be viewed as a map between these combinatorial objects. Our work entails new bijections between (square) lattice paths and ballot paths.

In recent years there has developed a satisfactory and coherent theory of orthogonal polynomials in several variables, attached to root systems, and depending on two or more parameters. These polynomials include as special cases: symmetric functions; zonal spherical functions on real and p-adic reductive Lie groups; the Jacobi polynomials of Heckman and Opdam; and the Askey–Wilson polynomials, which themselves include as special or limiting cases all the classical families of orthogonal polynomials in one variable. This book, first published in 2003, is a comprehensive and organised account of the subject aims to provide a unified foundation for this theory, to which the author has been a principal contributor. It is an essentially self-contained treatment, accessible to graduate students familiar with root systems and Weyl groups. The first four chapters are preparatory to Chapter V, which is the heart of the book and contains all the main results in full generality.

Let g be a complex simple Lie algebra, h its Cartan subalgebra, and Σ and \( \prod =\left\{ {{\beta }_{1}},\ldots ,{{\beta }_{i}} \right\} \)the root system of g and a basis of the root system, respectively. It is known that the Weyl group W(g) is a finite group generated by reflections [3] $${w_{{\rm B}i}}:\xi \in h,\xi \to \xi - \frac{{2\left( {\xi ,} \right.}}{{\left( {{\beta _{i,}}} \right.}}\frac{{\left. {{\beta _i}} \right)}}{{\left. {{\beta _i}} \right)}}{\beta _{i \bullet }}$$

The abelianizations of Weyl groups of root systems extended by abelian groups

ABSTRACTThe vertices of the four-dimensional 600-cell form a non-crystallographic root system whose corresponding symmetry group is the Coxeter group H4. There are two special coordinate representations of this root system in which they and their corresponding Coxeter groups involve only rational numbers and the golden ratio τ. The two are related by the conjugation τ↦τ′ = −1∕τ. This paper investigates what happens when the two root systems are combined and the group generated by both versions of H4 is allowed to operate on them. The result is a new, but infinite, ‘root system’ Σ which itself turns out to have a natural structure of the unitary group SU(2,ℛ) over the ring (called here golden numbers). Acting upon it is the naturally associated infinite reflection group H∞, which we prove is of index 2 in the orthogonal group O(4,ℛ). The paper makes extensive use of the quaternions over ℛ and leads to a highly structured discretized filtration of SU(2). We use this to offer a simple and effective way to approximate any element of SU(2) to any degree of accuracy required using the repeated actions of just five fixed reflections, a process that may find application in computational methods in quantum mechanics.