In [4, Section 3.2] Bauer, Catanese and Grunewald pose the general question “which [finite] **groups** admit Beauville structures?” as well as the more difficult question of which Beauville **groups** are strongly real Beauville **groups**. Many authors have investigated these questions for several classes of finite **groups** including abelian **groups** [7]; symmetric **groups** [4, 17] as well as decorations of simple **groups** more generally [11, 18, 13, 14, 19, 20]; characteristically simple **groups** [8, 10, 24, 25] and nilpotent **groups** [1, 2, 3, 8, 16, 27] (this list of references is by no means exhaustive!) Here we extend this list by considering **Coxeter** **groups**. First we classify which of the irreducible **Coxeter** **groups** are (strongly real) Beauville **groups**.

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In 1986, G.X. Viennot introduced the theory of heaps of pieces as a visualiza- tion of Cartier and Foata’s “partially commutative monoids”. These are es- sentially labeled posets satisfying a few additional properties, and one natural setting where they arise is as models of reduced words in **Coxeter** **groups**. In this paper, we introduce a cyclic version of a heap, which loosely speaking, can be thought of as taking a heap and wrapping it into a cylinder. We call this object a toric heap , because we formalize it as a labeled toric poset, which is a cyclic version of an ordinary poset. Defining the category of toric heaps leads to the notion of certain morphisms such as toric extensions. We study toric heaps in **Coxeter** theory, because a cyclic shift of a reduced word is simply a conjugate by an initial or terminal generator. As such, we formalize and study a framework that we call cyclic reducibility in **Coxeter** theory, which is closely related to conjugacy. We introduce what it means for ele- ments to be torically reduced, which is a stronger condition than simply being cyclically reduced. Along the way, we encounter a new class of elements that we call torically fully commutative (TFC), which are those that have a unique cyclic commutativity class, and comprise a strictly bigger class than the cycli- cally fully commutative (CFC) elements. We prove several cyclic analogues of results on fully commutative (FC) elements due to Stembridge. We conclude with how this framework fits into recent work in **Coxeter** **groups**, and we correct a minor flaw in a few recently published theorems.

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their conjugacy classes, by Specht [14] and Young [15], have also been known for a long time. In 1972, Carter [2] gave a uniform and systematic treatment of the conjugacy classes of Weyl **groups**. More recently, Geck and Pfeiffer [6] reworked Carter’s description from more of an algo- rithmic standpoint. Motivation for investigating the conjugacy classes of finite **Coxeter** **groups**, and principally those of the irreducible finite **Coxeter** **groups**, has come from many directions, for example in the representation theory of these **groups** and the classification of maximal tori in **groups** of Lie type (see [3]). The behaviour of length in a conjugacy class is frequently important. Of particular interest are those elements of minimal and maximal lengths in their class. Instrumental to Carter’s work was establishing the fact that in a finite **Coxeter** group every element is either an involution or a product of two involutions. Given the importance of the length function, it is natural to ask whether for an element w it is possible to choose two involutions σ and τ with w = στ in such a way that combining a reduced expression for σ with one for τ produces a reduced expression for w. That is, can we ensure that the length `(w) is given by `(w) = `(σ) + `(τ)? Not surprisingly, the answer to this is, in general, no. This naturally leads to introducing the concept of excess of w, denoted by e(w), and defined by

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there is at least one (usually many) elements of maximal length and excess zero in every conjugacy class. Finally it is easy to check that every element of the dihe- dral group has excess zero, so the result is trivially true. Thus Theorem 1.1 holds for every finite irreducible **Coxeter** group, and hence for all finite **Coxeter** **groups**. Theorem 1.1 shows the existence of at least one element of maximal length and excess zero in every conjugacy class of a finite **Coxeter** group. However, if one looks at some small examples in the classical Weyl **groups**, it appears that every element of maximal length in a conjugacy class has excess zero. It is natural to ask whether this holds in general. It turns out that it does not – although the number of elements for which it fails seems to be small. For example, if W is of type E 6 , then in 23 of

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these automorphisms. In Section III C, we consider affine extending the crystallographic **groups** by two nodes and show that these do not induce any further affine extensions. In Section IV A, we briefly review the novel Kac-Moody-type extensions of non-crystallographic **Coxeter** **groups** from a recent paper and compare the induced extensions with the classification scheme presented there (Section IV B). In Section V, we conclude that in a wide class of extensions (single extensions or simply-laced double extensions with trivial projection kernel), the ten induced cases consid- ered here are the only ones that are compatible with the projection. We also discuss how lifting affine extensions of non-crystallographic **groups** to the crystallographic setting, as well as sym- metrisability of the resulting matrices, motivate a study of Cartan matrices over extended number fields.

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generalizations of previously constructed ones in the sense that they are formulated in terms of general roots, that is representation independent and irrespective of a specific **Coxeter** group. Our derivation of the solutions is based on some general identities which we present in appendix A, together with some evidence of their validity. For self-consistency and easy reference we present some case-by-case data for **Coxeter** **groups** in appendix B. Possibility iii) is not yet formulated in a generic way in terms of root systems and we will therefore not comment on it here.

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One problem is to classify the quasiparabolic subgroups in all finite **Coxeter** **groups**. We manage to solve the problem for all finite classical **Coxeter** **groups**, and the quasiparabolic subgroups are listed in Theorem 16 of Chapter 3. In the original paper of Rains and Vazirani [14], the authors prove that all quasiparabolic subgroups are generated by rotations. Heading this direction, we first classify the rotation subgroups of finite classical **Coxeter** **groups**. Compared with the classification of reflection subgroups [6], the classification of rotation subgroups turn out to be much more compli- cated, and the results are given by Theorem 3, 8, 9 and 10. Then we exclude the non-quasiparabolic rotation subgroups, and confirm the quasiparabolic subgroups within the rotation subgroups. In particular, we prove the quasiparabolicity of a previously conjectural class of subgroups which have index 4 in the centralizer of the minimal fixed-point-free involutions of D 2n .

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Systems such as Calogero models and Toda ﬁeld theories can be related to root systems coupled to **Coxeter** **groups** or Weyl **groups** in complete gen- erality [38][39][40]. In single particle systems it is easy enough to obtain the symmetry of the system, which can and has been used to construct new models that are physically meaningful. However, in ﬁeld theories and multi- particle systems it is not always a straight forward procedure to observe the symmetries involved in these systems. Often it will involve elaborate trans- formations on the level of the dynamical variables. The generalized Calogero model takes the form

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More recently, Hart and Sbeiti Clarke considered the remaining classical Weyl **groups** (see [7], [6] and [11]). They showed that if G is a classical affine Weyl group, then if C (G, X) is connected, its diameter exceeds the rank of G by at most 1. The obvious next question is: what happens in the exceptional affine **groups**? The purpose of this short note is to establish some general results for commuting involution graphs in affine **Coxeter** **groups**, and to deal with types ˜ F 4 and ˜ G 2 . Types ˜ E 6 E ˜ 7 and ˜ E 8 are more substantial and

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Commutation Classes of Reduced Words 1.1 Finite Coxeter Groups and Root Systems We begin with a brief overview of basic Coxeter group theory, following [Humphreys].. No originality is cl[r]

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The GAP routine works as follows. Firstly, we generate the cellular decomposi- tion, cell stabilizers and boundary of the corresponding Davis complex. To do that, we have written functions to find the irreducible components of a **Coxeter** group and to decide whether the group it is finite (comparing each irreducible component with the finite **Coxeter** **groups** of the same rank). We generate a multidimensional list CHAINS such that CHAINS[i] is the list of all chains T 1 < . . . < T i of length i (<

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Proposition 5.1. The vertex set for each of the involution posets of **Coxeter** **groups** of type H , excluding the identity, is the set of order 2 elements of the group (or involutions of the group) Proof. In the case of the weak and strong Bruhat Involution posets, we are specifically taking the set of involutions as our vertex set. Also, note that if the weak involution poset has a vertex set made up of the order 2 elements of the group, then the strong involution poset will have the same vertex set. Ultimately, we only need to show that it’s true for the weak involution poset. By Lemmas 2.42 and 2.43, we know that based on our two operations, that our vertices will be involutions. However, we then have to ensure that we are not just getting a subset of the set of involutions. However, fortunately the order of the set of involutions of the **Coxeter** **groups** of type H is known. There are 5 elements of order 2 in H 2 , 31 elements

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It has been shown previously that solvability of certain types of Hamiltonians can be es- tablished [17, 18, 19, 20] by relating the differential operators inside the Hamiltonians, that is essentially the Laplace operator, to a representation of the gl(N)-Lie algebra. This formulation can be made very systematic by associating the differential structure to poly- nomial invariants of **Coxeter** **groups**. This led the authors of [22, 23, 24, 25] to propose a procedure which allows to construct solvable Hamiltonians by taking the structure of the polynomial invariants as a starting point. Here we showed that this procedure can be extended successfully to polynomial invariants of q-deformed **Coxeter** **groups**. We con- structed some potentials resulting from these type of invariants. Due to the fact that the Jacobian determinant can still be factorized in terms of linear polynomials the resulting potentials are of Calogero type.

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Theorem 9.3 (Pure quaternionic sub-root systems). The rank-3 **Coxeter** **groups** can be represented by the pure quaternion part of the representations of the rank-4 **groups** they generate as spinors if and only if they contain the central inversion. Proof. If the central inversion I is generated by the roots, the bivectors defined as the Hodge duals of the roots are pure quaternions by Proposition 4.8 and they have the same product as the root vectors by Proposition 9.2, thus giving the required sub- root system. Conversely, if the pure quaternionic representation of the rank-3 group I α k is contained in the spinors α i α j generating the rank-4 group, then I α k = α i α j

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Kac-Moody approach in Lie Theory. Under the assumption that the extended matrix is symmetric, a unique affine extension has been obtained in each case. In Ref. [8] it was shown that this approach is not sufficient for applications in virology: The structures of viruses follow several different extensions of I by translation operators. Motivated by these results, we revisit here affine extensions of the non-crystallographic **Coxeter** **groups** H 2 , H 3

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Noting that for the **Coxeter** **groups** we consider the Cartan matrices are symmetric, such that the relations (2.9) also hold with i ↔ j. As the right hand side involves only one inner product in ˜ ∆, this formula achieves our objective and we may now express inner products in ˜ ∆ in terms of those in ∆. Recalling that simple roots and fundamental weights λ i are

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classes, by Specht [11] and Young [12], have also been known for a long time. In 1972, Carter [2] gave a uniform and systematic treatment of the conju- gacy classes of Weyl **groups**. More recently, Geck and Pfeiffer [6] reworked Carter’s descriptions from more of an algorithmic standpoint. Motivation for investigating the conjugacy classes of finite **Coxeter** **groups**, and prin- cipally those of the irreducible finite **Coxeter** **groups**, has come from many directions, for example in the representation theory of these **groups** and the classification of maximal tori in **groups** of Lie type (see [3]). The behaviour of lengths in a conjugacy class is frequently important. Of particular interest are those elements of minimal and maximal lengths in their class. Minimal length elements have received considerable attention – see [6]. Here we look at maximal length elements. Now every finite irreducible **Coxeter** group W possesses a (unique) element w 0 of maximal length in W . For C a conjugacy

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The second motivation is that **Coxeter** **groups** have a very special relationship with involutions. They are generated by involutions (known as fundamental reflections). The set of reflections of a **Coxeter** group (conjugates of fundamental reflections) is in one-to- one correspondence with its set of positive roots. For a **Coxeter** group of rank n, every involution can be expressed as a product of at most n orthogonal reflections. Every ele- ment of a finite **Coxeter** group can be expressed as a product of at most two involutions [2]. Moreover, the conjugacy classes of involutions have a particularly nice structure. Due to a result of Richardson [9], it is straightforward to determine them from the **Coxeter** graph. The involutions of minimal and maximal length in a conjugacy class are well un- derstood [8], and the lengths of involutions behave well with respect to conjugation, in that if x is an involution and r is a fundamental reflection, then either `(rxr) = `(x) + 2, or `(rxr) = `(x) − 2, or rxr = x. The involutions of minimal length in a conjugacy class are central elements of parabolic subgroups, and so are fairly easily counted. So again it is natural to ask what can be determined about the length distribution of involutions, both in a conjugacy class and in a **Coxeter** group as a whole.

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We slightly generalize the notion of a ‘graph with a tail’ and, in doing so, provide symmetric presentations for all the simply laced irreducible ﬁnite **Coxeter** **groups** with the aid of little more than a single short relation. These in turn readily give rise to natural representations of these **groups**.

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After this initial description in 1976, little more was achieved, until 1984, when Etienne, [Eti84], proved a result from which the existence of the descent algebra of the **Coxeter** **groups** of type A, or tlie symmetric **groups**, could be deduced. This was followed by Atkinson in 1985 who reproved Theorem 1 using yet another approach, [Atk86]. 1985 also saw a simple proof of Theorem 1 pro duced by Gai sia and Remmel for tlie case of tlie symmetric **groups**. Section 5 [GR85]. However, tliis time tlie proof had implications. By their clever use of shuffle products in their proof, Garsia and Remmel showed how the structure constants ajKL given in Theorem 1 could be obtained from matrices. More precisely, ajKL is the number of matrices whose matrix entries satisfy various con ditions related to J, K and L. This was one of tlie first instances of what we shall therefore call a “matrix inteipretation” of Theorem 1, and it was tliis interpretation that was to start a revival of interest in tlie subject, by making it more accessible to manipulation.

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