A Clifford Algebraic Framework for Coxeter Group Theoretic Computations

Real physical systems with reflective and rotational symmetries such as viruses, fullerenes and quasicrystals have recently been modeled successfully in terms of three-dimensional (affine) Coxeter groups. Motivated by this progress, we explore here the benefits of performing the relevant computations in a Geometric Algebra framework, which is particularly suited to describing reflections. Starting from the Coxeter generators of the reflections, we describe how the relevant chiral (rotational), full (Coxeter) and binary polyhedral groups can be easily generated and treated in a unified way in a versor formalism. In particular, this yields a simple construction of the binary polyhedral groups as discrete spinor groups. These in turn are known to generate Lie and Coxeter groups in dimension four, notably the exceptional groups D ₄, F ₄ and H ₄. A Clifford algebra approach thus reveals an unexpected connection between Coxeter groups of ranks 3 and 4. We discuss how to extend these considerations and computations to the Conformal Geometric Algebra setup, in particular for the non-crystallographic groups, and construct root systems and quasicrystalline point arrays. We finally show how a Clifford versor framework sheds light on the geometry of the Coxeter element and the Coxeter plane for the examples of the twodimensional non-crystallographic Coxeter groups I ₂(n) and the threedimensional groups A ₃, B ₃, as well as the icosahedral group H ₃. IPPP/12/49, DCPT/12/98     

Reducible polar representations

In this paper the reducible polar representations of the compact connected Lie groups are classified. It turns out that there only exist “interesting” reducible polar representations of Lie groups of the types A ₃, A ₃×T ¹, B ₃, B ₃×T ¹, D ₄, D ₄×T ¹ and D ₄×A ₁. Up to equivalence, there is just one such representation of the first four Lie groups, there are three reducible polar representations of D ₄ and six of D ₄×T ¹ and D ₄×A ₁, respectively. From this follows immediately the classification of the compact connected subgroups of SO(n) which act transitively on products of spheres. Received: 28 April 2000     

Maximal subgroups of the Coxeter group W(H4) and quaternions

The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W(H₄) of order 14,400. Its derived subgroup is the largest finite subgroup W(H₄)/Z₂ of SO(4) of order 7200. Moreover, up to conjugacy, it has five non-normal maximal subgroups of orders 144, two 240, 400 and 576. Two groups [W(H₂) × W(H₂)] ⋊ Z₄ and W(H₃) × Z₂ possess non-crystallographic structures with orders 400 and 240 respectively. The groups of orders 144, 240 and 576 are the extensions of the Weyl groups of the root systems of SU(3) × SU(3), SU(5) and SO(8) respectively. We represent the maximal subgroups of W(H₄) with sets of quaternion pairs acting on the quaternionic root systems.     

Spineurs symplectiques purs et indice de Maslov de plans lagrangiens positifs

As is well known, each point of the closed generalized unit-disk X can be associated to a holomorphically induced representation of the Heisenberg group. First canonical intertwining operators are constructed between pairs of such representations. Next, after having introduced suitable definitions, it is noted that the classical correspondence between group extensions and 2-cocycles also makes sense when applied to transformation spaces. As an example of transformation space extension, the manifold of pure symplectic spinors is described. It is the analogue of the manifold of pure spinors when the spin representation of the Clifford algebra is replaced by the Stone-Von Neumann representation of the Heisenberg group. Then, the associated 2-cocycle m₂ is worked out, which is a $\text{T}$-valued function on X × X × X, and the composition law of the canonical intertwining operators is given. Lifting m₂, an $\text{R}$-valued 2-cocycle m is constructed whose restriction to the Shilov boundary of X takes integer values and coincides with the ordinary Maslov index. For this reason, it is called the generalized Maslov index. Finally, using these results, explicit realizations of the metaplectic group, its Shale-Weil representation, and the universal covering of the symplectic group are given.     

Clifford Theory for Glider Representations

Classical Clifford theory studies the decomposition of simple G-modules into simple H-modules for some normal subgroup H ? G. In this paper we deal with chains of normal subgroups 1?G ₁?· · ·?G _d = G, which allow to consider fragments and in particular glider representations. These are given by a descending chain of vector spaces over some field K and relate different representations of the groups appearing in the chain. Picking some normal subgroup H ? G one obtains a normal subchain and one can construct an induced fragment structure. Moreover, a notion of irreducibility of fragments is introduced, which completes the list of ingredients to perform a Clifford theory.     

On quadruples of Griffiths points

Tabov (Math Mag 68:61–64, 1995) has proved the following theorem: if points A ₁, A ₂, A ₃, A ₄ are on a circle and a line l passes through the centre of the circle, then four Griffiths points G ₁, G ₂, G ₃, G ₄ corresponding to pairs (Δ _i ,l) are on a line (Δ _i denotes the triangle A _j A _k A _l , j,k,l ≠ i). In this paper we present a strong generalisation of the result of Tabov. An analogous property for four arbitrary points A ₁, A ₂, A ₃, A ₄, is proved, with the help of the computer program “Mathematica”.     

Cellules de Kazhdan–Lusztig et correspondance de Robinson–Schensted

Kazhdan and Lusztig have introduced (left, right and two-sided) cells in an arbitrary Coxeter group. For the symmetric group, they showed that these cells are given by the Robinson–Schensted correspondence. Here, we describe a Robinson–Schensted correspondence for the complex reflection groups G(e,1,n). In a recent joint work with C. Bonnafé, we have shown that, in the case e=2 (where G(2,1,n) is the Coxeter group of type B_n), this correspondence determines the Kazhdan–Lusztig cells with respect to certain unequal parameters. To cite this article: L. Iancu, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On the Spectra of Left-Definite Operators

If A is a self-adjoint operator that is bounded below in a Hilbert space H, Littlejohn and Wellman (J Diff Equ 181(2):280–339, 2002) showed that, for each r > 0, there exists a unique Hilbert space H _r and a unique self-adjoint operator A _r in H _r satisfying certain conditions dependent on H and A. The space H _r and the operator A _r are called, respectively, the rth left-definite space and rth left-definite operator associated with (H, A). In this paper, we show that the operators A, A _r , and A _s (r, s > 0) are isometrically isomorphically equivalent and that the spaces H, H _r , and H _s (r, s > 0) are isometrically isomorphic. These results are then used to reproduce the left-definite spaces and left-definite operators. Furthermore, we will see that our new results imply that the spectra of A and A _r are equal, giving us another proof of this phenomenon that was first established in Littlejohn and Wellman (J Diff Equ 181(2):280–339, 2002).     

On Erdos' ten-point problem

Around 1994, Erdoset al. abstracted from their work the following problem: “Given ten pointsA _ij, 1≤i≤j≤5, on a plane and no three of them being collinear, if there are five pointsA _k, 1≤k≤5, on the plane, including points at infinity, with at least two points distinct, such thatA _i, A_j, A_ij are collinear, where 1≤i≤j≤5, is it true that there are only finitely many suchA _k's?” Erdoset al. obtained the result that generally there are at most 49 groups of suchA _k's. In this paper, using Clifford algebra and Wu's method, we obtain the results that generally there are at most 6 such groups ofA _k's.     

Matrix Mechanics of the Relativistic Point Particle and String in Clifford Space

We resolve the space-time canonical variables of the relativistic point particle into inner products of Weyl spinors with components in a Clifford algebra and find that these spinors themselves form a canonical system with generalized Poisson brackets. For N particles, the inner products of their Clifford coordinates and momenta form two N × N Hermitian matrices X and P which transform under a U(N) symmetry in the generating algebra. This is used as a starting point for defining matrix mechanics for a point particle in Clifford space. Next we consider the string. The Lorentz metric induces a metric and a scalar on the world sheet which we represent by a Jackiw–Teitelboim term in the action. The string is described by a polymomenta canonical system and we find the wave solutions to the classical equations of motion for a flat world sheet. Finally, we show that the \({SL(2.\mathbb{C})}\) charge and space-time momentum of the quantized string satisfy the Poincaré algebra.     

Generic Modules Over a Class of Drinfeld's Quantum Doubles

Let k be a field and A _n (ω) be the Taft's n ²-dimensional Hopf algebras. When n is odd, the Drinfeld quantum double D(A _n (ω)) of A _n (ω) is a Ribbon Hopf algebra. In the previous articles, we constructed an n ⁴-dimensional Hopf algebra H _n (p, q) which is isomorphic to D(A _n (ω)) if p ≠ 0 and q = ω^?1, and studied the finite dimensional representations of H _n (1, q). We showed that the basic algebra of any nonsimple block of H _n (1, q) is independent of n. In this article, we examine the infinite representations of H ₂(1, ? 1), or equivalently of H _n (1, q)?D(A _n (ω)) for any n ≥ 2. We investigate the indecomposable and algebraically compact modules over H ₂(1, ? 1), describe the structures of these modules and classify them under the elementary equivalence.     

On Huppert’s Conjecture for 3 D 4(q), q ≥ 3

Let G be a finite group and cd(G) be the set of all complex irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G???H × A, where A is an abelian group. In this paper, we verify the conjecture for the family of simple exceptional groups of Lie type ³ D ₄(q), when q?≥?3.     

Estimate for index of hypersurfaces in spheres with null higher order mean curvature

In a recent paper (Barros, Sousa in: Kodai Math. J. 2009) the authors proved that closed oriented non-totally geodesic minimal hypersurfaces of the Euclidean unit sphere have index of stability greater than or equal to n + 3 with equality occurring at only Clifford tori provided their second fundamental forms A satisfy the pinching: |A|² ≥ n. The natural generalization for this pinching is ?(r + 2)S _r+2 ≥ (n ? r)S _r  > 0. Under this condition we shall extend such result for closed oriented hypersurface Σ ⁿ of the Euclidean unit sphere ${\mathbb{S}^{n+1}}$ with null S _r+1 mean curvature by showing that the index of r-stability, ${Ind_{\Sigma^n}^{r}}$ , also satisfies ${Ind_{\Sigma^n}^{r}\ge n+3}$ . Instead of the previous hypothesis if we consider ${\frac{S_{r+2}}{{S_r}}}$ constant we have the same conclusion. Moreover, we shall prove that, up to Clifford tori, closed oriented hypersurfaces ${\Sigma^{n}\subset \mathbb{S}^{n+1}}$ with S _r+1 = 0 and S _r+2 < 0 have index of r-stability greater than or equal to 2n + 5.     

Bimeasure algebras on locally compact groups

For locally compact groups G and H, let BM(G, H) denote the Banach space of bounded bilinear forms on C₀(G) × C₀(H). Using a consequence of the fundamental inequality of A. Grothendieck. a multiplication and an adjoint operation are introduced on BM(G, H) which generalize the convolution structure of M(G × H) and which make BM(G, H) into a K_G²-Banach ${}^1$-algebra, where K_G is Grothendieck's universal constant. Various topics relating to the ideal structure of BM(G, H) and the lifting of unitary representations of G × H to ${}^1$-representations of BM(G, H) are investigated.     

Weak Cayley Table Groups II: Alternating Groups and Finite Coxeter Groups

A weak Cayley table isomorphism is a bijection φ: G → H of groups such that φ(xy) ～ φ(x)φ(y) for all x, y ∈ G. Here ～denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ: G → G forms a group 𝒲(G) that contains the automorphism group Aut(G) and the inverse map I: G → G, x → x ^?1. Let 𝒲₀(G) = ?Aut(G), I? ≤ 𝒲(G) and say that G has trivial weak Cayley table group if 𝒲(G) = 𝒲₀(G). We show that all finite irreducible Coxeter groups (except possibly E ₈) have trivial weak Cayley table group, as well as most alternating groups. We also consider some sporadic simple groups.     

Bilinear Forms and Fierz Identities for Real Spin Representations

Given a real representation of the Clifford algebra corresponding to ${\mathbb{R}^{p+q}}$ with metric of signature (p, q), we demonstrate the existence of two natural bilinear forms on the space of spinors. With the Clifford action of k-forms on spinors, the bilinear forms allow us to relate two spinors with elements of the exterior algebra. From manipulations of a rank four spinorial tensor introduced in [1], we are able to find a general class of identities which, upon specializing from four spinors to two spinors and one spinor in signatures (1,3) and (10,1), yield some well-known Fierz identities. We will see, surprisingly, that the identities we construct are partly encoded in certain involutory real matrices that resemble the Krawtchouk matrices [2][3].     

Hecke-Kiselman monoids of small cardinality

In this paper, we give a characterization of digraphs Q, |Q|≤4 such that the associated Hecke-Kiselman monoids H _Q are finite. In general, a necessary condition for H _Q to be a finite monoid is that Q is acyclic and its Coxeter components are Dynkin diagrams. We show, by constructing examples, that such conditions are not sufficient.     

Cauchy Integral Formulae in Quaternionic Hermitean Clifford Analysis

The theory of complex Hermitean Clifford analysis was developed recently as a refinement of Euclidean Clifford analysis; it focusses on the simultaneous null solutions, called Hermitean monogenic functions, of two Hermitean Dirac operators constituting a splitting of the traditional Dirac operator. In this function theory, the fundamental integral representation formulae, such as the Borel?CPompeiu and the Clifford?CCauchy formula have been obtained by using a (2 ×?2) circulant matrix formulation. In the meantime, the basic setting has been established for so-called quaternionic Hermitean Clifford analysis, a theory centred around the simultaneous null solutions, called q-Hermitean monogenic functions, of four Hermitean Dirac operators in a quaternionic Clifford algebra setting. In this paper we address the problem of establishing a quaternionic Hermitean Clifford?CCauchy integral formula, by following a (4?× 4) circulant matrix approach.     

Coxeter cochain complexes

We define the Coxeter cochain complex of a Coxeter group (G, S) with coefficients in a ?[G]-module A. This is closely related to the complex of simplicial cochains on the abstract simplicial complex I(S) of the commuting subsets of S. We give some representative computations of Coxeter cohomology and explain the connection between the Coxeter cohomology for groups of type A, the (singular) homology of certain configuration spaces, and the (Tor) homology of certain local Artin rings.