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].     

Generalized Killing spinors on spheres

We study generalized Killing spinors on round spheres \(\mathbb {S}^n\) . We show that on the standard sphere \(\mathbb {S}^8\) any generalized Killing spinor has to be an ordinary Killing spinor. Moreover, we classify generalized Killing spinors on \(\mathbb {S}^n\) whose associated symmetric endomorphism has at most two eigenvalues and recover in particular Agricola–Friedrich’s canonical spinor on 3-Sasakian manifolds of dimension 7. Finally, we show that it is not possible to deform Killing spinors on standard spheres into genuine generalized Killing spinors.     

A Few Comments on the Clifford Algebra {C\ell_2} of the Standard Euclidean Plane

This paper is a continuation of the author’s plenary lecture given at ICCA 9 which was held in Weimar at the Bauhaus University, 15–20 July, 2011. We want to study on both the mathematical and the epistemological levels the thought of the brilliant geometer W. K. Clifford by presenting a few comments on the structure of the Clifford algebra ${C\ell_2}$ associated with the standard Euclidean plane ${\mathbb{R}^2}$ . Miquel’s theorem will be given in the algebraic context of the even Clifford algebra ${C\ell^+_2}$ isomorphic to the real algebra ${\mathbb{C}}$ . The proof of this theorem will be based on the cross ratio (the anharmonic ratio) of four complex numbers. It will lead to a group of homographies of the standard projective line ${\mathbb{C}P^1 = P(\mathbb{C}^2)}$ which appeared so attractive to W. K. Clifford in his overview of a general theory of anharmonics. In conclusion it will be shown how the classical Clifford-Hopf fibration S ¹ → S ³ → S ² leads to the space of spinors ${\mathbb{C}^2}$ of the Euclidean space ${\mathbb{R}^3}$ and to the isomorphism ${{\rm {PU}(1) = \rm {SU}(2)/\{I,-I\} \simeq SO(3)}}$ .     

Full automorphism group of the generalized symplectic graph

Let Fq be a finite field of odd characteristic, m, ν the integers with 1≤m≤ν and Ka 2ν× 2ν nonsingular alternate matrix over Fq. In this paper, the generalized symplectic graph GSp2ν (q, m) relative to K over Fq is introduced. It is the graph with m-dimensional totally isotropic subspaces of the 2ν-dimensional symplectic space F(2ν)q as its vertices and two vertices P and Q are adjacent if and only if the rank of PKQT is 1 and the dimension of P ∩ Q is m-1. It is proved that the full automorphism group of the graph GSp2ν(q, m) is the projective semilinear symplectic group PΣp(2ν, q).     

Collineation group as a subgroup of the symmetric group

Let ψ be the projectivization (i.e., the set of one-dimensional vector subspaces) of a vector space of dimension ≥ 3 over a field. Let H be a closed (in the pointwise convergence topology) subgroup of the permutation group $\mathfrak{S}_\psi $ of the set ψ. Suppose that H contains the projective group and an arbitrary self-bijection of ψ transforming a triple of collinear points to a non-collinear triple. It is well known from [Kantor W.M., McDonough T.P., On the maximality of PSL(d+1,q), d ≥ 2, J. London Math. Soc., 1974, 8(3), 426] that if ψ is finite then H contains the alternating subgroup $\mathfrak{A}_\psi $ of $\mathfrak{S}_\psi $ . We show in Theorem 3.1 that H = $\mathfrak{S}_\psi $ , if ψ is infinite.     

Zhelobenko invariants, Bernstein-Gelfand-Gelfand operators and the analogue Kostant Clifford algebra conjecture

Let $ \mathfrak{g} $ be a complex simple Lie algebra and $ \mathfrak{h} $ a Cartan subalgebra. The Clifford algebra C( $ \mathfrak{g} $ ) of g admits a Harish-Chandra map. Kostant conjectured (as communicated to Bazlov in about 1997) that the value of this map on a (suitably chosen) fundamental invariant of degree 2?m?+?1 is just the zero weight vector of the simple (2?m?+?1)-dimensional module of the principal s-triple obtained from the Langlands dual $ {\mathfrak{g}^\vee } $ . Bazlov [1] settled this conjecture positively in type A. The hard part of the Kostant Clifford algebra conjecture is a question concerning the Harish-Chandra map for the enveloping algebra U( $ \mathfrak{g} $ ) composed with evaluation at the half sum ?? of the positive roots. The analogue Kostant conjecture is obtained by replacing the Harish-Chandra map by a ??generalized Harish-Chandra?? map. This map had been studied notably by Zhelobenko [15]. The proof given here involves a symmetric algebra version of the Kostant conjecture, the Zhelobenko invariants in the adjoint case, and, surprisingly, the Bernstein-Gelfand-Gelfand operators introduced in their study [3] of the cohomology of the flag variety.     

Spin geometry and grand unification

Gravitation becomes unified with quantum mechanics when we recognize that the spacetime tetrads and the matter fields of Fermions are the integral and half-integral spin representations of theEinstein group, E, the global extension of the Poincaré group to a curved spacetimeM. There are8 fundamental spinor representations of theE group, interchanged byP, T, andC: the degree-one maps of spin space overM. Tensor products of2 spinor fields buildClifford vectors or 1 forms, e.g. the spacetime tetrads. It takes tensor products of all8 spinor fields to build a natural 4 form; in particular, ourE-invariant Lagrangian density . We propose a simple form for : the8-spinor factorization of theMaurer-Cartan 4-form, Ω⁴. Thespin connections Ω_α step off the conjoined left and right internalgl (2, ?) phase increments over aspacetime incremente _α. Our actionS _g measures the covering number of the spinor phases over spacetimeM∪D _J; theD _J aresingular domains or caustics, whereJ=1, 2, and 3 chiral pairs of spin waves cross. Here, the massive Dirac equations emerge to govern the mass scattering that keep the “null zig-zags” of a bispinor particle confined to a timelike worldtube. We identify the coupled envelopes of 1, 2, and 3 chiral bispinor pairs as the leptons, mesons, and hadrons, respectively. These source topologically —nontrivialgl (2,C) phase distributions in the far-field region, which appear aseffective vector potentials. Their vorticities are thespin curvatures, whose Hermitian parts —thegravitational curvatures —specify how our spacetime manifoldM must expand and curve to accommodate such anholonomic differentials. The anti-Hermitian parts reproduce the standard electroweak and strong fields, together with their actions. also contains some new cross terms between electroweak potentials and gravitational curvatures. Do these signal a failure of unification, or predict new phenomena?     

The Complex Algebra of Physical Space: A Framework for Relativity

A complex and, equivalently, hyperbolic extension of the algebra of physical space (APS) is discussed that allows one to distinguish space-time vectors from paravectors of APS, while preserving the natural origin of the Minkowski space-time metric. The CAPS formalism is Lorentz covariant and gives expression to persistent vectors in physical space as time-like planes in space-time. Commuting projectors ${P_{\pm} = \frac{1}{2} (1 \pm h)}$ project CAPS onto two-sided ideals, one of which is APS. CAPS has the same dimension as the space-time algebra (STA) if both are considered real algebras, and it distinguishes covariant roles of elements, as does STA. Its structure, however, is closer to APS, with a volume element that belongs to the center of the algebra and a simple relation between space-times of opposite signature. Furthermore, CAPS, unlike STA, distinguishes point-like space-time inversion of a Dirac spinor from a physical rotation. To illustrate its use, CAPS is applied to the Dirac equation and to the fundamental symmetry transformations of the equation and Dirac spinors. The physical interpretations of both the equation and the spinor are clarified, and it is seen that the space-time frame ${\{\gamma_{\mu}\}}$ arises fully from relative vectors and does not imply the existence of an absolute space-time frame.     

Spinorial Formulations of Graph Problems

Beginning with an arbitrary finite graph, various spinor spaces are constructed within Clifford algebras of appropriate dimension. Properties of spinors within these spaces then reveal information about the structure of the graph. Spinor polynomials are introduced and the notions of degrees of polynomials and Fock subspace dimensions are tied together with matchings, cliques, independent sets, and cycle covers of arbitrary finite graphs. In particular, matchings, independent sets, cliques, cycle covers, and cycles of arbitrary length are all enumerated by dimensions of spinor subspaces, while sizes of maximal cliques and independent sets are revealed by degrees of spinor polynomials. The spinor adjacency operator is introduced and used to enumerate cycles of arbitrary length and to compute graph circumference and girth.     

Clifford parallelism: old and new definitions, and their use

Parallelity in the real elliptic 3-space was defined by W. K. Clifford in 1873 and by F. Klein in 1890; we compare the two concepts. A Clifford parallelism consists of all regular spreads of the real projective 3-space ${{\rm PG}(3,\mathbb{R})}$ whose (complex) focal lines (=directrices) form a regulus contained in an imaginary quadric (D1 = Klein??s definition). Our new access to the topic ??Clifford parallelism?? is free of complexification and involves Klein??s correspondence ?? of line geometry together with a bijective map ?? from all regular spreads of ${{\rm PG}(3,\mathbb{R})}$ onto those lines of ${{\rm PG}(5,\mathbb{R})}$ having no common point with the Klein quadric; a regular parallelism P of ${{\rm PG}(3,\mathbb{R})}$ is Clifford, if the spreads of P are mapped by ?? onto a plane of lines (D2 = planarity definition). We prove the equivalence of (D1) and (D2). Associated with ?? is a simple dimension concept for regular parallelisms which allows us to say instead of (D2): the 2-dimensional regular parallelisms of ${{\rm PG}(3,\mathbb{R})}$ are Clifford (D3 = dimensionality definition). Submission of (D2) to ??^?1 yields a complexification free definition of a Clifford parallelism which uses only elements of ${{\rm PG}(3,\mathbb{R})}$ : A regular parallelism P is Clifford, if the union of any two distinct spreads of P is contained in a general linear complex of lines (D4 = line geometric definition). In order to see (D1) and (D2) simultaneously at work we discuss the following two examples using, at the one hand, complexification and (D1) and, at the other hand, (D2) under avoidance of complexification. Example 1. In the projectively extended real Euclidean 3-space a rotational regular spread with center o is submitted to the group of all rotations about o; we prove, that a Clifford parallelism is generated. Example 2. We determine the group ${Aut_e({\bf P}_{\bf C})}$ of all automorphic collineations and dualities of the Clifford parallelism P _C and show ${Aut_e({\bf P}_{\bf C})\hspace{1.5mm} \cong ({\rm SO}_3\mathbb{R} \times {\rm SO}_3\mathbb{R})\rtimes \mathbb{Z}_2}$ .     

Bounds for codes and designs in complex subspaces

We introduce the concepts of complex Grassmannian codes and designs. Let $\mathcal{G}_{m,n}$ denote the set of m-dimensional subspaces of ? ⁿ : then a code is a finite subset of $\mathcal{G}_{m,n}$ in which few distances occur, while a design is a finite subset of $\mathcal{G}_{m,n}$ that polynomially approximates the entire set. Using Delsarte’s linear programming techniques, we find upper bounds for the size of a code and lower bounds for the size of a design, and we show that association schemes can occur when the bounds are tight. These results are motivated by the bounds for real subspaces recently found by Bachoc, Bannai, Coulangeon and Nebe, and the bounds generalize those of Delsarte, Goethals and Seidel for codes and designs on the complex unit sphere.     

A search for higher-dimensional arc planes

The concept of arc planes introduced byH. Groh (see [3]) has been extended to higher dimensions in [9]. By definition, a stable plane with point space ?^2l is an arc plane if all vector space translations are automorphisms. The lines are either affine subspaces or graphs of non-affine functions. We examine these functions, especially their domains, and obtain information restricting the search for examples. Roughly speaking, one should start with a translation plane and substitute some lines with the graphs of suitable functions.     

Mixed Invariant Subspaces over the Bidisk

It is known that the structure of invariant subspaces I of the Hardy space H ² over the bidisk is extremely complicated. One reason is that it is difficult to describe infinite dimensional wandering spaces ${I\ominus zI}$ completely. In this paper, we study the structure of nontrivial closed subspaces N of H ² with ${T_zN\subset N}$ and ${T^*_wN\subset N}$ , which are called mixed invariant subspaces under T _z and ${T^*_w}$ . We know that the dimension of ${N\ominus zN}$ ranges from 1 to ??. If ${T^*_w(N\ominus zN)\subset N\ominus zN}$ , we may describe N completely. If ${T^*_w(N\ominus zN)\not\subset N\ominus zN}$ , it seems difficult to describe N generally. So we study N under the condition ${dim\,(N\ominus zN)=1}$ . Write ${M=H^2\ominus N}$ . We describe ${M\ominus wM}$ precisely. We give a characterization of N for which there is a nonzero function ${\varphi}$ in ${M\ominus wM}$ satisfying ${z^k\varphi\in M\ominus wM}$ for every k ?? 0. We also see that the space ${M\ominus wM}$ has a deep connection with the de Branges?CRovnyak spaces studied by Sarason.     

Tangent-Point Repulsive Potentials for a Class of Non-smooth m-dimensional Sets in ℝ n . Part I: Smoothing and Self-avoidance Effects

We consider repulsive potential energies $\mathcal {E}_{q}(\Sigma)$ , whose integrand measures tangent-point interactions, on a large class of non-smooth m-dimensional sets Σ in ? ⁿ . Finiteness of the energy $\mathcal {E}_{q}(\Sigma)$ has three sorts of effects for the set Σ: topological effects excluding all kinds of (a priori admissible) self-intersections, geometric and measure-theoretic effects, providing large projections of Σ onto suitable m-planes and therefore large m-dimensional Hausdorff measure of Σ within small balls up to a uniformly controlled scale, and finally, regularizing effects culminating in a geometric variant of the Morrey–Sobolev embedding theorem: Any admissible set Σ with finite $\mathcal {E}_{q}$ -energy, for any exponent q>2m, is, in fact, a C ¹-manifold whose tangent planes vary in a Hölder continuous manner with the optimal Hölder exponent μ=1?(2m)/q. Moreover, the patch size of the local C ^1,μ -graph representations is uniformly controlled from below only in terms of the energy value $\mathcal {E}_{q}(\Sigma)$ .     

Hyperflock determining line sets and totally regular parallelisms of PG {(3,\,{\mathbb{R}})}

By a totally regular parallelism of the real projective 3-space ${\Pi_3:={{\rm PG}}(3, \mathbb {R})}$ we mean a family T of regular spreads such that each line of Π ₃ is contained in exactly one spread of T. For the investigation of totally regular parallelisms the authors mainly employ Klein’s correspondence λ of line geometry and the polarity π ₅ associated with the Klein quadric H ₅ (for details see Chaps. 1 and 3). The λ-image of a totally regular parallelism T is a hyperflock of H ₅, i.e., a family H of elliptic subquadrics of H ₅ such that each point of H ₅ is on exactly one subquadric of H. Moreover, ${\{\pi_5({{\rm span}} \,\lambda(\mathcal {X}))\vert\mathcal {X}\in\bf{T}\}=:\mathcal {H}_{\bf{T}}}$ is a hyperflock determining line set, i.e., a set ${\mathcal {Z}}$ of 0-secants of H ₅ such that each tangential hyperplane of H ₅ contains exactly one line of ${\mathcal {Z}}$ . We say that ${{{\rm dim}}({{\rm span}}\,\mathcal {H}_{\bf{T}})=:d_{\bf{T}}}$ is the dimension of T and that T is a d _T - parallelism. Clifford parallelisms and 2-parallelisms coincide. The examples of non-Clifford parallelisms exhibited in Betten and Riesinger [Result Math 47:226–241, 2004; Adv Geom 8:11–32, 2008; J Geom (to appear)] are totally regular and of dimension 3. If ${\mathcal{G}}$ is a hyperflock determining line set, then ${\{\lambda^{-1}\,{\rm (}\pi_5(X){\,\cap H_5)\,|\, X\in\mathcal{G}\}}}$ is a totally regular parallelism. In the present paper the authors construct examples of topological (see Definition 1.1) 4- and 5-parallelisms via hyperflock determining line sets.     

Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster \mathfrak{D}^ \bot-parallel structure Jacobi operator

Regarding the generalized Tanaka-Webster connection, we considered a new notion of \(\mathfrak{D}^ \bot\) -parallel structure Jacobi operator for a real hypersurface in a complex two-plane Grassmannian G ₂(? ^m+2) and proved that a real hypersurface in G ₂(? ^m+2) with generalized Tanaka-Webster \(\mathfrak{D}^ \bot\) -parallel structure Jacobi operator is locally congruent to an open part of a tube around a totally geodesic quaternionic projective space ?P ⁿ in G ₂(? ^m+2), where m = 2n.     

Weakly Convex Biharmonic Hypersurfaces in Nonpositive Curvature Space Forms are Minimal

A submanifold M ^m of a Euclidean space R ^m+p is said to have harmonic mean curvature vector field if ${\Delta \vec{H}=0}$ , where ${\vec{H}}$ is the mean curvature vector field of ${M\hookrightarrow R^{m+p}}$ and Δ is the rough Laplacian on M. There is a famous conjecture named after Bangyen Chen which states that submanifolds of Euclidean spaces with harmonic mean curvature vector fields are minimal. In this paper we prove that weakly convex hypersurfaces (i.e. hypersurfaces whose principle curvatures are nonnegative) with harmonic mean curvature vector fields in Euclidean spaces are minimal. Furthermore we prove that weakly convex biharmonic hypersurfaces in nonpositively curved space forms are minimal.     

The supertail of a subspace partition

Let V = V(n, q) be a vector space of dimension n over the finite field with q elements, and let d ₁ < d ₂ < ... < d _m be the dimensions that occur in a subspace partition ${\mathcal{P}}$ of V. Let σ _q (n, t) denote the minimum size of a subspace partition ${\mathcal P}$ of V, in which t is the largest dimension of a subspace. For any integer s, with 1 < s ≤ m, the set of subspaces in ${\mathcal{P}}$ of dimension less than d _s is called the s-supertail of ${\mathcal{P}}$ . The main result is that the number of spaces in an s-supertail is at least σ _q (d _s , d _s?1).     

The maximum order of adjacency matrices of graphs with a given rank

We look for the maximum order m(r) of the adjacency matrix A of a graph G with a fixed rank r, provided A has no repeated rows or all-zero row. Akbari, Cameron and Khosrovshahi conjecture that m(r)?=?2^(r+2)/2 ? 2 if r is even, and m(r)?=?5 · 2^(r?3)/2 ? 2 if r is odd. We prove the conjecture and characterize G in the case that G contains an induced subgraph ${\frac{r}{2}K_2}$ or ${\frac{r-3}{2}K_2+K_3}$ .