Invariant Quadratic Form Research Articles

Orthogonal representations of $${{\,\textrm{SL}\,}}_2(q)$$ in defining characteristic

AbstractThis paper determines the types of the invariant quadratic forms over their respective (finite) fields of definition for all irreducible modules of the groups $${{\,\textrm{SL}\,}}_2(q)$$ SL 2 ( q ) in defining characteristic. We prove that for $$q>2$$ q > 2 any absolutely irreducible even dimensional orthogonal $${{\,\textrm{SL}\,}}_2(q)$$ SL 2 ( q ) -module W in defining characteristic carries a split invariant quadratic form (i.e. it is of $$+$$ + type) unless $$\dim (W) \equiv 4 \pmod {8}$$ dim ( W ) ≡ 4 ( mod 8 ) and the field of definition of W is the subfield of index 2 in $${\mathbb {F}}_q$$ F q ; in the latter case the type of the invariant quadratic forms is −.

3d super hyperbolic geometry

Wigner's unitary representation of the Lorentz group is extended to a representation of the complex orthosymplectic Lie super group OSpC(1|2) acting on Minkowski (3,1|4)-dimensional super space essentially by Hermitean conjugation. The invariant quadratic form is x1x2−xx¯+ϕψ+ϕ¯ψ¯ in Wigner's coordinates x1,x2∈R and x∈C, where ϕ,ψ are Dirac fermions. The extended action is linear in the super space variables, but not quadratic in the odd group variables, and is described by an explicit even purely imaginary auxiliary parameter defined on the product of the Lie super group with its Lie algebra. This extension of Wigner's representation opens the door to studying geometry in super hyperbolic three-space with this metric, initiated here with foundations on super geodesics, triangles and tetrahedra, and culminating in a proof of divergence of the volume of a typical ideal tetrahedron and a discussion of the non-zero fermionic correction to the Schläfli formula.

Invariants of quadratic forms and applications in design theory

The study of regular incidence structures, such as projective planes and symmetric block designs, is a well established topic in discrete mathematics. The work of Bruck, Ryser and Chowla in the mid-twentieth century applied the Hasse-Minkowski local-global theory for quadratic forms to derive non-existence results for certain design parameters.Several combinatorialists have provided alternative proofs of this result, replacing conceptual arguments with algorithmic ones. In this paper, we show that the methods required are purely linear-algebraic and are no more difficult conceptually than the theory of the Jordan Canonical Form. Computationally, they are rather easier. We conclude with some classical and recent applications to design theory, including a novel application to the decomposition of incidence matrices of symmetric designs.

Orthogonal determinants of characters

For an irreducible orthogonal character chi of even degree, there is a unique square class det ({chi }) in the character field such that the invariant quadratic forms in any L-representation affording chi have determinant in det ({chi })(L^{times })^2.

Quadratic principal indecomposable modules and strongly real elements of finite groups

Let $P$ be a principal indecomposable module of a finite group $G$ in characteristic $2$ and let $\varphi$ be the Brauer character of the corresponding simple $G$-module. We show that $P$ affords a non-degenerate $G$-invariant quadratic form if and only if there are involutions $s,t\in G$ such that $st$ has odd order and $\varphi(st)/2$ is not an algebraic integer. We then show that the number of isomorphism classes of quadratic principal indecomposable $G$-modules is equal to the number of strongly real conjugacy classes of odd order elements of $G$.

The orthogonal character table of SL2(q)

The rational invariants of the SL2(q)-invariant quadratic forms on the real irreducible representations are determined. There is still one open question (see Remark 6.5) if q is an even square.

Invariant forms on irreducible modules of simple algebraic groups

Let G be a simple linear algebraic group over an algebraically closed field K of characteristic p≥0 and let V be an irreducible rational G-module with highest weight λ. When V is self-dual, a basic question to ask is whether V has a non-degenerate G-invariant alternating bilinear form or a non-degenerate G-invariant quadratic form.If p≠2, the answer is well known and easily described in terms of λ. In the case where p=2, we know that if V is self-dual, it always has a non-degenerate G-invariant alternating bilinear form. However, determining when V has a non-degenerate G-invariant quadratic form is a classical problem that still remains open. We solve the problem in the case where G is of classical type and λ is a fundamental highest weight ωi, and in the case where G is of type Al and λ=ωr+ωs for 1≤r<s≤l. We also give a solution in some specific cases when G is of exceptional type.As an application of our results, we refine Seitz's 1987 description of maximal subgroups of simple algebraic groups of classical type. One consequence of this is the following result. If X<Y<SL(V) are simple algebraic groups and V↓X is irreducible, then one of the following holds: (1) V↓Y is not self-dual; (2) both or neither of the modules V↓Y and V↓X have a non-degenerate invariant quadratic form; (3) p=2, X=SO(V), and Y=Sp(V).

On the algebraic structure of rotationally invariant two-dimensional Hamiltonians on the noncommutative phase space

We study two-dimensional Hamiltonians in phase space with noncommutativity both in coordinates and momenta. We consider the generator of rotations on the noncommutative plane and the Lie algebra generated by Hermitian rotationally invariant quadratic forms of noncommutative dynamical variables. We show that two quantum phases are possible, characterized by the Lie algebras or su(2) according to the relation between the noncommutativity parameters, with the rotation generator related with the Casimir operator. From this algebraic perspective, we analyze the spectrum of some simple models with nonrelativistic rotationally invariant Hamiltonians in this noncommutative phase space, such as the isotropic harmonic oscillator, the Landau problem and the cylindrical well potential.

The cohesiveness of G-symplectic methods

General linear methods are multistage multivalue methods. This large family of numerical methods for ordinary differential equations, includes Runge---Kutta and linear multistep methods as special cases. G-symplectic general linear methods are multivalue methods which preserve a generalization of quadratic invariants. If Q is an invariant quadratic form then symplectic Runge---Kutta methods preserve this invariant. In the case of a G-symplectic general linear method, there exists a non-singular symmetric r×r matrix G such that G?Q is an invariant quadratic form for this method. Although the numerical results can be corrupted by parasitic behaviour, it is possible to overcome the effect of parasitic growth by imposing additional constraints on the method. For G-symmetric methods satisfying these additional conditions, numerical experiments give excellent results. A new concept known as "cohesiveness" is introduced in an attempt to explain this favourable numerical behaviour. It is shown that the deviation from perfect cohesiveness grows slowly as steps of the method are carried out.

On interaction of symplectic and orthogonal Hecke–Shimura rings of one-class quadratic forms

Transformation formulas of theta-series with harmonic polynomials of one-class quadratic forms under Hecke operators are interpreted as a result of interaction of the standard representation of the symplectic Hecke-Shimura ring on theta-series with the natural representation of the orthogonal Hecke-Shimura ring on the same theta-series considered as invariants of quadratic forms. Properties of the interaction rnaps and their relations to the action of Hecke operators are considered. Bibliography: 9 titles.

On a cosmological invariant as an observational probe in the early universe

k-essence scalar field models are usually taken to have lagrangians of the form ${\mathcal L}=-V(\phi)F(X)$ with $F$ some general function of $X=\nabla_{\mu}\phi\nabla^{\mu}\phi$. Under certain conditions this lagrangian in the context of the early universe can take the form of that of an oscillator with time dependent frequency. The Ermakov invariant for a time dependent oscillator in a cosmological scenario then leads to an invariant quadratic form involving the Hubble parameter and the logarithm of the scale factor. In principle, this invariant can lead to further observational probes for the early universe. Moreover, if such an invariant can be observationally verified then the presence of dark energy will also be indirectly confirmed.

Clifford modules and invariants of quadratic forms

AbstractWe construct new invariants of quadratic forms over commutative rings, using ideas from Topology. More precisely, we define a hermitian analog of the Bott class with target algebraic K-theory, based on the classification of Clifford modules. These invariants of quadratic forms go beyond the classical invariants defined via the Clifford algebra. An appendix by J.-P. Serre, of independent interest, describes the “square root” of the Bott class in the general framework of lambda rings.

Generation of vector boson masses in Electroweak Model without dynamical symmetry breaking

The vector boson masses are generated by transformation of the free (without any potential term) Lagrangian of the Electroweak Model from Cartesian coordinates to a coordinates on the sphere S3, which is defined by the gauge invariant quadratic form ϕ†ϕ = ρ2 in the matter field space Φ2. This transformation corresponds to transition from linear representation of the gauge group in the space Φ2 to its nonlinear representation in the space of functions on S3. Such modified Electroweak Model keep all experimentally verified fields of the standard Electroweak Model and does not include massive scalar field (Higgs boson), if the sphere radius does not depend on the space-time coordinates ρ = R = const. The concept of generation masses for vector bosons in Electroweak Model by transformation to radial coordinates is further developed in context of nonlinearly realized gauge groups, as well as in context of nonlinear sigma models. The limiting case of the modified Electroweak Model which corresponds to the contracted gauge group is discussed.

Invariant quadratic forms on spaces of holomorphic functions

Existence and uniqueness of invariant quadratic forms on spaces of holomorphic functions is studied. Invariance is with respect to the action of the automorphism group of the underlying domain given by , where ϕ is an automorphism and J ϕ is its Jacobian determinant. Quadratic forms invariant under this action are called q-invariant forms. For q > 0, it is shown, under mild assumptions on the Bergman kernal, that there is a unique q-invariant form, which is positive definite and determines a unique functional Hilbert space. In the case of the ball, there is a unique q-invariant form for all real q, which is positive definite for q > 0, and positive semi-definite for q = 0 and a discrete set of negative q. In those cases when the form is positive semi-definite, integral formulas are obtained for the associated (semi) norms.

FIXED POINTS OF HOLOMORPHIC TRANSFORMATIONS OF OPERATOR BALLS

A new technique for proving fixed-point theorems for families of holomorphic transformations of operator balls is developed. One of these theorems is used to show that a bounded group representation in a real or complex Hilbert space is orthogonalizable or unitarizable (that is similar to an orthogonal or unitary representation), respectively, provided the representation has an invariant indefinite quadratic form with finitely many negative squares.

Unitarizable representations and fixed points of groups of biholomorphic transformations of operator balls

We show that the open unit ball of the space of operators from a finite-dimensional Hilbert space into a separable Hilbert space (we call it “operator ball”) has a restricted form of normal structure if we endow it with a hyperbolic metric (which is an analogue of the standard hyperbolic metric on the unit disc in the complex plane). We use this result to get a fixed point theorem for groups of biholomorphic automorphisms of the operator ball. The fixed point theorem is used to show that a bounded representation in a separable Hilbert space which has an invariant indefinite quadratic form with finitely many negative squares is unitarizable (equivalent to a unitary representation). We apply this result to find dual pairs of invariant subspaces in Pontryagin spaces. In Appendix A we present results of Itai Shafrir about hyperbolic metrics on the operator ball.

Hyperbolic surfaces in the Grassmannian

In this article we study real 2-dimensional surfaces in the Grassmannian of 2-planes in a 4-dimensional vector space. These surfaces occur naturally as the fibers of jet bundles of partial differential equations. On the Grassmannian there is an invariant conformal quadratic form and we will use the structure induced by this quadratic form to study the surfaces. We give a topological classification of compact hyperbolic surfaces similar to the classification by Gluck and Warner [Duke Math. J. 50 (1) (1983)] of compact elliptic surfaces. In contrast with elliptic surfaces there are several topological possibilities for hyperbolic surfaces. We make a calculation of the differential invariants under the action of the group of conformal isometries. Finally, we analyze a class of surfaces called geometrically flat and show that within this class there exist many examples of non-trivial compact surfaces.

PONTRJAGIN FORMS AND INVARIANT OBJECTS RELATED TO THE Q-CURVATURE

It was shown by Chern and Simons that the Pontrjagin forms are conformally invariant. We show them to be the Pontrjagin forms of the conformally invariant tractor connection. The Q-curvature is intimately related to the Pfaffian. Working on even-dimensional manifolds, we show how the k-form operators Qkof [12], which generalize the Q-curvature, retain a key aspect of the Q-curvature's relation to the Pfaffian, by obstructing certain representations of natural operators on closed forms. In a closely related direction, we show that the Qkgive rise to conformally invariant quadratic forms Θkon cohomology that interpolate, in a suitable sense, between the integrated metric pairing (at k = n/2) and the Pfaffian (at k = 0). Using a different construction, we show that the Qkoperators yield a map from conformal structures to Lagrangian subspaces of the direct sum Hk⊕ Hk(where Hkis the dual of the de Rham cohomology space Hk); in an appropriate sense this generalizes the period map. We couple the Qkoperators with the Pontrjagin forms to construct new natural densities that have many properties in common with the original Q-curvature; in particular these integrate to global conformal invariants. We also work out a relevant example, and show that the proof of the invariance of the (nonlinear) action functional whose critical metrics have constant Q-curvature extends to the action functionals for these new Q-like objects. Finally we set up eigenvalue problems that generalize to Qk-operators the Q-curvature prescription problem.

NON-ABELIAN TENSOR GAUGE FIELDS II

In recent papers we suggested an infinite-dimensional extension of gauge transformations which includes non-Abelian tensor gauge fields. Extended gauge transformations of non-Abelian tensor gauge fields form a new large group which has a natural geometrical interpretation in terms of extended current algebra associated with compact Lie group. We shall demonstrate that one can construct two infinite series of gauge invariant quadratic forms, so that a linear combination of them comprises the general Lagrangian. The general Lagrangian exhibits enhanced local gauge invariance with double number of gauge parameters and allows to eliminate all negative norm states of the nonsymmetric second-rank tensor gauge field. Therefore it describes two polarizations of helicity-two massless charged tensor gauge boson and the helicity-zero Kalb–Ramond field. The geometrical interpretation of the enhanced gauge symmetry with double number of gauge parameters is not yet known.

Gauge Invariant Forms for Non-Abelian Tensor Gauge Fields of Fourth Rank

Using generalized field strength tensors we constructed all possible Lorentz invariant quadratic forms for the rank-4 non-Abelian tensor gauge field and demonstrated that there exist only two linearly independent combinations of them which are gauge invariant. Together with the previous result for the rank-2 and rank-3 tensor gauge fields this construction proves the uniqueness of early proposed Lagrangian up to rank-4 tensor fields.