An algebra determined by a binary cubic form

Heerema, Nickolas

doi:10.1215/s0012-7094-54-02143-2

Cited by 20 publications

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“…As explained in the introduction, we characterize this problem in terms of finite dimensional representations of an algebra with involution. We became aware that a similar approach has been used for the problem of linearizing forms, by Heerema [6], Roby [17] and Childs [4], among others. A solution to their problem implies a determinantal representation for the polynomial, but not necessarily a hermitian one, and also without the matrix M 0 being positive semidefinite.…”

Section: The Generalized Clifford Algebra Associated With a Real Zeromentioning

confidence: 99%

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Polynomials with and without determinantal representations

Netzer

Thom

2012

Linear Algebra and its Applications

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The problem of writing real zero polynomials as determinants of linear matrix polynomials has recently attracted a lot of attention. Helton and Vinnikov [9] have proved that any real zero polynomial in two variables has a determinantal representation. Brändén [2] has shown that the result does not extend to arbitrary numbers of variables, disproving the generalized Lax conjecture. We prove that in fact almost no real zero polynomial admits a determinantal representation; there are dimensional differences between the two sets. So the generalized Lax conjecture fails badly. The result follows from a general upper bound on the size of linear matrix polynomials. We then provide a large class of surprisingly simple explicit real zero polynomials that do not have a determinantal representation, improving upon Brändén's mostly unconstructive result. We finally characterize polynomials of which some power has a determinantal representation, in terms of an algebra with involution having a finite dimensional representation. We use the characterization to prove that any quadratic real zero polynomial has a determinantal representation, after taking a high enough power. Taking powers is thereby really necessary in general. The representations emerge explicitly, and we characterize them up to unitary equivalence.

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Section: The Generalized Clifford Algebra Associated With a Real Zeromentioning

confidence: 99%

“…Similar algebras have been used before by different authors (see e.g. [4,6,14,17]), in attempts to linearize forms and realize polynomials as minimal polynomials of matrix pencils. Their results relate to our problem, but do not take into account the desire for hermitian representations.…”

Section: Introductionmentioning

confidence: 98%

Polynomials with and without determinantal representations

Netzer

Thom

2012

Linear Algebra and its Applications

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“…The case of n = 2 and d = 3 was first considered by Heerema in [12]. Haile studied these algebras in the series of papers [6,7,8,9].…”

Section: Introductionmentioning

confidence: 99%

On the generalized Clifford algebra of a monic polynomial

Chapman

Kuo

2015

Linear Algebra and its Applications

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In this paper we study the generalized Clifford algebra defined by Pappacena of a monic (with respect to the first variable) homogeneous polynomial Φ(Z,variables over some field F . We completely determine its structure in the following cases: n = 2 and d = 3 and either char (F ) = 3, f 1 = 0 and f 2 (X 1 , X 2 ) = eX 1 X 2 for some e ∈ F , or char (F ) = 3, f 1 (X 1 , X 2 ) = rX 2 and f 2 (X 1 , X 2 ) = eX 1 X 2 + tX 2 2 for some r, t, e ∈ F . Except for a few exceptions, this algebra is an Azumaya algebra of rank nine whose center is the coordinate ring of an affine elliptic curve. We also discuss representations of arbitrary generalized Clifford algebras assuming the base field F is algebraically closed of characteristic zero.

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“…Therefore we can deal with the functions satisfying equation (23) independently of whether or not we know that they arose from a projective representation, forming a group of them, and introducing the equivalence described by equation (27). In dealing with multipliers, arising from a projective representation with n x n matrices,, we could divide each matrix by the nth root of its determinant.…”

Section: Conversely Any Complex Valued Functionmentioning

confidence: 99%

“…A much more exhaustive treatment of this realm was given by HarishChandra [23] and Rao [24], the former also applying similar techniques [25] to the closely related Kemmer matrices [26]. Although Dirac's use of anticommuting operators in 1928 to factor a quadratic form into linear factors suggests a similar technique to be used for other forms, it was not until 1954 that Heerema [27] published a comprehensive analysis of the use of hypercomplex numbers to factor a binary cubic. Beyond this, most attention seems to have centered on quadratic forms, Witt's paper having appeared in 1937, at the same time as another by Mordell [28].…”

Section: Introductionmentioning

confidence: 99%

Symmetry adapted functions belonging to the dirac groups

Deepak

Dulock

Thomas

et al. 1969

Int J of Quantum Chemistry

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AbstractsAn earlier analysis of the canonical form of a pair of invertible operators obeying the exchange rule A B = w B Ais extended to cover a set of operators, between each pair of which a relation of this type exists; and for which a power of each operator is the unit matrix. Such relations define a system which may be regarded as a generalization of the Dirac matrices of relativistic quantum mechanics. We concentrate upon the group theoretic aspects of such a system and its matrix representations. Applications arise from the fact that all projective representations of finite abelian groups take the form of a Dirac Group. In particular, the representations of the magnetic space groups, which are projective representations of the lattice groups, arise in this manner.On gtntralise une analyse anttrieure de la forme canonique d'une paire d'optrateurs ayant des inverses et satisfaisant la relation A B = w B AA un ensemble d'optrateurs satisfaisant a une relation de ce type et pour lesquels une certaine puissance de chaque operateur est l'optrateur d'identitt. Ces relations dtfinissent un systeme qui peut Stre regardt comme une gtntralisation des matrices de Dirac dans la mtcanique quantique relativiste. Nous nous inttressons surtout aux aspects se rapportant a la thtorie des groupes. Des applications proviennent du fait que toutes les reprksentations projectives des groupes abtliens finis prennent la forme d'un groupe de Dirac. En particulier les reprtsentations des groupes d'espace magnttiques qui sont des representations projectives des groupes de rtseau, surgissent de cette facon-ci.

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An algebra determined by a binary cubic form

Cited by 20 publications

References 0 publications

Polynomials with and without determinantal representations

Polynomials with and without determinantal representations

On the generalized Clifford algebra of a monic polynomial

Symmetry adapted functions belonging to the dirac groups

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