arXiv subjects

Shayne Waldron

Publications and source records attributed to Shayne Waldron.

At least 19 recordsLinked to original sources

arXiv2026

The lattice of normal reflection subgroups of an irreducible reflection group

The reflection subgroups of a reflection group have a natural lattice structure given by the reflections that they contain. By considering the conjugation action orbits of the reflection subgroups for a given root line, we are able to give an essentially combinatorial way to calculate the lattice of all the normal reflection subgroups of a given (finite irreducible) reflection group, and natural generators for them. Moreover, we observe that every complex reflection group is a normal subgroup of the unique maximal reflection group which shares its collineation group. Hence, we are able to present the Shephard-Todd classification of the complex reflection groups as a collection of maximal reflection groups, together with appropriate (collineation preserving) normal reflection subgroups. We investigate the quotients by the normal reflection subgroups, which are known to be reflection groups. We also consider the action of the collineation group on some appropriate small systems of lines, and how these results extend to quaternionic reflection groups. Some novel techniques are introduced, including the notion of a "hidden reflection", a combinatorial-geometric description of the reflection subgroups and the size of their conjugacy class, and the role played by the abelianisation of the reflection group.

math.GR↗

arXiv2026

The quaternionic systems of imprimitivity for the reflection groups of rank two

Given an explicit presentation of a reflection group of rank two (or any rank two group for that matter), we give a simple procedure for calculating all its systems of imprimitivity, when viewed as a matrix group over the quaternions. This is applied to all the reflection groups, in particular the quaternionic reflection groups, thereby unifying a number of results and ideas in the literature. For example, a primitive complex reflection group of rank two has either uncountably many quaternionic systems of imprimitivity (3 cases) or none (16 cases).

math.GR↗

arXiv2025

Real and complex spherical designs and their Gramian

If a (weighted) spherical design is defined as an integration (cubature) rule for a unitarily invariant space P of polynomials (on the sphere), then any unitary image of it is also such a spherical design. It therefore follows that such spherical designs are determined by their Gramian (Gram matrix). We outline a general method to obtain such a characterisation as the minima of a function of the Gramian, which we call a potential. This characterisation can be used for the numerical and analytic construction of spherical designs. When the space P of polynomials is not irreducible under the action of the unitary group, then the potential is not unique. In several cases of interest, e.g., spherical t-designs and half-designs, we use this flexibility to provide potentials with a very simple form. We then use our results to develop certain aspects of the theory of real and complex spherical designs for unitarily invariant polynomial spaces.

math.GM↗

arXiv2025

An elementary classification of the quaternionic reflection groups of rank two

We give an elementary classification and presentation of the finite quaternionic reflection groups of rank two, based on the notion of a``reflection system''. This simplifies the existing classification, which is shown to be incomplete, e.g., there exist four imprimitive quaternionic reflection groups of order 192 with 22 reflections which are not isomorphic (one of which was previously unknown).

math.GR↗

arXiv2025

Quaternionic MUBs in H^2 and their reflection symmetries

We consider the primitive quaternionic reflection groups of type P for H^2 that are obtained from Blichfeldt's collineation groups for C^4.These are seen to be intimately related to the maximal set of five quaternionic mutually unbiased bases (MUBs) in H2 , for which they are symmetries. From these groups, we construct other interesting sets of lines that they fix, including a new quaternionic spherical 3-design of 16 lines in H^2 with angles {1/5,3/5}, which meets the special bound. Some interesting consequences of this investigation include finding imprimitive quaternionic reflection groups with several systems of imprimitivity, and finding a nontrivial reducible subgroup which has a continuous family of eigenvectors.

math.RT↗

arXiv2024

The geometry of the six quaternionic equiangular lines in $\mathbb{H}^2$

We give a simple presentation of the six quaternionic equiangular lines in $\mathbb{H}^2$ as an orbit of the primitive quaternionic reflection group of order 720 (which is isomorphic to 2.A_6 the double cover of $A_6)$. Other orbits of this group are also seen to give optimal spherical designs (packings) of 10, 15 and 20 lines in $\mathbb{H}^2$, with angles { 1/3, 2/3 }, { 1/4, 5/8 } and { 0, 1/3, 2/3 }, respectively. We consider the origins of this reflection group as one of Blichfeldt's "finite collineation groups" for lines in $\mathbb{C}^4$, and general methods for finding nice systems of quaternionic lines.

math.GT↗

arXiv2024

Equations for the overlaps of a SIC

We give a holomorphic quartic polynomial in the overlap variables whose zeros on the torus are precisely the Weyl-Heisenberg SICs (symmetric informationally complete positive operator valued measures). By way of comparison, all the other known systems of equations that determine a Weyl-Heisenberg SIC involve variables and their complex conjugates. We also give a related interesting result about the powers of the projective Fourier transform of the group G = Z d x Z d .

cs.IT↗

arXiv2024

Putatively optimal projective spherical designs with little apparent symmetry

We give some new explicit examples of putatively optimal projective spherical designs. i.e., ones for which there is numerical evidence that they are of minimal size. These form continuous families, and so have little apparent symmetry in general, which requires the introduction of new techniques for their construction. New examples of interest include an 11-point spherical (3, 3)-design for R 3 , and a 12-point spherical (2, 2)-design for R 4 given by four Mercedes-Benz frames that lie on equi-isoclinic planes. We also give results of an extensive numerical study to determine the nature of the real algebraic variety of optimal projective real spherical designs, and in particular when it is a single point (a unique design) or corresponds to an infinite family of designs.

math.CO↗

arXiv2024

An explicit construction of the unitarily invariant quaternionic polynomial spaces on the sphere

The decomposition of the polynomials on the quaternionic unit sphere in $\Hd$ into irreducible modules under the action of the quaternionic unitary (symplectic) group and quaternionic scalar multiplication has been studied by several authors. Typically, these abstract decompositions into ``quaternionic spherical harmonics'' specify the irreducible representations involved and their multiplicities. The elementary constructive approach taken here gives an orthogonal direct sum of irreducibles, which can be described by some low-dimensional subspaces, to which commuting linear operators $L$ and $R$ are applied. These operators map harmonic polynomials to harmonic polynomials, and zonal polynomials to zonal polynomials. We give explicit formulas for the relevant ``zonal polynomials'' and describe the symmetries, dimensions, and ``complexity'' of the subspaces involved. Possible applications include the construction and analysis of desirable sets of points in quaternionic space, such as equiangular lines, lattices and spherical designs (cubature rules).

math.RT↗

arXiv2023

Complex spherical designs from group orbits

We consider the general question of when all orbits under the unitary action of a finite group give a complex spherical design. Those orbits which have large stabilisers are then good candidates for being optimal complex spherical designs. This is done by developing the general theory of complex designs and associated (harmonic) Molien series for group actions. As an application, we give explicit constructions of some putatively optimal real and complex spherical t-designs.

math.CO↗

arXiv2023

On Waldron Interpolation on a Simplex in $\mathbb{R}^d$

We introduce explicit families of good interpolation points for interpolation on a triangle in $\mathbb{R}^2$ that may be used for either polynomial interpolation or a certain rational interpolation for which we give explicit formulas.

math.NA↗

arXiv2021

Testing isomorphism between tuples of subspaces

Given two tuples of subspaces, can you tell whether the tuples are isomorphic? We develop theory and algorithms to address this fundamental question. We focus on isomorphisms in which the ambient vector space is acted on by either a unitary group or general linear group. If isomorphism also allows permutations of the subspaces, then the problem is at least as hard as graph isomorphism. Otherwise, we provide a variety of polynomial-time algorithms with Matlab implementations to test for isomorphism. Keywords: subspace isomorphism, Grassmannian, Bargmann invariants, $H^\ast$-algebras, quivers, graph isomorphism

math.MG↗

arXiv2020

A variational characterisation of projective spherical designs over the quaternions

We give an inequality on the packing of vectors/lines in quaternionic Hilbert space $\Hd$, which generalises those of Sidelnikov and Welch for unit vectors in $\Rd$ and $\Cd$. This has a parameter $t$, and depends only on the vectors up to projective unitary equivalence. The sequences of vectors in ${\mathbb{F}}^d={\mathbb{R}}^d,{\mathbb{C}}^d,{\mathbb{H}}^d$ that give equality, which we call spherical $(t,t)$-designs, are seen to satisfy a cubature rule on the unit sphere in ${\mathbb{F}}^d$ for a suitable polynomial space $\Hom_{\Fd}(t,t)$. Using this, we show that the projective spherical $t$-designs on the Delsarte spaces $\FF P^{d-1}$ coincide with the spherical $(t,t)$-designs of unit vectors in ${\mathbb{F}}^d$. We then explore a number of examples in quaternionic space. The unitarily invariant polynomial space ${\mathop{\rm Hom}\nolimits}_{\mathbb{H}^d}(t,t)$ and the inner product that we define on it so the reproducing kernel has a simple form are of independent interest.

cs.IT↗

arXiv2020

Multivariate Lagrange interpolation and polynomials of one quaternionic variable

This paper considers the extension of classical Lagrange interpolation in one real or complex variable to "polynomials of one quaternionic variable". To do this we develop some aspects of the theory of such polynomials. We then give a number of related multivariate polynomial interpolation schemes for ${\mathbb{R}}^4$ and ${\mathbb{C}}^2$ with good geometric properties, and some aspects of least interpolation and of Kergin interpolation.

math.CA↗

arXiv2020

Tight frames over the quaternions and equiangular lines

We show that much of the theory of finite tight frames can be generalised to vector spaces over the quaternions. This includes the variational characterisation, group frames, and the characterisations of projective and unitary equivalence. We are particularly interested in sets of equiangular lines (equi-isoclinic subspaces) and the groups associated with them, and how to move them between the spaces $\Rd$, $\Cd$ and $\Hd$. We discuss what the analogue of Zauner's conjecture for equiangular lines in $\Hd$ might be.

math.FA↗

arXiv2019

Constructing high order spherical designs as a union of two of lower order

We show how the variational characterisation of spherical designs can be used to take a union of spherical designs to obtain a spherical design of higher order (degree, precision, exactness) with a small number of points. The examples that we consider involve taking the orbits of two vectors under the action of a complex reflection group to obtain a weighted spherical $(t,t)$-design. These designs have a high degree of symmetry (compared to the number of points), and many are the first known construction of such a design, e.g., a $32$ point $(9,9)$-design for $\mathbb{C}^2$, a $48$ point $(4,4)$-design for $\mathbb{C}^3$, and a $400$ point $(5,5)$-design for $\mathbb{C}^4$.From a real reflection group, we construct a $360$ point $(9,9)$-design for $\mathbb{R}^4$ (spherical half-design of order $18$), i.e., a $720$ point spherical $19$-design for $\mathbb{R}^4$.

math.MG↗

arXiv2018

The Fourier transform of a projective group frame

Many tight frames of interest are constructed via their Gramian matrix (which determines the frame up to unitary equivalence). Given such a Gramian, it can be determined whether or not the tight frame is projective group frame, i.e., is the projective orbit of some group $G$ (which may not be unique). On the other hand, there is complete description of the projective group frames in terms of the irreducible projective representations of $G$. Here we consider the inverse problem of taking the Gramian of a projective group frame for a group $G$, and identifying the cocycle and constructing the frame explicitly as the projective group orbit of a vector $v$ (decomposed in terms of the irreducibles). The key idea is to recognise that the Gramian is a group matrix given by a vector $f\in\mathbb{C}^G$, and to take the Fourier transform of $f$ to obtain the components of $v$ as orthogonal projections. This requires the development of a theory of group matrices and the Fourier transform for projective representations. Of particular interest, we give a block diagonalisation of (projective) group matrices. This leads to a unique Fourier decomposition of the group matrices, and a further fine-scale decomposition into low rank group matrices.

math.RT↗

arXiv2018

Constructing exact symmetric informationally complete measurements from numerical solutions

Recently, several intriguing conjectures have been proposed connecting symmetric informationally complete quantum measurements (SIC POVMs, or SICs) and algebraic number theory. These conjectures relate the SICs and their minimal defining algebraic number field. Testing or sharpening these conjectures requires that the SICs are expressed exactly, rather than as numerical approximations. While many exact solutions of SICs have been constructed previously using Gröbner bases, this method has probably been taken as far as is possible with current computer technology (except in special cases where there are additional symmetries). Here we describe a method for converting high-precision numerical solutions into exact ones using an integer relation algorithm in conjunction with the Galois symmetries of a SIC. Using this method we have calculated 69 new exact solutions, including 9 new dimensions where previously only numerical solutions were known, which more than triples the number of known exact solutions. In some cases the solutions require number fields with degrees as high as 12,288. We use these solutions to confirm that they obey the number-theoretic conjectures and we address two questions suggested by the previous work.

quant-ph↗