The monomial representations of the Clifford group

We show that the Clifford group - the normaliser of the Weyl-Heisenberg group - can be represented by monomial phase-permutation matrices if and only if the dimension is a square number. This simplifies expressions for SIC vectors, and has other applications to SICs and to Mutually Unbiased Bases. Exact solutions for SICs in dimension 16 are presented for the first time.Comment: Additional author and exact solutions to the SIC problem in dimension 16 adde

Twin bent functions, strongly regular Cayley graphs, and Hurwitz-Radon theory

The real monomial representations of Clifford algebras give rise to two sequences of bent functions. For each of these sequences, the corresponding Cayley graphs are strongly regular graphs, and the corresponding sequences of strongly regular graph parameters coincide. Even so, the corresponding graphs in the two sequences are not isomorphic, except in the first 3 cases. The proof of this non-isomorphism is a simple consequence of a theorem of Radon.Comment: 13 pages. Addressed one reviewer's questions in the Discussion section, including more references. Resubmitted to JACODES Math, with updated affiliation (I am now an Honorary Fellow of the University of Melbourne

Spin Hecke algebras of finite and affine types

We introduce the spin Hecke algebra, which is a q-deformation of the spin symmetric group algebra, and its affine generalization. We establish an algebra isomorphism which relates our spin (affine) Hecke algebras to the (affine) Hecke-Clifford algebras of Olshanski and Jones-Nazarov. Relation between the spin (affine) Hecke algebra and a nonstandard presentation of the usual (affine) Hecke algebra is displayed, and the notion of covering (affine) Hecke algebra is introduced to provide a link between these algebras. Various algebraic structures for the spin (affine) Hecke algebra are established.Comment: 24 pages, to appear in Adv. in Mat

SIC~POVMs and Clifford groups in prime dimensions

We show that in prime dimensions not equal to three, each group covariant symmetric informationally complete positive operator valued measure (SIC~POVM) is covariant with respect to a unique Heisenberg--Weyl (HW) group. Moreover, the symmetry group of the SIC~POVM is a subgroup of the Clifford group. Hence, two SIC~POVMs covariant with respect to the HW group are unitarily or antiunitarily equivalent if and only if they are on the same orbit of the extended Clifford group. In dimension three, each group covariant SIC~POVM may be covariant with respect to three or nine HW groups, and the symmetry group of the SIC~POVM is a subgroup of at least one of the Clifford groups of these HW groups respectively. There may exist two or three orbits of equivalent SIC~POVMs for each group covariant SIC~POVM, depending on the order of its symmetry group. We then establish a complete equivalence relation among group covariant SIC~POVMs in dimension three, and classify inequivalent ones according to the geometric phases associated with fiducial vectors. Finally, we uncover additional SIC~POVMs by regrouping of the fiducial vectors from different SIC~POVMs which may or may not be on the same orbit of the extended Clifford group.Comment: 30 pages, 1 figure, section 4 revised and extended, published in J. Phys. A: Math. Theor. 43, 305305 (2010

Hecke-Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type

We introduce an odd double affine Hecke algebra (DaHa) generated by a classical Weyl group W and two skew-polynomial subalgebras of anticommuting generators. This algebra is shown to be Morita equivalent to another new DaHa which are generated by W and two polynomial-Clifford subalgebras. There is yet a third algebra containing a spin Weyl group algebra which is Morita (super)equivalent to the above two algebras. We establish the PBW properties and construct Verma-type representations via Dunkl operators for these algebras

New solutions to the Hurwitz problem on square identities

The Hurwitz problem of composition of quadratic forms, or of "sum of squares identity" is tackled with the help of a particular class of $(\mathbb{Z}_2)^n$-graded non-associative algebras generalizing the octonions. This method provides an explicit formula for the classical Hurwitz-Radon identity and leads to new solutions in a neighborhood of the Hurwitz-Radon identity.Comment: 13 pages, 2 figures, final version to appear in J. Pure Appl. Al

Grobner Bases for Finite-temperature Quantum Computing and their Complexity

Following the recent approach of using order domains to construct Grobner bases from general projective varieties, we examine the parity and time-reversal arguments relating de Witt and Lyman's assertion that all path weights associated with homotopy in dimensions d <= 2 form a faithful representation of the fundamental group of a quantum system. We then show how the most general polynomial ring obtained for a fermionic quantum system does not, in fact, admit a faithful representation, and so give a general prescription for calcluating Grobner bases for finite temperature many-body quantum system and show that their complexity class is BQP