The Carroll algebra is constructed as the $c\to0$ limit of Poincare and associated to symmetries on generic null surfaces. In this paper, we begin investigations Carrollian fermions or defined Due availability two different (degenerate) metrics spacetimes, there possibility versions Clifford algebras. We consider both possibilities construct explicit representations gamma matrices show how buil...

This paper briefly reviews the conventional method of obtaining the canonical form of an antisymmetric (skewsymmetric, alternating) matrix. Conventionally a vector space over the complex field has to be introduced. After a short introduction to the universal mathematical “language” Geometric Calculus, its fundamentals, i.e. its “grammar” Geometric Algebra (Clifford Algebra) is explained. This l...

Square Complex Orthogonal Space Time Block Codes (COSTBC) with no zero entries are known for 2,4 and 8 transmit antennas. But, for 16 transmit antennas COSTBCs with no zero entries have not been reported. Zero entries in the design lead to the problem of switching off of transmit antenna, imposing severe constraints on hardware implementation. The zeros also result in the increase of the Peak t...

This book aims to introduce the reader to the geometry of surfaces and submanifolds in the conformal n-sphere. Various models for Möbius geometry are presented: the classical projective model, the quaternionic approach, and an approach that uses the Clifford algebra of the space of homogeneous coordinates of the classical model — the use of 2-by-2 matrices in this context is elaborated. For eac...

As a natural generalisation of q-Schur algebras associated with the Hecke algebra Hr,R (of the symmetric group), we introduce the Q-q-Schur superalgebra associated with the Hecke-Clifford superalgebra H r,R, which, by definition, is the endomorphism algebra of the induced H r,R-module from certain q-permutation modules over Hr,R. We will describe certain integral bases for these superalgebras i...

We study the algebra of complex polynomials which remain invariant under the action of the local Clifford group under conjugation. Within this algebra, we consider the linear spaces of homogeneous polynomials degree by degree and construct bases for these vector spaces for each degree, thereby obtaining a generating set of polynomial invariants. Our approach is based on the description of Cliff...

Matsumoto and Amano (2008) showed that every single-qubit Clifford+T operator can be uniquely written of a particular form, which we call the Matsumoto-Amano normal form. In this mostly expository paper, we give a detailed and streamlined presentation of Matsumoto and Amano’s results, simplifying some proofs along the way. We also point out some corollaries to Matsumoto and Amano’s work, includ...

After a brief discussion of the computational complexity of Clifford algebras, we present a new basis for even Clifford algebra Cl 2m that simplifies greatly the actual calculations and, without resorting to the conventional matrix isomorphism formulation, obtains the same complexity. In the last part we apply these results to the Clifford algebra formulation of the NP-complete problem of the m...