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Orthogonal sets of idempotents are used to design sets of unitary matrices, known as constellations, such that the modulus of the determinant of the difference of any two distinct elements is greater than $0$. It is shown that unitary…
The spatial symmetry of matter - including finite objects like molecules or atomic clusters, and extended objects like periodic or aperiodic crystals - is described using point groups and space groups. Magnetic point groups and space groups…
We define three combinatorial models for \hat{sl(n)} crystals, parametrized by partitions, configurations of beads on an `abacus', and cylindric plane partitions, respectively. These are reducible, but we can identify an irreducible…
The theory of spectral methods for partial differential equations leads to infinite-dimensional matrices which represent the derivative operator with respect to an underlying orthonormal basis. Favourable properties of such differentiation…
The basis of spinors in three-dimensional Euclidean space is expressed by differential forms. Its expression is found from the spectral decomposition of the modified Hamiltonian describing Weyl semimetals where the wavenumber parameters are…
We obtain some criteria for a symmetric square-central element of a totally decomposable algebra with orthogonal involution in characteristic two, to be contained in an invariant quaternion subalgebra.
For the quantum group $GL_{p,q}(2)$ and the corresponding quantum algebra $U_{p,q}(gl(2))$ Fronsdal and Galindo explicitly constructed the so-called universal $T$-matrix. In a previous paper we showed how this universal $T$-matrix can be…
Consider rectangular matrices over a local ring R. In the previous work we have obtained criteria for block-diagonalization of such matrices, i.e. U A V=A_1\oplus A_2, where U,V are invertible matrices over R. In this short note we extend…
The goals of this article are as follows: (1) To determine the irreducible components of the affine varieties parametrizing the representations of $ \Lambda $ with dimension vector d, where $ \Lambda $ traces a major class of finite…
We construct real and complex matrices in terms of Kronecker products of a Witt basis of 2n null vectors in the geometric algebra over the real and complex numbers. In this basis, every matrix is represented by a unique sum of products of…
Algorithmic methods for the explicit inversion of the indefinite double covering maps are proposed. These are based on either the Givens decomposition or the polar decomposition of the given matrix in the proper, indefinite orthogonal group…
We present a new interpretation for reduced density matrices of secondary variables in relativistic systems via an analysis of Wigner's method to construct the irreducible unitary representations of the Poincar\'e group. We argue that the…
Spin bases of relevance for quantum systems with cyclic symmetry as well as for quantum information and quantum computation are constructed from the theory of angular momentum. This approach is connected to the use of generalized Pauli…
Inspired by the Capelli identities for group determinants obtained by T\^oru Umeda, we give a basis of the center of the group algebra of any finite group by using Capelli identities for irreducible representations. The Capelli identities…
A group theoretical understanding of the two dimensional fractional supersymmetry is given in terms of the quantum Poincare group at roots of unity. The fractional supersymmetry algebra and the quantum group dual to it are presented and the…
Matrix generators for the general and special linear groups, the symplectic groups and the general and special unitary groups over finite fields. For the most part the generators have been obtained by translating Steinberg's generators for…
According to the Ringel-Green Theorem([G],[R1]), the generic composition algebra of the Hall algebra provides a realization of the positive part of the quantum group. Furthermore, its Drinfeld double can be identified with the whole quantum…
We study the modular representation theory of the symmetric and alternating groups. One of the most natural ways to label the irreducible representations of a given group or algebra in the modular case is to show the unitriangularity of the…
Mutually unbiased bases that can be cyclically generated by a single unitary operator are of special interest, since they can be readily implemented in practice. We show that, for a system of qubits, finding such a generator can be cast as…
We calculate for the first time all four-quark hadronic matrix elements of local operators possibly contributing to $K\to\pi\pi$ decays and in particular to the ratio $\epsilon'/\epsilon$ beyond the Standard Model (BSM). To this end we use…