Fischer provided a new type of binomial determinant for the number alternating sign matrices involving third root unity. In this paper we prove that her formula, when replacing unity by an indeterminate q, actually gives (q?1+2+q)-enumeration matrices. By evaluating generalisation are able to reprove conjecture Mills, Robbins and Rumsey stating Q-enumeration is product two polynomials in Q. Fur...

Finite Groups 20Dxx [1] A. Adem, J. F. Carlson, D. B. Karagueuzian, and R. James Milgram, The cohomology of the Sylow 2-subgroup of the Higman-Sims group, J. Pure Appl. Algebra 164 (2001), no. 3, 275–305. MR MR1857743 (2002g:20089) [2] Faryad Ali and Jamshid Moori, Fischer-Clifford matrices of the non-split group extension 2 · U4(2), Quaest. Math. 31 (2008), no. 1, 27–36. MR MR2404644 [3] Habib...

For a positive definite fundamental tensor all known examples of Osserman algebraic curvature tensors have a typical structure. They can be produced from a metric tensor and a finite set of skew-symmetric matrices which fulfil Clifford commutation relations. We show by means of Young symmetrizers and a theorem of S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins that every algebraic c...

We express Maxwell’s equations as a single equation, first using the divergence of a special type of matrix field to obtain the four current, and then the divergence of a special matrix to obtain the Electromagnetic field. These two equations give rise to a remarkable dual set of equations in which the operators become the matrices and the vectors become the fields. The decoupling of the equati...

In recent work of T. Cassidy and the author, a notion of complete intersection was defined for (non-commutative) regular skew polynomial rings, defining it using both algebraic and geometric tools, where the commutative definition is a special case. In this article, we extend the definition to a larger class of algebras that contains regular graded skew Clifford algebras, the coordinate ring of...

Finite Groups 20Dxx [1] A. Adem, J. F. Carlson, D. B. Karagueuzian, and R. James Milgram, The cohomology of the Sylow 2-subgroup of the Higman-Sims group, J. Pure Appl. Algebra 164 (2001), no. 3, 275–305. MR MR1857743 (2002g:20089) [2] Faryad Ali and Jamshid Moori, Fischer-Clifford matrices of the non-split group extension 2 · U4(2), Quaest. Math. 31 (2008), no. 1, 27–36. MR MR2404644 [3] Habib...

The ZX-calculus is a graphical calculus for reasoning about quantum systems and processes. It is known to be universal for pure state qubit quantum mechanics (QM), meaning any pure state, unitary operation and post-selected pure projective measurement can be expressed in the ZX-calculus. The calculus is also sound, i.e. any equality that can be derived graphically can also be derived using matr...

We deduce the periodicity 8 for the type of Pin and Spin representations of the orthogonal groups O(n) from simple combinatorial properties of the finite Clifford groups generated by the gamma matrices. We also include the case of arbitrary signature O(p, q). The changes in the type of representation can be seen as a rotation in the complex plane. The essential result is that adding a (+) dimen...

In  this paper we define the $p$-analog of the restericted reperesentations and also the $p$-analog of the Fourier--Stieltjes algebras on the inverse semigroups . We improve some results about Herz algebras on Clifford semigroups. At the end of this paper we give the necessary and sufficient condition for amenability of these algebras on Clifford semigroups.

We are going to give a systematic presentation of spinors in various spacetime dimensions which is a prerequisite for understanding the supersymmetry. (For general reviews see [1,2,3].) To set up the conventions, we will consider in what follows the Dirac matrices that satisfy the following Clifford algebra {γ, γ} = 2η1 , η = diag(+ · · ·+,− · · ·−) , (1) the sign +, − appear t, s times which a...