Kravchuk Polynomials and Induced/Reduced Operators on Clifford Algebras

Kravchuk polynomials arise as orthogonal polynomials with respect to the binomial distribution and have numerous applications in harmonic analysis, statistics, coding theory, and quantum probability. The relationship between Kravchuk polynomials and Clifford algebras is multifaceted. In this paper, Kravchuk polynomials are discovered as traces of conjugation operators in Clifford algebras, and appear in Clifford Berezin integrals of Clifford polynomials. Regarding Kravchuk matrices as linear operators on a vector space V, the action induced on the Clifford algebra over V is equivalent to blade conjugation, i.e., reflections across sets of orthogonal hyperplanes. Such operators also have a natural interpretation in terms of raising and lowering operators on the algebra. On the other hand, beginning with particular linear operators on the Clifford algebra ClQ(V), one obtains Kravchuk matrices as operators on the paravector space V* through a process of operator grade-reduction. Symmetric Kravchuk matrices are recovered as representations of grade-reductions of maps induced by negative-definite quadratic forms on V

On Representations of Semigroups Having Hypercube-like Cayley Graphs

The $n-dimensional hypercube, or n-cube, is the Cayley graph of the Abelian group Z2n. A number of combinatorially-interesting groups and semigroups arise from modified hypercubes. The inherent combinatorial properties of these groups and semigroups make them useful in a number of contexts, including coding theory, graph theory, stochastic processes, and even quantum mechanics. In this paper, particular groups and semigroups whose Cayley graphs are generalizations of hypercubes are described, and their irreducible representations are characterized. Constructions of faithful representations are also presented for each semigroup. The associated semigroup algebras are realized within the context of Clifford algebras

Clifford Algebra Decompositions of Conformal Orthogonal Group Elements

Beginning with a finite-dimensional vector space V equipped with a nondegenerate quadratic form Q, we consider the decompositions of elements of the conformal orthogonal group COQ(V), defined as the direct product of the orthogonal group OQ(V) with dilations. Utilizing the correspondence between conformal orthogonal group elements and ``decomposable\u27\u27 elements of the associated Clifford algebra, ClQ(V), a decomposition algorithm is developed. Preliminary results on complexity reductions that can be realized passing from additive to multiplicative representations of invertible elements are also presented with examples. The approach here is based on group actions in the conformal orthogonal group. Algorithms are implemented in Mathematica using the CliffMath package

Zeon Roots

Zeon algebras can be thought of as commutative analogues of fermion algebras, and they can be constructed as subalgebras within Clifford algebras of appropriate signature. Their inherent combinatorial properties make them useful for applications in graph enumeration problems and evaluating functions defined on partitions. In this paper, kth roots of invertible zeon elements are considered. More specifically, conditions for existence of roots are established, numbers of existing roots are determined, and computational methods for constructing roots are developed. Recursive and closed formulas are presented, and specific low-dimensional examples are computed with Mathematica. Interestingly, Stirling numbers of the first kind appear among coefficients in the closed formulas

On the complexity of cycle enumeration using Zeons

Nilpotent adjacency matrix methods are employed to enumerate $k$-cycles in simple graphs on $n$ vertices for any $k\le n$. The worst-case time complexity of counting $k$-cycles in an $n$-vertex simple graph is shown to be $\mathcal{O}(n^{\alpha+1} 2^{n})$, where $\alpha\le 3$ is the exponent representing the complexity of matrix multiplication. When $k$ is fixed, the enumeration of all $k$-cycles in an $n$-vertex graph is of time complexity $\mathcal{O}(n^{\alpha+k-1})$. Letting $\Omega=\binom{n}{2}$, the average-case time complexity of counting $k$-cycles in an $n$-vertex, $e$-edge graph where $e\le \displaystyle q\left(\frac{\Omega}{k}-1\right)$ for fixed $0<q<1$ is found to be $\mathcal{O}(n^{4} (1+q)^{n})$. The storage complexity of the approach detailed herein is $\mathcal{O}(n2 2^{n})$. For reference, experimental results detailing computation times (in seconds) are included alongside similar computations performed with algorithms based on the approaches of Bax and Tarjan

ZEONS, LATTICES OF PARTITIONS, AND FREE PROBABILITY

International audienceCentral to the theory of free probability is the notion of summing multiplicative functionals on the lattice of non-crossing partitions. In this paper, a graph-theoretic perspective of partitions is investigated in which independent sets in graphs correspond to non-crossing partitions. By associating particular graphs with elements of “zeon” algebras (commutative subalgebras of fermion algebras), multiplicative functions can be summed over segments of lattices of partitions by employing methods of “zeon-Berezin” operator calculus. In particular, properties of the algebra are used to “sieve out” the appropriate segments and sub-lattices. The work concludes with an application to joint moments of quantum random variables

Nilpotent adjacency matrices, random graphs, and quantum random variables

International audienceFor fixed $n>0$, the space of finite graphs on $n$ vertices is canonically associated with an abelian, nilpotent-generated subalgebra of the $2n$-particle fermion algebra. using the generators of the subalgebra, an algebraic probability space of "nilpotent adjacency matrices" associated with finite graphs is defined. Each nilpotent adjacency matrix is a quantum random variable whose $m^th$ moment corresponds to the number of $m$-cycles in the graph $G$. Each matrix admits a canonical "quantum decomposition" into a sum of three algebraic random variables: $a = a^\Delta+ a^\Upsilon+a^Lambda$, where $a^\Delta$ is classical while $a^\Upsilon and$a^\Lambda$are quantum. Moreover, within the algebraic context, the NP problem of cycle enumeration is reduced to matrix multiplication, requiring no more than$n^4$ multiplications within the algebra

Cycles and components in geometric graphs: adjacency operator approach

Nilpotent and idempotent adjacency operator methods are applied to the study of random geometric graphs in a discretized, $d$-dimensional unit cube$[0; 1]$^d. Cycles are enumerated, sizes of maximal connected compo- nents are computed, and closed formulas are obtained for graph circumfer- ence and girth. Expected numbers of $k$-cycles, expected sizes of maximal components, and expected circumference and girth are also computed by considering powers of adjacency operators

Dynamic random walks on Clifford algebras

Multiplicative random walks with dynamic transitions are defined on Clifford algebras of arbitrary signature. These multiplicative walks are then summed to induce additive walks on the algebra. Properties of both types of walks are considered, and limit theorems are developed