Let K be an algebraically closed field of characteristic 0, and let E be the infinite dimensional Grassmann (or exterior) algebra over K. Denote by Pn the vector space of the multilinear polynomials of degree n in x1, . . . , xn in the free associative algebra K(X). The symmetric group Sn acts on the left-hand side on Pn, thus turning it into an Sn-module. This fact, although simple, plays an i...

We study a natural generalization of the noncrossing relation between pairs of elements in [n] to k-tuples in [n] that was first considered by Petersen et al. [J. Algebra 324(5) (2010), 951–969]. We give an alternative approach to their result that the flag simplicial complex on ([n] k ) induced by this relation is a regular, unimodular and flag triangulation of the order polytope of the poset ...

In this paper we show that a quadratic lagrangian, with no constraints, containing ordinary time derivatives up to the order $m$ of $N$ dynamical variables, has $2mN$ symmetries consisting in translation variables solutions equations motion. We construct explicitly generators these transformations and prove they satisfy Heisenberg algebra. also analyse other specific cases which are not include...

Let g be a Lie algebra over a field of characteristic zero equipped with a vector space decomposition g = g ⊕ g+, and let s and t be commuting formal variables commuting with g. We prove that the map C : sg[[s, t]] × tg+[[s, t]] −→ sg[[s, t]] ⊕ tg+[[s, t]] defined by the Campbell-Baker-Hausdorff formula and given by e − e + = eC(sg ,tg) for g ∈ g[[s, t]] is a bijection, as is well known when g ...

We show that the Kazhdan-Lusztig basis elements Cw of the Hecke algebra of the symmetric group, when w ∈ Sn corresponds to a Schubert subvariety of a Grassmann variety, can be written as a product of factors of the form Ti + fj(v), where fj are rational functions.

We present a simple and new method of constructing superdistributions on superspace over a Grassmann-Banach algebra, which close to the de Rham’s “currents” defined as dual objects to differential forms. The paper also contains the extension of the Hörmander’s description of the singularity structure (wavefront set) of a distribution to include the supersymmetric case. PACS numbers: 11.10.-z, 1...

We present a new treament of 2-spinors and twistors, using the spacetime algebra. The key rôle of bilinear covariants is emphasized. As a by-product, an explicit representation is found, composed entirely of real spacetime vectors, for the Grassmann entities of supersymmetric field theory.

We introduce Z3-graded objects which are the generalization of the more familiar Z2-graded objects that are used in supersymmetric theories and in many models of non-commutative geometry. First, we introduce the Z3graded Grassmann algebra, and we use this object to construct the Z3matrices, which are the generalizations of the supermatrices. Then, we generalize the concepts of supertrace and su...

This paper integrates statistical reasoning and Grassmann-Cayley algebra for making 2D and 3D geometric reasoning practical. The multi-linearity of the forms allows rigorous error propagation and statistical testing of geometric relations. This is achieved by representing all objects in homogeneous coordinates and expressing all relations using standard matrix calculus. 1 Motivation Many Comput...