we survey some recent results on cocharacters of upper triangular matrices. in particular, we deal both with ordinary and graded cocharacter sequence; we list the principal combinatorial results; we show di erent tech-niques in order to solve similar problems.

Here we demonstrate the emergence of Grassmann variables in matrix models based on the exceptional Jordan algebra. The Grassmann algebras are built naturally using the octonion algebra. We argue the appearance of Grassmann variables solidifies the relationship between supersymmetry and triality.

We prove that the m-generated Grassmann algebra can be embedded into a 2 × 2 matrix algebra over a factor of a commutative polynomial algebra in m indeterminates. Cayley–Hamilton and standard identities for n× n matrices over the m-generated Grassmann algebra are derived from this embedding. Other related embedding results are also presented.

The present study introduces and investigates a new type of equation which is called Grassmann integral equation in analogy to integral equations studied in real analysis. A Grassmann integral equation is an equation which involves Grassmann (Berezin) integrations and which is to be obeyed by an unknown function over a (finite-dimensional) Grassmann algebra Gm (i.e., a sought after element of t...

A close relationship between the determinant, the pfaffian, and the Grassmann algebra is well-known. In this paper, a similar relation between the permanent, the hafnian, and a commutative analogue of the Grassmann algebra is described. Using the latter, some new properties of the hafnian are proved.

This paper presents some tools for least square computation in Grassmann-Cayley algebra, more specifically for elements expressed in homogeneous coordinates. We show that building objects with the outer product from k-vectors of same grade presents some properties that can be expressed in term of linear algebra and can be treated as a least square problem. This paper mainly focuses on line and ...

1. Historical Developments About 150 years ago, in 1844, the German high school teacher Hermann Grassmann published an ambitious work entitled The Linear Extension Theory, A New Branch of Mathematics. For Grassmann this was indeed The Branch of mathematics, which in his own words “far surpasses” all others. His subsequent work Geometric Algebra won the prize of 45 gold ducats set out by the Pri...