On the Concept of Representations of Prime Numbers and Prime Products

The author used to think that the only representation of prime numbers, etc. was the prime number double helix representation. However, in 2011, the author published a paper which puzzled him and it eventually dawned upon him that there was more than one representation of prime numbers. The second representation will be referred to as the hyperbolic representation. The third representation will be called the parabolic representation and is related to the hyperbolic representation. The fourth representation is the triangular representation and is related to the hyperbolic representation. Both of the primary representations, the double helix and the hyperbolic both reject that 2 and 3 are prime numbers. The hyperbolic representation is shown to be related to Lorentz-like transformations.

The First Triangular Representation of The Symmetric Groups over a field K of characteristic p=0

In this paper we will study the new type of triangular representations of the symmetric groups which is called the first triangular representations of the symmetric groups over a field K of characteristic p=0.

The First Triangular Representation of The Symmetric Groups over a field K of characteristic pdivides (n-2)

In this paper we will study the new type of triangular representations of the symmetric groups which is called the first triangular representations of the symmetric groups over a field K of characteristic pdivides(n-2).

FINITE POSETS AND THEIR REPRESENTATION ALGEBRAS

A finite poset (P ≼ S) determines a finite dimensional algebra TP over the field 𝔽 of two elements, with an upper triangular representation. We determine the structure of the radical of the representation algebra A of the monoid (TP,·) over a field of characteristic different from 2. We also consider degenerations of A over a complete discrete valuation ring with residue field of characteristic 2.

CONSTRUCTING FULL BLOCK TRIANGULAR REPRESENTATIONS OF ALGEBRAS

Every finite dimensional representation of an algebra is equivalent to a finite direct sum of indecomposable representations. Hence, the classification of indecomposable representations of algebras is a relevant (and usually complicated) task. In this note we study the existence of full block triangular representations, an interesting example of indecomposable representations, from a computational perspective. We describe an algorithm for determining whether or not an associative finitely presented k-algebra R has a full block triangular representation over [Formula: see text].

ON THE KROHN–RHODES COMPLEXITY OF SEMIGROUPS OF UPPER TRIANGULAR MATRICES

We consider the Krohn–Rhodes complexity of certain semigroups of upper triangular matrices over finite fields. We show that for any n > 1 and finite field k, the semigroups of all n × n upper triangular matrices over k and of all n × n unitriangular matrices over k have complexity n - 1. A consequence is that the complexity c > 1 of a finite semigroup places a lower bound of c + 1 on the dimension of any faithful triangular representation of that semigroup over a finite field.