Some factorizations in the twisted group algebra of symmetric groups

In this paper we will give a similar factorization as in \cite{4}, \cite{5}, where the autors Svrtan and Meljanac examined certain matrix factorizations on Fock-like representation of a multiparametric quon algebra on the free associative algebra of noncommuting polynomials equiped with multiparametric partial derivatives. In order to replace these matrix factorizations (given from the right) by twisted algebra computation, we first consider the natural action of the symmetric group $S_{n}$ on the polynomial ring $R_{n}$ in $n^2$ commuting variables $X_{a\,b}$ and also introduce a twisted group algebra (defined by the action of $S_{n}$ on $R_{n}$) which we denote by ${\mathcal{A}(S_{n})}$. Here we consider some factorizations given from the left because they will be more suitable in calculating the constants (= the elements which are annihilated by all multiparametric partial derivatives) in the free algebra of noncommuting polynomials

A note on square-free factorizations

We analyze properties of various square-free factorizations in greatest common divisor domains (GCD-domains) and domains satisfying the ascending chain condition for principal ideals (ACCP-domains)

Isolated factorizations and their applications in simplicial affine semigroups

We introduce the concept of isolated factorizations of an element of a commutative monoid and study its properties. We give several bounds for the number of isolated factorizations of simplicial affine semigroups and numerical semigroups. We also generalize $\alpha$-rectangular numerical semigroups to the context of simplicial affine semigroups and study their isolated factorizations. As a consequence of our results, we characterize those complete intersection simplicial affine semigroups with only one Betti minimal element in several ways. Moreover, we define Betti sorted and Betti divisible simplicial affine semigroups and characterize them in terms of gluings and their minimal presentations. Finally, we determine all the Betti divisible numerical semigroups, which turn out to be those numerical semigroups that are free for any arrangement of their minimal generators

Noncommutative Symmetric Functions Associated with a Code, Lazard Elimination, and Witt Vectors

The construction of the universal ring of Witt vectors is related to Lazard's factorizations of free monoids by means of a noncommutative analogue. This is done by associating to a code a specialization of noncommutative symmetric functions

Transitive Hall sets

We give the definition of Lazard and Hall sets in the context of transitive factorizations of free monoids. The equivalence of the two properties is proved. This allows to build new effective bases of free partially commutative Lie algebras. The commutation graphs for which such sets exist are completely characterized and we explicit, in this context, the classical PBW rewriting process

Factorizations of Matrices Over Projective-free Rings

An element of a ring $R$ is called strongly $J^{\#}$-clean provided that it can be written as the sum of an idempotent and an element in $J^{\#}(R)$ that commute. We characterize, in this article, the strongly $J^{\#}$-cleanness of matrices over projective-free rings. These extend many known results on strongly clean matrices over commutative local rings