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It is well known that over an infinite field the ring of symmetric functions in a finite number of variables is isomorphic to the one of polynomial functions on matrices that are invariants by the action of conjugation by general linear…

For a mixing shift of finite type, the associated automorphism group has a rich algebraic structure, and yet we have few criteria to distinguish when two such groups are isomorphic. We introduce a stabilization of the automorphism group,…

We give a new explicit construction for the simplicial group $K(A,n)$. We explain the topological interpretation and discuss some possible applications.

Let A and B be finite dimensional simple real algebras with division gradings by an abelian group G. In this paper we give necessary and sufficient conditions for the coincidence of the graded identities of A and B. We also prove that every…

We determine all finite subgroups of simple algebraic groups that have irreducible centralizers - that is, centralizers whose connected component does not lie in a parabolic subgroup.

First we prove that any inner automorphism in the stabilizer of a graded-simple unital associative algebra whose grading group is abelian is the conjugation by a homogeneous element. Now consider a grading by an abelian group on an…

We construct two combinatorially equivalent line arrangements in the complex projective plane such that the fundamental groups of their complements are not isomorphic. The proof uses a new invariant of the fundamental group of the…

A derived version of Maschke's theorem for finite groups is proved: the derived categories, bounded or unbounded, of all blocks of the group algebra of a finite group are simple, in the sense that they admit no nontrivial recollements. This…

Frucht showed that, for any finite group $G$, there exists a cubic graph such that its automorphism group is isomorphic to $G$. For groups generated by two elements we simplify his construction to a graph with fewer nodes. In the general…

We study the codegree isomorphism problem for finite simple groups. In particular, we show that such a group is determined by the codegrees (counting multiplicity) of its irreducible characters. The proof is uniform for all simple groups…

Let $KG$ denote the group algebra of the group $G$ over the field $K$ and let $U(KG)$ denote its group of units. Here without the use of a computer we give presentations for the unit groups of all group algebras $KG$, where the size of $KG$…

The article studies power complexes and generalized power complexes, and investigates the algebraic structure of their automorphism groups. The combinatorial incidence structures involved are cube-like, in the sense that they have many…

This paper extends the study of group algebras of finite groups in which the socle of the center is an ideal. We provide a detailed analysis of the structure of these groups. In a particular case, we reach a complete characterization of the…

Passing from arithmetic schemes to algebraic schemes, in a similar manner we will have the computation of the \'etale fundamental group of an algebraic scheme and then will define and discuss the qc fundamental group of an algebraic scheme…

Let $M$ be an irreducible smooth projective variety, defined over an algebraically closed field, equipped with an action of a connected reductive affine algebraic group $G$, and let ${\mathcal L}$ be a $G$--equivariant very ample line…

According to the classical theorem, every irreducible algebraic variety endowed with a nontrivial rational action of a connected linear algebraic group is birationally isomorphic to a product of another algebraic variety and ${\bf P}^s$…

In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.

We study fundamental groups of projective varieties with normal crossing singularities and of germs of complex singularities. We prove that for every finitely-presented group G there is a complex projective surface S with simple normal…

We adapt the abstract concepts of abelianness and centrality of universal algebra to the context of inverse semigroups. We characterize abelian and central congruences in terms of the corresponding congruence pairs. We relate centrality to…

We give a new proof that compact infra-solvmanifolds with isomorphic fundamental groups are smoothly diffeomorphic. More generally, we prove rigidity results for manifolds which are constructed using affine actions of virtually polycyclic…