Related papers: Free functions with symmetry

A fundamental result in the study of graph homomorphisms is Lov\'asz's theorem that two graphs are isomorphic if and only if they admit the same number of homomorphisms from every graph. A line of work extending Lov\'asz's result to more…

Since the end of the XIXth century, we know that each birational map of the complex projective plane is the product of a finite number of quadratic birational maps of the projective plane; this motivates our work which essentially deals…

We prove that the subset of quasisymmetric polynomials conjectured by Bergeron and Reutenauer to be a basis for the coinvariant space of quasisymmetric polynomials is indeed a basis. This provides the first constructive proof of the…

We study Hilbert's fourteenth problem from a geometric point of view. Nagata's celebrated counterexample demonstrates that for an arbitrary group action on a variety the ring of invariant functions need not be isomorphic to the ring of…

Let $L_{n}$ be the free Lie algebra, $F_{n}$ be the free metabelian Lie algebra, and $L_{n,c}$ be the free metabelian nilpotent of class $c$ Lie algebra of rank $n$ generated by $x_1,\ldots,x_n$ over a field $K$ of characteristic zero. We…

In 2006 Sommers and Tymoczko defined so called arrangements of ideal type A_I stemming from ideals I in the set of positive roots of a reduced root system. They showed in a case by case argument that A_I is free if the root system is of…

We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several…

We define a new family of noncommutative Bell polynomials in the algebra of free quasi-symmetric functions and relate it to the dual immaculate basis of quasi-symmetric functions. We obtain noncommutative versions of Grinberg's results…

The number of square-free integers in $x$ consecutive values of any polynomial $f$ is conjectured to be $c_fx$, where the constant $c_f$ depends only on the polynomial $f$. This has been proven for degrees less or equal to 3. Granville was…

We introduce analogues of the Hopf algebra of Free quasi-symmetric functions with bases labelled by colored permutations. As applications, we recover in a simple way the descent algebras associated with wreath products $\Gamma\wr\SG_n$ and…

Let GF(q)[x,y] be the polynomial algebra in two variables over the finite field GF(q) with q elements. We give an exact formula and the asymptotics for the number p(n) of automorphisms (f,g) of GF(q)[x,y] such that max{deg(f),deg(g)}=n. We…

This chapter is a tutorial on techniques and results in free convex algebraic geometry and free real algebraic geometry (RAG). The term free refers to the central role played by algebras of noncommutative polynomials R<x> in free (freely…

This paper develops from scratch a theory of Galois rings and orders over arbitrary fields. Our approach is different from others in the literature in that there is no non-modularity assumption. We prove, when the field is algebraically…

In [14] we introduced a new class of algebras, which we named \textit{quantum generalized Heisenberg algebras} and which depend on a parameter $q$ and two polynomials $f,g$. We have shown that this class includes all generalized Heisenberg…

For any Lie algebra L over a field, its universal enveloping algebra U(L) can be embedded in a division ring D(L) constructed by Lichtman. If U(L) is an Ore domain, D(L) coincides with its ring of fractions. It is well known that the…

We show that the Hilbert space with basis indexed by infinite permutations and the cohomology ring of the infinite flag variety can be seen as representations of the Heisenberg algebra, which are isomorphic using the back-stable Schubert…

Let $\operatorname{K}_0(\operatorname{Var}_k)$ denote the Grothendieck ring of $k$-varieties over an algebraically closed field $k$. Larsen and Lunts asked if two $k$-varieties having the same class in $\operatorname{K}_0…

A free-by-cyclic group can often be viewed as a mapping torus of a free group automorphism (monodromy) in multiple ways. What dynamical properties must these monodromies share, and to what extent are they invariant under quasi-isometries?…

Consider the polynomial ring in countably infinitely many variables over a field of characteristic zero, together with its natural action of the infinite general linear group G. We study the algebraic and homological properties of finitely…

In this paper, we give a description of polynomial functors from (finitely generated free) groups to abelian groups in terms of non-linear Mackey functors generalizing those given in a paper of Baues-Dreckmann-Franjou-Pirashvili published…