Related papers: A step towards Twist Conjecture

This paper explores the orbit structure and homomesy (constant averages over orbits) properties of certain actions of toggle groups on the collection of independent sets of a path graph. In particular we prove a generalization of a homomesy…

We formulate two conjectures about etale cohomology and fundamental groups motivated by categoricity conjectures in model theory. One conjecture says that there is a unique Z-form of the etale cohomology of complex algebraic varieties, up…

A classical theorem of Scheunert on $G$-color Lie algebras, asserts in the case of finitely generated abelian groups, one can twist the algebra structure and the commutation bicharacter on $G$ by a 2-cocycle twist to a super-Lie $G$ graded,…

Let $H$ be a Hopf algebra that is $\mathbb Z$-graded as an algebra. We provide sufficient conditions for a 2-cocycle twist of $H$ to be a Zhang twist of $H$. In particular, we introduce the notion of a twisting pair for $H$ such that the…

We conjecture a characterization of a cluster automorphism as an algebra homomorphism from the cluster algebra to itself that restricts to a bijection between two clusters. This formulation does not require that the map commutes with…

We prove Lusztig's conjectures P1-P15 for Coxeter groups with complete graph, using deceasing induction on $ \mathbf{a} $-values and a kind of decomposition formula of Kazhdan-Lusztig basis elements. As a byproduct, we give a description of…

In 1979 Frankl conjectured that in a finite non-trivial union-closed collection of sets there has to be an element that belongs to at least half the sets. We show that this is equivalent to the conjecture that in a finite non-trivial graph…

We prove the strong Atiyah conjecture for right-angled Artin groups and right-angled Coxeter groups. More generally, we prove it for groups which are certain finite extensions or elementary amenable extensions of such groups.

Based on Nijenhuis-Richardson bracket and bidegree on the cohomology complex for a Lie conformal algebra, we develop a twisting theory of Lie conformal algebras. By using derived bracket constructions, we construct $L_\infty$-algebras from…

Hadwiger's conjecture asserts that any graph contains a clique minor with order no less than the chromatic number of the graph. We prove that this well-known conjecture is true for all graphs if and only if it is true for squares of split…

We show the analogue of Muhlherr's [Coxeter groups in Coxeter groups, Finite Geom. and Combinatorics, Cambridge Univ. Press (1993), 277-287] for Artin-Tits monoids, and for Artin-Tits groups of spherical type. That is, the submonoid (resp.…

The ``Flux conjecture'' for symplectic manifolds states that the group of Hamiltonian diffeomorphisms is C^1-closed in the group of all symplectic diffeomorphisms. We prove the conjecture for spherically rational manifolds and for those…

We show that every Coxeter group that is not virtually abelian and for which all labels in the corresponding Coxeter graph are powers of 2 or infinity can be mapped onto uncountably many infinite 2-groups which, in addition, may be chosen…

Frankl's union-closed sets conjecture states that in every finite union-closed set of sets, there is an element that is contained in at least half of the member-sets (provided there are at least two members). The conjecture has an…

We combinatorially characterize the number $\mathrm{cc}_2$ of conjugacy classes of involutions in any Coxeter group in terms of higher rank odd graphs. This notion naturally generalizes the concept of odd graphs, used previously to count…

We prove a conjecture about the vertices and edges of the exchange graph of a cluster algebra $\A$ in two cases: when $\A$ is of geometric type and when $\A$ is arbitrary and its exchange matrix is nondegenerate. In the second case we also…

We will present an algebra related to the Coxeter group of type I2n which can be taken as a twisted subalgebra in Brauer algebra of type A_{n-1}. Also we will describe some properties of this algebra.

The class of cographs is one of the most well-known graph classes, which is also known to be equivalent to the class of $P_4$-free graphs. We show that Mader's conjecture is true if we restrict ourselves to cographs, that is, for any tree…

We study the stable category of the factor algebra of the preprojective algebra associated with an element $w$ of the Coxeter group of a quiver. We show that there exists a silting object $M(\bf{w})$ of this category associated with each…

The open intervals in the Bruhat order on twisted involutions in a Coxeter group are shown to be PL spheres. This implies results conjectured by F. Incitti and sharpens the known fact that these posets are Gorenstein* over Z_2. We also…