Related papers: Bruhat Preclosure

We describe an algorithm to identify a minimal set of "braid relations" which span and preserve all sets of involution words for twisted Coxeter systems of finite or affine type. We classify the cases in which adding the smallest possible…

We answer an open question in the theory of transducer degrees initially posed in [1] on the existence of a diamond structure in the transducer hierarchy. Transducer degrees are the equivalence classes formed by word transformations which…

A group G has property FA if G fixes a point of every tree on which G acts without inversions. We prove that every Coxeter system of finite rank has a visual JSJ decomposition over subgroups with property FA. As an application, we reduce…

The Frankl or Union-Closed Sets conjecture states that for any finite union-closed family of sets $\mathcal{F}$ containing some nonempty set, there is some element $i$ in the ground set $U(\mathcal F) := \bigcup_{S \in \mathcal{F}} S$ of…

Motivated by recent work with Mazorchuk, we characterize the conditions under which the intersection of two principal order ideals in the Bruhat order is boolean. That characterization is presented in three versions: in terms of reduced…

The F-threshold $c^J(\a)$ of an ideal $\a$ with respect to the ideal $J$ is a positive characteristic invariant obtained by comparing the powers of $\a$ with the Frobenius powers of $J$. We show that under mild assumptions, we can detect…

We consider the question of determining whether a given group (especially one generated by involutions) is a right-angled Coxeter group. We describe a group invariant, the involution graph, and we characterize the involution graphs of…

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,…

Given two affine permutations, some results of Lascoux and Deodhar, and independently Jacon-Lecouvey, allow to decide if they are comparable for the strong Bruhat order. These permutations are associated with tuples of core partitions, and…

A family of sets is union-closed (UC) if the union of any two sets in the family is also in the family. Frankl's UC sets conjecture states that for any nonempty UC family $\mathcal{F} \subseteq 2^{[n]}$ such that $\mathcal{F} \neq…

Let $N$ be a normal subgroup of a finite group $G$. For a faithful $N$-set $\Delta$, applying the university embedding theorem one can construct a faithful $G$-set $\Omega$. In this short note, it is proved that if the $2$-closure of $N$ in…

The Alperin-McKay conjecture is a longstanding open conjecture in the representation theory of finite groups. Sp\"ath showed that the Alperin-McKay conjecture holds if the so-called inductive Alperin-McKay (iAM) condition holds for all…

Let $\bullet^{\dag}$ be the map in sense of the Losev, which sends the set of two sided ideals of a finite W-algebras to that of the universal enveloping algebra of corresponding Lie algebras. The Premet conjecture which was proved in…

Let $\nat^\nat$ be the semigroup of all mappings on the natural numbers $\nat$, and let $U$ and $V$ be subsets of $\nat^\nat$. We write $U\preccurlyeq V$ if there exists a countable subset $C$ of $\nat^\nat$ such that $U$ is contained in…

Let $G$ be a nontrivial permutation group of degree $n$. If $G$ is transitive, then a theorem of Jordan states that $G$ has a derangement. Equivalently, a finite group is never the union of conjugates of a proper subgroup. If $G$ is…

The Bruhat stratification for Shimura varieties of PEL type is studied. In the Siegel case this stratification is a scheme-theoretic variant of the stratification by the a-number. We show that all Bruhat strata are smooth and determine…

In this paper we address the problem of generating all elements obtained by the saturation of an initial set by some operations. More precisely, we prove that we can generate the closure by polymorphisms of a boolean relation with a…

This is the first in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global confor- mal invariants"; these are defined to be conformally invariant integrals of geometric scalars.…

This short note contains random thoughts about a factorization theorem for closure/interior operators on a powerset which is reminiscent to the notion of resolution for a monad/comonad. The question originated from formal topology but is…

We prove that the conjecture made by Peter Frankl in the late 1970s is true. In other words for every finite union-closed family which contains a non?empty set, there is an element that belongs to at least half of its m