Related papers: Root systems, affine subspaces, and projections
The notion of a "root base" together with its geometry plays a crucial role in the theory of finite and affine Lie theory. However, it is known that such a notion does not exist for the recent generalizations of finite and affine root…
Let $\Phi$ be a finite crystallographic irreducible root system and $\mathcal P_{\Phi}$ be the convex hull of the roots in $\Phi$. We give a uniform explicit description of the polytope $\mathcal P_{\Phi}$, analyze the…
A root systems in Carroll spaces with degenerate metric are defined. It is shown that their Cartan matrices and reflection groups are affine. With the help of the geometric consideration the root system structure of affine algebras is…
The interaction of a Lie algebra $\LL,$ having a weight space decomposition with respect to a nonzero toral subalgebra, with its corresponding root system forms a powerful tool in the study of the structure of $\LL.$ This, in particular,…
We study a combinatorial object, which we call a GRRS (generalized reflection root system); the classical root systems and GRSs introduced by V. Serganova are examples of finite GRRSs. A GRRS is finite if it contains a finite number of…
We examine certain maps from root systems to vector spaces over finite fields. By choosing appropriate bases, the images of these maps can turn out to have nice combinatorial properties, which reflect the structure of the underlying root…
Extended affine root systems appear as the root systems of extended affine Lie algebras. A subclass of extended affine root systems, whose elements are called ``minimal" turns out to be of special interest mostly because of the geometric…
We discuss the classification of reflection subgroups of finite and affine Weyl groups from the point of view of their root systems. A short case free proof is given of the well known classification of the isomorphism classes of reflection…
Root systems are sets with remarkable symmetries and therefore they appear in many situations in mathematics. Among others, denominator formulae of root systems are very beautiful and mysterious equations which have several meanings from a…
Let $W$ be a finitely generated infinite Coxeter group, with $\Phi$ and $\Pi$ being the corresponding root system and set of simple roots respectively. It has been observed by Hohlweg et la that the projections of elements of $\Phi$ onto…
The (extended) Linial arrangement $\mathcal{L}_{\Phi}^m$ is a certain finite truncation of the affine Weyl arrangement of a root system $\Phi$ with a parameter $m$. Postnikov and Stanley conjectured that all roots of the characteristic…
Any symmetric closed subset of a finite crystallographic root system must be a closed subroot system. This is not, in general, true for real affine root systems. In this paper, we determine when this is true and also give a very explicit…
Let $\Phi$ be an irreducible crystallographic root system with Weyl group $W$ and coroot lattice $\check{Q}$, spanning a Euclidean space $V$. Let $m$ be a positive integer and $\aA^m_\Phi$ be the arrangement of hyperplanes in $V$ of the…
We study the root polytope $\mathcal P_\Phi$ of a finite irreducible crystallographic root system $\Phi$ using its relation with the abelian ideals of a Borel subalgebra of a simple Lie algebra with root system $\Phi$. We determine the…
Splint of root system of simple Lie algebra appears naturally in the study of (regular) embeddings of reductive subalgebras. It can be used to derive branching rules. Application of splint properties drastically simplifies calculations of…
We endow the set of complements of a fixed subspace of a projective space with the structure of an affine space, and show that certain lines of such an affine space are affine reguli or cones over affine reguli. Moreover, we apply our…
Given any polynomial system with fixed monomial term structure, we give explicit formulae for the generic number of roots with specified coordinate vanishing restrictions. For the case of affine space minus an arbitrary union of coordinate…
We suggest a (conjectural) construction of a basis in the plus part of the affine Lie algebra of type ADE indexed by irreducible components of certain quiver varieties. This construction is closely related to a string-theoretic construction…
We consider a tuple $\Phi = (\phi_1,\ldots,\phi_m)$ of commuting maps on a finitary matroid $X$. We show that if $\Phi$ satisfies certain conditions, then for any finite set $A\subseteq X$, the rank of $\{\phi_1^{r_1}\cdots\phi_m^{r_m}(a):a…
The Kochen-Specker theorem states that a 3-dimensional complex Euclidean space admits a finite configuration of projective lines such that the corresponding quantum observables (the orthogonal projectors) cannot be assigned with 0 and 1…