scholarly journals The Weak Order on Weyl Posets

2019 ◽  
Vol 72 (4) ◽  
pp. 867-899
Author(s):  
Joël Gay ◽  
Vincent Pilaud
Keyword(s):  
Weyl Group ◽  
Root System ◽  
Coxeter Group ◽  
Lattice Structure ◽  
Root Systems ◽  
Weak Order ◽  
Generalized Associahedra

AbstractWe define a natural lattice structure on all subsets of a finite root system that extends the weak order on the elements of the corresponding Coxeter group. For crystallographic root systems, we show that the subposet of this lattice induced by antisymmetric closed subsets of roots is again a lattice. We then study further subposets of this lattice that naturally correspond to the elements, the intervals, and the faces of the permutahedron and the generalized associahedra of the corresponding Weyl group. These results extend to arbitrary finite crystallographic root systems the recent results of G. Chatel, V. Pilaud, and V. Pons on the weak order on posets and its induced subposets.

On Closed Subsets of Root Systems

1994 ◽  
Vol 37 (3) ◽  
pp. 338-345 ◽  
Author(s):  
D. Ž. Doković ◽  
P. Check ◽  
J.-Y. Hée
Keyword(s):  
Weyl Group ◽  
Root System ◽  
Irreducible Component ◽  
Product Space ◽  
Root Systems ◽  
Inner Product Space ◽  
Inner Product ◽  
Closed Sets ◽  
Finite Dimensional ◽  
Real Inner Product Space

AbstractLet R be a root system (in the sense of Bourbaki) in a finite dimensional real inner product space V. A subset P ⊂ R is closed if α, β ∊ P and α + β ∊ R imply that α + β ∊ P. In this paper we shall classify, up to conjugacy by the Weyl group W of R, all closed sets P ⊂ R such that R\P is also closed. We also show that if θ:R —> R′ is a bijection between two root systems such that both θ and θ-1 preserve closed sets, and if R has at most one irreducible component of type A1, then θ is an isomorphism of root systems.

scholarly journals Lattice structure of Weyl groups via representation theory of preprojective algebras

2018 ◽  
Vol 154 (6) ◽  
pp. 1269-1305 ◽  
Author(s):  
Osamu Iyama ◽  
Nathan Reading ◽  
Idun Reiten ◽  
Hugh Thomas
Keyword(s):  
Representation Theory ◽  
Weyl Group ◽  
Lattice Structure ◽  
Weak Order ◽  
Weyl Groups ◽  
Preprojective Algebra ◽  
Combinatorial Description ◽  
Irreducible Elements ◽  
Preprojective Algebras

This paper studies the combinatorics of lattice congruences of the weak order on a finite Weyl group $W$, using representation theory of the corresponding preprojective algebra $\unicode[STIX]{x1D6F1}$. Natural bijections are constructed between important objects including join-irreducible congruences, join-irreducible (respectively, meet-irreducible) elements of $W$, indecomposable $\unicode[STIX]{x1D70F}$-rigid (respectively, $\unicode[STIX]{x1D70F}^{-}$-rigid) modules and layers of $\unicode[STIX]{x1D6F1}$. The lattice-theoretically natural labelling of the Hasse quiver by join-irreducible elements of $W$ is shown to coincide with the algebraically natural labelling by layers of $\unicode[STIX]{x1D6F1}$. We show that layers of $\unicode[STIX]{x1D6F1}$ are nothing but bricks (or equivalently stones, or 2-spherical modules). The forcing order on join-irreducible elements of $W$ (arising from the study of lattice congruences) is described algebraically in terms of the doubleton extension order. We give a combinatorial description of indecomposable $\unicode[STIX]{x1D70F}^{-}$-rigid modules for type $A$ and $D$.

scholarly journals Root System Chip-Firing II: Central-Firing

2019 ◽  
Author(s):  
Pavel Galashin ◽  
Sam Hopkins ◽  
Thomas McConville ◽  
Alexander Postnikov
Keyword(s):  
Weyl Group ◽  
Root System ◽  
Dynkin Diagram ◽  
Positive Root ◽  
Root Systems ◽  
Initial Weight ◽  
Fundamental Weight ◽  
Chip Firing ◽  
Number Game

Abstract Jim Propp recently proposed a labeled version of chip-firing on a line and conjectured that this process is confluent from some initial configurations. This was proved by Hopkins–McConville–Propp. We reinterpret Propp’s labeled chip-firing moves in terms of root systems; a “central-firing” move consists of replacing a weight $\lambda$ by $\lambda +\alpha$ for any positive root $\alpha$ that is orthogonal to $\lambda$. We show that central-firing is always confluent from any initial weight after modding out by the Weyl group, giving a generalization of unlabeled chip-firing on a line to other types. For simply-laced root systems we describe this unlabeled chip-firing as a number game on the Dynkin diagram. We also offer a conjectural classification of when central-firing is confluent from the origin or a fundamental weight.

Prenilpotent pairs in the E10 root lattice

2017 ◽  
Vol 164 (3) ◽  
pp. 473-483
Author(s):  
DANIEL ALLCOCK
Keyword(s):  
Weyl Group ◽  
Root System ◽  
Root Systems ◽  
Commutative Rings ◽  
Inner Product ◽  
Root Lattice ◽  
Group Presentation ◽  
Central Concept ◽  
Real Roots ◽  
The One

AbstractTits has defined Kac–Moody groups for all root systems, over all commutative rings with unit. A central concept is the idea of a prenilpotent pair of (real) roots. In particular, writing down his group presentation explicitly would require knowing all the Weyl-group orbits of such pairs. We show that for the hyperbolic root system E10 there are so many orbits that any attempt at direct enumeration is impractical. Namely, the number of orbits of prenilpotent pairs having inner product k grows at least as fast as (constant) ⋅ k7 as k → ∞. Our purpose is to motivate alternate approaches to Tits' groups, such as the one in [2].

scholarly journals Clifford Spinors and Root System Induction: $$H_4$$ and the Grand Antiprism

2021 ◽  
Vol 31 (3) ◽  
Author(s):  
Pierre-Philippe Dechant
Keyword(s):  
Root System ◽  
Root Systems ◽  
Invariant Sets ◽  
Simple Construction ◽  
Reflection Groups ◽  
Computational Work ◽  
Algebraic Framework ◽  
Underlying Geometry ◽  
Coxeter Plane ◽  
Insight Into

AbstractRecent work has shown that every 3D root system allows the construction of a corresponding 4D root system via an ‘induction theorem’. In this paper, we look at the icosahedral case of $$H_3\rightarrow H_4$$ H 3 → H 4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan–Dieudonné theorem, giving a simple construction of the $${\mathrm {Pin}}$$ Pin and $${\mathrm {Spin}}$$ Spin covers. Using this connection with $$H_3$$ H 3 via the induction theorem sheds light on geometric aspects of the $$H_4$$ H 4 root system (the 600-cell) as well as other related polytopes and their symmetries, such as the famous Grand Antiprism and the snub 24-cell. The uniform construction of root systems from 3D and the uniform procedure of splitting root systems with respect to subrootsystems into separate invariant sets allows further systematic insight into the underlying geometry. All calculations are performed in the even subalgebra of $${\mathrm {Cl}}(3)$$ Cl ( 3 ) , including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes, and are shared as supplementary computational work sheets. This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework.

scholarly journals The Action of the Weyl Group on the $$E_8$$ Root System

2021 ◽  
Author(s):  
Rosa Winter ◽  
Ronald van Luijk
Keyword(s):  
Automorphism Group ◽  
Weyl Group ◽  
Root System ◽  
Sufficient Conditions ◽  
Maximal Clique ◽  
Necessary And Sufficient Conditions ◽  
Inner Product ◽  
Colored Graphs ◽  
Necessary And Sufficient ◽  
Root Polytope

AbstractLet $$\varGamma $$ Γ be the graph on the roots of the $$E_8$$ E 8 root system, where any two distinct vertices e and f are connected by an edge with color equal to the inner product of e and f. For any set c of colors, let $$\varGamma _c$$ Γ c be the subgraph of $$\varGamma $$ Γ consisting of all the 240 vertices, and all the edges whose color lies in c. We consider cliques, i.e., complete subgraphs, of $$\varGamma $$ Γ that are either monochromatic, or of size at most 3, or a maximal clique in $$\varGamma _c$$ Γ c for some color set c, or whose vertices are the vertices of a face of the $$E_8$$ E 8 root polytope. We prove that, apart from two exceptions, two such cliques are conjugate under the automorphism group of $$\varGamma $$ Γ if and only if they are isomorphic as colored graphs. Moreover, for an isomorphism f from one such clique K to another, we give necessary and sufficient conditions for f to extend to an automorphism of $$\varGamma $$ Γ , in terms of the restrictions of f to certain special subgraphs of K of size at most 7.

Extension and Longitudinal Growth During the Development of Red Pine Root Systems

1975 ◽  
Vol 5 (1) ◽  
pp. 109-121 ◽  
Author(s):  
D. C. F. Fayle
Keyword(s):  
Root System ◽  
Root Systems ◽  
Red Pine ◽  
Longitudinal Growth ◽  
Rooting Depth ◽  
Soil Conditions ◽  
Moisture Availability ◽  
Below Ground

Extension of the root system and stem during the first 30 years of growth of plantation-grown red pine (Pinusresinosa Ait.) on four sites was deduced by root and stem analyses. Maximum rooting depth was reached in the first decade and maximum horizontal extension of roots was virtually complete between years 15 and 20. The main horizontal roots of red pine seldom exceed 11 m in length. Elongation of vertical and horizontal roots was examined in relation to moisture availability and some physical soil conditions. The changing relations within the tree in lineal dimensions and annual elongation of the roots and stem are illustrated. The development of intertree competition above and below ground is considered.

Effects of Temperature on Root Morphology and Ectendomycorrhizal Development in Pinusresinosa Ait.

1975 ◽  
Vol 5 (2) ◽  
pp. 171-175 ◽  
Author(s):  
Hugh E. Wilcox ◽  
Ruth Ganmore-Neumann
Keyword(s):  
Root Growth ◽  
Root System ◽  
Root Morphology ◽  
Root Systems ◽  
Second Order ◽  
Root Temperature ◽  
First Order ◽  
Effects Of Temperature

Seedlings of Pinusresinosa were grown at root temperatures of 16, 21 and 27 °C, both aseptically and after inoculation with the ectendomycorrhizal fungus BDG-58. Growth after 3 months was significantly influenced by the presence of the fungus at all 3 temperatures. The influence of the fungus on root growth was obscured by the effects of root temperature on morphology. The root system at 16 and at 21 °C possessed many first-order laterals with numerous, well developed second-order branches, but those at 27 °C had only a few, relatively long, unbranched first-order laterals. Although the root systems of infected seedlings were larger, the fungus increased root growth in the same pattern as determined by the temperature.

scholarly journals Contemporary Concepts of Root System Architecture of Urban Trees

2010 ◽  
Vol 36 (4) ◽  
pp. 149-159
Author(s):  
Susan Day ◽  
P. Eric Wiseman ◽  
Sarah Dickinson ◽  
J. Roger Harris
Keyword(s):  
Built Environment ◽  
Root Growth ◽  
Root System ◽  
Root Architecture ◽  
Root Systems ◽  
Growth Patterns ◽  
Urban Trees ◽  
Rooting Depth ◽  
Similar Species ◽  
Urban Settings

Knowledge of the extent and distribution of tree root systems is essential for managing trees in the built environment. Despite recent advances in root detection tools, published research on tree root architecture in urban settings has been limited and only partially synthesized. Root growth patterns of urban trees may differ considerably from similar species in forested or agricultural environments. This paper reviews literature documenting tree root growth in urban settings as well as literature addressing root architecture in nonurban settings that may contribute to present understanding of tree roots in built environments. Although tree species may have the genetic potential for generating deep root systems (>2 m), rooting depth in urban situations is frequently restricted by impenetrable or inhospitable soil layers or by underground infrastructure. Lateral root extent is likewise subject to restriction by dense soils under hardscape or by absence of irrigation in dry areas. By combining results of numerous studies, the authors of this paper estimated the radius of an unrestricted root system initially increases at a rate of approximately 38 to 1, compared to trunk diameter; however, this ratio likely considerably declines as trees mature. Roots are often irregularly distributed around the tree and may be influenced by cardinal direction, terrain, tree lean, or obstacles in the built environment. Buttress roots, tap roots, and other root types are also discussed.

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