On the Importance of the Weyl Group for Kac-Moody and String Algebras

1990 ◽  
pp. 531-534
Author(s):  
N. Gorman ◽  
W. McGlinn ◽  
L. O’Raifeartaigh
Keyword(s):  
Weyl Group ◽  
String Algebras

scholarly journals Symmetry of Narayana Numbers and Rowvacuation of Root Posets

2021 ◽  
Vol 9 ◽  
Author(s):  
Colin Defant ◽  
Sam Hopkins
Keyword(s):  
Weyl Group ◽  
Catalan Number ◽  
Recent Past ◽  
Positive Roots ◽  
Narayana Numbers ◽  
The Subject

Abstract For a Weyl group W of rank r, the W-Catalan number is the number of antichains of the poset of positive roots, and the W-Narayana numbers refine the W-Catalan number by keeping track of the cardinalities of these antichains. The W-Narayana numbers are symmetric – that is, the number of antichains of cardinality k is the same as the number of cardinality $r-k$ . However, this symmetry is far from obvious. Panyushev posed the problem of defining an involution on root poset antichains that exhibits the symmetry of the W-Narayana numbers. Rowmotion and rowvacuation are two related operators, defined as compositions of toggles, that give a dihedral action on the set of antichains of any ranked poset. Rowmotion acting on root posets has been the subject of a significant amount of research in the recent past. We prove that for the root posets of classical types, rowvacuation is Panyushev’s desired involution.

scholarly journals Affine Weyl Group Symmetry of the Garnier System

2005 ◽  
Vol 48 (2) ◽  
pp. 203-230 ◽  
Author(s):  
Takao Suzuki
Keyword(s):  
Weyl Group ◽  
Group Symmetry ◽  
Affine Weyl Group ◽  
Garnier System

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.

scholarly journals Partial orders on conjugacy classes in the Weyl group and on unipotent conjugacy classes

2021 ◽  
Vol 383 ◽  
pp. 107688
Author(s):  
Jeffrey Adams ◽  
Xuhua He ◽  
Sian Nie
Keyword(s):  
Weyl Group ◽  
Partial Orders ◽  
Conjugacy Classes

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.

Lie algebra automorphisms, the Weyl group, and tables of shift vectors

1989 ◽  
Vol 30 (3) ◽  
pp. 571-584 ◽  
Author(s):  
Peter Bouwknegt
Keyword(s):  
Weyl Group ◽  
Lie Algebra

The affine Weyl group and modular invariant partition functions

1988 ◽  
Vol 205 (2-3) ◽  
pp. 281-284 ◽  
Author(s):  
D. Altschüler ◽  
J. Lacki ◽  
Ph. Zaugg
Keyword(s):  
Weyl Group ◽  
Partition Functions ◽  
Modular Invariant ◽  
Affine Weyl Group
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