Distributive Lattices Defined for Representations of Rank Two Semisimple Lie Algebras
Search citation statements
Paper Sections
Citation Types
Publication Types
Relationship
Authors
Journals
Cited by 1 publication
References 9 publications
The numbers game is a one-player game played on a finite simple graph with certain "amplitudes" assigned to its edges and with an initial assignment of real numbers to its nodes. The moves of the game successively transform the numbers at the nodes using the amplitudes in a certain way. This game and its interactions with Coxeter/Weyl group theory and Lie theory have been studied by many authors. In particular, Eriksson connects certain geometric representations of Coxeter groups with games on graphs with certain real number amplitudes. Games played on such graphs are "E-games." Here we investigate various finiteness aspects of E-game play: We extend Eriksson's work relating moves of the game to reduced decompositions of elements of a Coxeter group naturally associated to the game graph. We use Stembridge's theory of fully commutative Coxeter group elements to classify what we call here the "adjacency-free" initial positions for finite E-games. We characterize when the positive roots for certain geometric representations of finite Coxeter groups can be obtained from E-game play. Finally, we provide a new Dynkin diagram classification result of E-game graphs meeting a certain finiteness requirement. Definitions and preliminary resultsFix a positive integer n and a totally ordered set I n with n elements (usually I n := {1 < . . . < n}). An E-generalized Cartan matrix or E-GCM * is an n × n matrix M = (M ij ) i,j∈In with real entries satisfying the requirements that each main diagonal matrix entry is 2, that all other matrix entries are nonpositive, that if a matrix entry M ij is nonzero then its transpose entry M ji is also nonzero, and that if M ij M ji is nonzero then M ij M ji ≥ 4 or M ij M ji = 4 cos 2 (π/k ij ) for some integer k ij ≥ 3. These peculiar constraints on products of transpose pairs of matrix entries are precisely those required in order to guarantee "strong convergence" for E-games, cf. Theorem 2.1 below, Theorem 3.6 of [Erik2], Theorem 3.1 of [Erik6]. To an n × n E-generalized Cartan matrix M = (M ij ) i,j∈In we associate a finite graph Γ (which has undirected edges, no loops, and no multiple edges) as follows:The nodes (γ i ) i∈In of Γ are indexed by the set I n , and an edge is placed between nodes γ i and γ j * Motivation for terminology: E-GCMs with integer entries are just generalized Cartan matrices, which are the starting point for the study of Kac-Moody algebras: beginning with a GCM, one can write down a list of the defining relations for a Kac-Moody algebra as well as its associated Weyl group ([Kac], [Kum]). Here we use the modifier "E" because of the relationship between these matrices and the combinatorics of Eriksson's E-games. Eriksson uses "E" for edge; he also allows for "N-games" where, in addition, nodes can be weighted. For finite Coxeter groups, the next result strengthens Part (1) of Eriksson's Reduced Word Result. At this time it is an open question whether the finiteness hypothesis for W can be relaxed.Corollary 3.4 Let J ⊆ I n and let λ be any J c -dominant position. Sup...
The numbers game is a one-player game played on a finite simple graph with certain "amplitudes" assigned to its edges and with an initial assignment of real numbers to its nodes. The moves of the game successively transform the numbers at the nodes using the amplitudes in a certain way. This game and its interactions with Coxeter/Weyl group theory and Lie theory have been studied by many authors. In particular, Eriksson connects certain geometric representations of Coxeter groups with games on graphs with certain real number amplitudes. Games played on such graphs are "E-games." Here we investigate various finiteness aspects of E-game play: We extend Eriksson's work relating moves of the game to reduced decompositions of elements of a Coxeter group naturally associated to the game graph. We use Stembridge's theory of fully commutative Coxeter group elements to classify what we call here the "adjacency-free" initial positions for finite E-games. We characterize when the positive roots for certain geometric representations of finite Coxeter groups can be obtained from E-game play. Finally, we provide a new Dynkin diagram classification result of E-game graphs meeting a certain finiteness requirement. Definitions and preliminary resultsFix a positive integer n and a totally ordered set I n with n elements (usually I n := {1 < . . . < n}). An E-generalized Cartan matrix or E-GCM * is an n × n matrix M = (M ij ) i,j∈In with real entries satisfying the requirements that each main diagonal matrix entry is 2, that all other matrix entries are nonpositive, that if a matrix entry M ij is nonzero then its transpose entry M ji is also nonzero, and that if M ij M ji is nonzero then M ij M ji ≥ 4 or M ij M ji = 4 cos 2 (π/k ij ) for some integer k ij ≥ 3. These peculiar constraints on products of transpose pairs of matrix entries are precisely those required in order to guarantee "strong convergence" for E-games, cf. Theorem 2.1 below, Theorem 3.6 of [Erik2], Theorem 3.1 of [Erik6]. To an n × n E-generalized Cartan matrix M = (M ij ) i,j∈In we associate a finite graph Γ (which has undirected edges, no loops, and no multiple edges) as follows:The nodes (γ i ) i∈In of Γ are indexed by the set I n , and an edge is placed between nodes γ i and γ j * Motivation for terminology: E-GCMs with integer entries are just generalized Cartan matrices, which are the starting point for the study of Kac-Moody algebras: beginning with a GCM, one can write down a list of the defining relations for a Kac-Moody algebra as well as its associated Weyl group ([Kac], [Kum]). Here we use the modifier "E" because of the relationship between these matrices and the combinatorics of Eriksson's E-games. Eriksson uses "E" for edge; he also allows for "N-games" where, in addition, nodes can be weighted. For finite Coxeter groups, the next result strengthens Part (1) of Eriksson's Reduced Word Result. At this time it is an open question whether the finiteness hypothesis for W can be relaxed.Corollary 3.4 Let J ⊆ I n and let λ be any J c -dominant position. Sup...
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
Contact Info
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Product
Resources
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.
