Eriksson’s numbers game and finite Coxeter groups

Abstract

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...

On dominance and minuscule Weyl group elements

“…In particular, the configuration v must converge to some limiting allowed configuration (which is zero on Γ ′ ), and one could continue the numbers game from this limit if desired. Note that, in the case that c ij = c ji for all odd n ij , we must also have α · v > −1 8 Also, this observation easily implies the main results (Theorems 2.1 and 4.1) of [DE08]: if vi ≤ 0 for all i and v = 0, then the usual numbers game can only terminate if Γ, C are associated to a finite Coxeter group: otherwise (assuming Γ is connected), infinitely many elements β ∈ ∆+ which are not multiples of each other satisfy β · v < 0: note that, for each i ∈ I, the set P(W αi) essentially does not depend on the choice of C for a given Coxeter group.…”

Poset models for Weyl group analogs of symmetric functions and Schur functions

“…Semistandard lattices [DW]; for a combinatorial description of the adjacency-free fundamental weights, see [Don6] We obtain, in a uniform way, splitting distributive lattices for the following (root system, dominant weight) pairs: (An, λ), where λ is any dominant weight; (Bn, aω 1 + bωn); (Cn, aω 1 + bωn); (Dn, aω 1 + bω n−1 + cωn); (E 6 , aω 1 + bω 6 ); (E 7 , aω 7 ); (G 2 , aω 1 + bω 2 ). The "KN" and "De Concini" symplectic lattices Defined in [Don1] and again in [Don2] These are two families of distributive lattice supporting graphs for the fundamental representations of the symplectic Lie algebras.…”

“…Using a Coxeter group viewpoint, it is possible to show that for any λ = i∈I a i ω i ∈ Λ + such that J = {i ∈ I | a i = 0}, then the stablizer W λ of λ under the Weyl group action is just W J ∼ = W Φ J (see for example §3 of [Don6]). In Corollary 3.10, we say how weight diagrams with respect to root subsystems can be understood in terms of weight diagrams with respect to the "parent" root system.…”

“…, which is closely related to moves of the so-called "numbers game" (see e.g. Appendix B, or [Don6] and references therein). With a root system / Cartan matrix / weight lattice / Weyl group in hand, one could begin a concrete theory of W -symmetric functions using the above numbers-game-…”

“…So we begin the exposition of this appendix with a discussion of Weyl groups and Weyl symmetric functions as presented in that ∼2008 paper. This discussion borrows from [Don6], [Don8], and [ADLMPPW] as well as standard treatments like [Hum1], [Hum2], [Bour],…”