609,816 research outputs found
Eriksson's numbers game and finite Coxeter groups
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.Comment: 18 page
Impartial avoidance games for generating finite groups
We study an impartial avoidance game introduced by Anderson and Harary. The
game is played by two players who alternately select previously unselected
elements of a finite group. The first player who cannot select an element
without making the set of jointly-selected elements into a generating set for
the group loses the game. We develop criteria on the maximal subgroups that
determine the nim-numbers of these games and use our criteria to study our game
for several families of groups, including nilpotent, sporadic, and symmetric
groups.Comment: 14 pages, 4 figures. Revised in response to comments from refere
An explicit bound on " for nonemptiness of "-cores of games
We consider parameterized collections of games without side payments and determine a bound on E so that all suffciently large games in the collection have non-empty E-cores. Our result makes explicit the relationship between the required size of E for non-emptiness of the E-core, the parameters describing the collection of games, and the size of the total player set. Given the parameters describing the collection, the larger the game, the smaller the E that can bechosen
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