The Two-Step Nilpotent Representations of the Extended Affine Hecke Algebra of Type A

We define a set of cell modules for the extended affine Hecke algebra of type A which are parametrised by SL_n()-conjugacy classes of pairs (s, N), where s SL_n() is semisimple and N is a nilpotent element of the Lie algebra which has at most two Jordan blocks and satisfies Ad(s)·N=q ² N. When q ²–1, each of these has irreducible head, and the irreducible representations of the affine Hecke algebra so obtained are precisely those which factor through its Temperley–Lieb quotient. When q ²=–1, the above remarks apply to a subset of the cell modules. Using our work on the cellular nature of those quotients, we are able to obtain complete information on the decomposition of the cell modules in all cases, even when q is a root of unity. They turn out to be multiplicity free, and the composition factors may be precisely described in terms of a partial order on the pairs (s, N). These results give explicit formulae for the dimensions of the irreducibles. Assuming our modules are identified with the standard modules earlier defined by Bernstein–Zelevinski, Kazhdan–Lusztig and others, our results may be interpreted as the determination of certain Kazhdan–Lusztig polynomials. [This has now been proved and will appear in a subsequent work of the authors.]The second author thanks the Australian Research Council and the Alexander von Humboldt Stiftung for support and the Universität Bielefeld for hospitality during the preparation of this work.     

Nash equilibria of n-player repeated games with semi-standard information

The folk theorem is extended here to the case where after each stage of the repeated game each player is informed only about the equivalence classes of the pure actions which were used by the other players. The sets of upper equilibrium payoffs and of lower equilibrium payoffs are characterized here, and they are found to be different.     

Developing conceptions of statistics by designing measures of distribution

Students often learn procedures for measuring, but rarely do they grapple with the foundational conceptual problem of generating and validating coordination between a measure and the phenomenon being measured. Coordinating measures with phenomenon involves developing an appreciation of the objects and relations in each as well as establishing their mutual correspondence. We supported students?? developing conceptions of statistics by positioning them to design measures of center and of variability for distributions that they had generated through repeated measure of a length. After students invented and explored the viability of their measures individually, they participated in a public (whole-class conversation) forum featuring justification and reflection about the viability of their designed measures. We illustrate how individual invention enticed students to attend to, and to make explicit, characteristics of distribution not initially noticed or known only tacitly. Conceptions of statistics and of relevant characteristics of distribution were further expanded as students justified and argued about the utility and prospective generalization of particular inventions. Teachers supported student learning by highlighting prospective relations between characteristics of measures and characteristics of distribution as they emerged during the course of activity in each setting.     

Global games

Global games are real-valued functions defined on partitions (rather than subsets) of the set of players. They capture “public good” aspects of cooperation, i.e., situations where the payoff is naturally defined for all players (“the globe”) together, as is the case with issues of environmental clean-up, medical research, and so forth. We analyze the more general concept of lattice functions and apply it to partition functions, set functions and the interrelation between the two. We then use this analysis to define and characterize the Shapley value and the core of global games.     

Mediated talk

The notion of mediated talk in which players communicate through a mediator before starting the game is introduced. It is shown that a deterministic mechanized mediator receiving private inputs and producing public outputs can generate any rational correlated equilibrium with rational probabilities by a self-enforcing procedure.