Annotated references (books, papers, etc)

Unit 1: The numbers game

ABSTRACT:
The numbers game is a one-player game played on any finite simple graph whose edges are allowed to be weighted in certain ways. This game has been independently invented several times, but we will consider the version studied by Kimmo Eriksson. We will see in Unit 3 of these talks that the game is a model for certain geometric representations of Coxeter groups. (These groups are named after H.S.M. Coxeter, a Canadian and great 20th century geometer who is famous for his work with regular polytopes.) This interplay between combinatorics and algebra will help us answer, by way of a classification result, a finiteness question about the numbers game. The answer to this finiteness question has also helped answer related questions about finite posets and distributive lattices that arise in the study of Weyl characters, cf. Unit 4.

Unit 2: More on generators and relations

        Notes for Unit 2 Part 2, 10 pp.
        Part 2 of Unit 2 focusses on linear algebra.

        Notes for Unit 2 Part 3, 2 pp.
        Part 3 of Unit 2 focusses a bit more on linear algebra.

ABSTRACT:
This unit will serve as a reminder/reintroduction to how algebraic objects can somtimes be usefully and succinctly described in terms of generators and relations, and how such descriptions can be very helpful in constructing morphisms between algebraic structures.

Unit 3: Asymmetric geometric representations of Coxeter groups

        Notes for Unit 3 Part 2, 16 pp.
        Part 2 of Unit 3 focusses on the Tits cone and connections with the numbers game.
        Classification of admissible SC-graphs, 8 pp.
        This presentation of a proof of a finiteness result stated on page 15 of the Unit 3 Part 2 notes was written by seminar student Evan Trevathan.

ABSTRACT:
We will show how there are many ways to view an arbitrary Coxeter group as a collection of invertible linear transformations on a real vector space whose geometry is given by a possibly asymmetric bilinear form. These representations were discovered independently by Vinberg (1970's) and Eriksson (1990's). One object of interest for us will be a convex cone (the so-called Tits cone, named after Jacques Tits, a Belgian/French mathematician, Abel prize winner in 2008, and progenitor of much of the basic theory of Coxeter groups) created by an associated action of the Coxeter group on a certain "polyhedral" fundamental domain. We will connect these Coxeter group representations/actions to the numbers game of Unit 1.

Unit 4: Poset models for Weyl characters

ABSTRACT:
A large subfamily of Coxeter groups consists of the so-called Weyl groups (named after Hermann Weyl, a German mathematician and one of the titans of 20th century mathematics). These have certain integrality properties not shared by all Coxeter groups in general. In the finite cases, Weyl groups arise naturally in the study of finite-dimensional complex semisimple Lie algebras/groups. Weyl characters are certain multivariate "Laurent" polynomials (negative exponents are allowed) which are preserved by the actions of these finite Weyl groups and which are invariants for representations of the associated Lie algebras/groups. A poset-theoretic approach to the study of Weyl characters will be presented, emphasizing Weyl characters as objects that are manifested in natural, attractive, combinatorial ways.