# Canonical bases, reduced expressions and normal forms

## EPSRC Grant GR/R17546/01

This project ran from 22/1/01 to 21/9/03, and was funded by the UK Engineering and Physical Sciences Research Council, 52708 pounds.

Research Asssociate: Ruth Corran.

Summary of Project (non-specialist)

A group is a way of measuring the symmetries of a mathematical object. For example a square has a symmetry group with 8 elements: 4 reflections and 4 rotations (including 0 degrees). The symmetries of the set of numbers 1,2,...,n, which has no structure, are just the possible reorderings (permutations) of the numbers. Nevertheless, these form a group which is of interest in many fields of mathematics. The most basic examples of these symmetries are those which just switch the numbers 1 and 2, or just switch 2 and 3, and so on. All elements of the symmetry group can be written as combinations of such elementary symmetries (known as reduced words) and there is a unique element w0 which needs the most elementary symmetries, known as the fundamental element of the symmetry group. The collection of all reduced words for w0 (and generalisations to other symmetry groups) formed the main object of study of this project. We have found a way to describe the geometry of this collection, in particular, showing that in some cases a beautiful sphere can be found inside it; we conjecture that this should hold in all symmetry groups in the same family (known as finite Coxeter groups).

We have also found natural ways of writing down reduced words for w0 (normal forms). Other parts of the project have included the study of certain monoids (symmetries, in a sense, related to the above symmetries, and arising from the consideration of braiding) from the point of view of automata - i.e. describing them using theoretical computing devices. We have also developed connections between the reduced expressions described above and the theory of quantum groups and their canonical bases - algebraic objects arising originally from quantum theory but with many applications in the area of Lie theory in mathematics.

Objectives

• To calculate the regions of linearity of reparametrization functions arising from Lusztig and Kashiwara's approaches to the canonical basis for a quantum group
• To generalise the canonical forms for reduced expressions for the fundamental element in a Weyl group of finite type to arbitrary finite type, in a natural way.
• To find the topological structure of the graph of commutation classes for the fundamental element in a Weyl group of finite type.
• To find Garside structures for braid groups associated to complex reflection groups.
• To develop connections between the multiplicativity properties of the dual canonical basis and Lusztig cones and the parametrizations of the (dual) canonical basis.
• To show that singular braid monoids are automatic.

Papers

The unpublished article above by R. Corran and D. Bessis led to the following article:

[ University of Leeds ] [ School of Mathematics ] [ Department of Pure Maths ]
Robert Marsh. Last modified 5 February 2008.