## What is a representation?Symmetry is a fundamental principle of nature, and behind anything symmetric is a mathematical object called a Lie group. Lie groups are ubiquitous, and mathematicians study Lie groups both for their inherent beauty and with applications to other fields in mind.Every Lie group can appear in various guises: these are called representations. More precisely, a representation of a group is a realization of that group inside a group of matrices. For example here is the circle sitting inside 2x2 matrices;
Mathematicians would like to understand all
representations of E
There are very precise intricate relationships between these 453,060
representations. This is sometimes referred to as character theory, by
analogy with the character table of a finite group.
This information is encoded in a matrix of size 453,060, with
205,263,363,600 (205 billion) entries. These are known to
mathematicians as To find this information mathematicians must solve a fantastically complicated set of equations. Part of the challenge was not knowing beforehand the size of the answer (the number of non-zero entries). Until recently finding this matrix was considered to be impractical, even with the world's most powerful computers. Recent advances in mathematics and computer science have converged and brought this problem to the very edge of what it is possible to compute.
The Lie group G |