Structure and representations of exceptional groups

From Cartan and Killing’s original classification of simple Lie groups in the 1890s, these groups have been understood to be of two rather different types: the infinite families of classical groups (related to classical linear algebra and geometry); and a finite number of exceptional groups, ranging from the 14-dimensional groups of typeG2 to the 248-dimensional groups of typeE8. Often it is possible to study all simple Lie groups at once, without reference to the classification; but for many fundamental problems, it is still necessary to treat each simple group separately. For the classical groups, such case-by-case analysis often leads to arguments by induction on the dimension (as for instance in Gauss’s method for solving systems of linear equations). This kind of structure and representation theory for classical groups brings tools from combinatorics (like the Robinson-Schensted algorithm), and leads to many beautiful and powerful results. For the exceptional groups, such arguments are not available. The groups are not directly connected to classical combinatorics. A typical example of odd phenomena associated to the exceptional groups is the non-integrable almost complex structure on the six-dimensional sphere S, derived from the group G2. What makes mathematics possible in this world is that there are only finitely many exceptional groups: some questions can be answered one group at a time, by hand or computer calculation. The same peculiarity makes the possibility of connecting the exceptional groups to physics an extraordinarily appealing one. The geometry of special relativity is governed by the ten-dimensional Lorentz group of the quadratic form of signature (3, 1). Mathematically this group is part of a family of Lorentz groups attached to signatures (p, q), for any non-negative integers p and q; there is no obvious mathematical reason to prefer the signature (3, 1). A physical theory attached to an exceptional group best of all, to the largest exceptional groups of type E8 would have no such mathematical cousins. There is only one E8.