Just Four Dimensions?

Wednesday, March 21st, 2007

On E8 and symmetry

At the most basic level, the E8 calculation is an investigation of symmetry. Mathematicians invented the Lie groups to capture the essence of symmetry: underlying any symmetrical object, such as a sphere, is a Lie group.

Lie groups come in families. The classical groups A1, A2, A3, … B1, B2, B3, … C1, C2, C3, … and D1, D2, D3, … rise like gentle rolling hills towards the horizon. Jutting out of this mathematical landscape are the jagged peaks of the exceptional groups G2, F4, E6, E7 and, towering above them all, E8. E8 is an extraordinarily complicated group: it is the symmetries of a particular 57-dimensional object, and E8 itself is 248-dimensional!

To describe the new result requires one more level of abstraction. The ways that E8 manifests itself as a symmetry group are called representations. The goal is to describe all the possible representations of E8. These representations are extremely complicated, but mathematicians describe them in terms of basic building blocks. The new result is a complete list of these building blocks for the representations of E8, and a precise description of the relations between them, all encoded in a matrix with 205,263,363,600 entries.

Nobody seems to be writing about what all this means.

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