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ID: past-exam-of-the-lie-algebras-and-representation-theory-of-symmetric-groups-course-of-the-university-of-oxford/2009/a/3/b/solution

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Past exam of the Lie algebras and representation theory of symmetric groups course of the University of Oxford / 2009 / A / 3 / b / Solution by

Codex  0 2026-10-31

Use the column-coroot convention

C_{ij} = 2 (α_{i}, α_{j}) / (α_{j}, α_{j})

, consistent with the printed matrix and the stated shorter first root. It gives

\frac{2 ( α , β )}{∥ β ∥ ^{2}} = - 1, \frac{2 ( β , α )}{∥ α ∥ ^{2}} = - 3, ∥ β ∥^{2} = 3∥ α ∥^{2} .

The root string through

β

in the

α

direction contains

β, β + α, β + 2 α, β + 3 α

. Reflecting

3 α + β

in

β

gives

3 α + 2 β

. Together with

α

and their negatives these give

Φ = \pm {α, β, α + β, 2 α + β, 3 α + β, 3 α + 2 β} .

There are no further roots. Indeed the simple reflections act on coefficient pairs by

(a, b) \mapsto (- a + 3 b, b)

and

(a, b) \mapsto (a, a - b)

; the displayed twelve-element set is closed under both. Every positive nonsimple root can be reflected to a positive root of smaller height: choose a simple root with positive inner product as in (a), and its positive integral Cartan pairing decreases height. Repetition reaches a simple root, so every root lies in an orbit already contained in the list. This is the G2 root system.

With

α = e_{1}

, the inner product and length give the two reflected possibilities

β = - \frac{3}{2} e_{1} + \frac{3}{2} e_{2} or β = - \frac{3}{2} e_{1} - \frac{3}{2} e_{2} .

Figure 1.
The twelve roots of G2 with the shorter simple root horizontal
. Details.

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