Reflection Symmetry

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Section 5.4 Reflection Symmetry

We have shown in Sections 5.2–5.3 that each pair of roots \(\pm\alpha\) generates a copy of \(\su(2)\) (really \(\sl(2,\RR)\)), so that the root diagram consists of a collection of representations of \(\su(2)\text{.}\) But we know from Section 4.2 what all such representations are, namely one-dimensional spaces with eigenvalues separated by integers and centered around zero.

Each such representation corresponds to a line in the root diagram parallel to the corresponding copy of \(\su(2)\text{,}\) that is, parallel to the root whose eigenvalues are being considered. What does it mean for these lines to be “centered” around zero?

Construct the perpendicular bisector of the line representing the pair of roots \(\pm\alpha\text{,}\) which is therefore also perpendicular to each representation of the copy of \(\su(2)\) corresponding to \(\alpha\text{.}\) The “zero” of the representation (which may or may not correspond to a root) is where these lines intersect. In other words, each representation must be symmetric about this perpendicular line.

We have therefore shown that the reflection of any root \(\beta\) about the line perpendicular to any other root \(\alpha\) must also be a root! Using orthogonal projections, we conclude that

\begin{equation} r_\alpha(\beta) = \beta - 2\frac{\beta\cdot\alpha}{\alpha\cdot\alpha} \alpha\tag{5.4.1} \end{equation}

must also be a root. Furthermore, since \(\alpha\) must raise the “\(\alpha\)” eigenvalue by \(1\text{,}\) the roots in the representation must be separated by integer multiples of \(\alpha\text{,}\) so that

\begin{equation} 2\frac{\beta\cdot\alpha}{\alpha\cdot\alpha} \in \ZZ .\tag{5.4.2} \end{equation}

Moreover, the representation theory of \(\su(2)\) tells us that all of the steps (if any) “between” \(\beta\) and \(r_\alpha(\beta)\) must also be roots. In particular, if \(\alpha\cdot\beta\ge0\text{,}\) then \(\beta-\alpha\) must be a root, whereas if \(\alpha\cdot\beta\le0\text{,}\) then \(\beta+\alpha\) must be a root. (If \(\alpha\cdot\beta=0\text{,}\) then \(r_\alpha(\beta)=\beta\text{,}\) and there are no nontrivial reflections.)