Editing Root system (section)

==Elementary consequences of the root system axioms==
[[Image:Integrality of root systems.svg|class=skin-invert-image|thumb|500px|right|The integrality condition for <math>\langle\beta, \alpha \rangle</math> is fulfilled only for ''β'' on one of the vertical lines, while the integrality condition for <math>\langle\alpha, \beta \rangle</math> is fulfilled only for ''β'' on one of the red circles. Any β perpendicular to ''α'' (on the ''Y'' axis) trivially fulfills both with 0, but does not define an irreducible root system. <br>Modulo reflection, for a given ''α'' there are only 5 nontrivial possibilities for ''β'', and 3 possible angles between ''α'' and ''β'' in a set of simple roots. Subscript letters correspond to the series of root systems for which the given ''β'' can serve as the first root and α as the second root (or in ''F''<sub>4</sub> as the middle 2 roots).]]

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The integrality condition also means that the ratio of the lengths (magnitudes) of any two non-perpendicular roots cannot be 2 or greater, since otherwise either the projection of the shorter root onto the longer root will be less than half as long as the longer root, or the shorter root will be exactly half the longer root or its negative.
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The cosine of the angle between two roots is constrained to be one-half of the square root of a positive integer. This is because <math> \langle \beta, \alpha \rangle</math> and <math>\langle \alpha, \beta \rangle</math> are both integers, by assumption, and
<math display=block> \begin{align}
\langle \beta, \alpha \rangle  \langle \alpha, \beta \rangle
&= 2 \frac{(\alpha,\beta)}{(\alpha,\alpha)} \cdot 2 \frac{(\alpha,\beta)}{(\beta,\beta)} \\
&= 4 \frac{(\alpha,\beta)^2}{\vert \alpha \vert^2 \vert \beta \vert^2} \\
&= 4 \cos^2(\theta)\\
&= (2\cos(\theta))^2 \in \mathbb{Z}.
\end{align}</math>

Since <math>2\cos(\theta) \in [-2,2]</math>, the only possible values for <math>\cos(\theta)</math> are <math>0, \pm \tfrac{1}{2}, \pm\tfrac{\sqrt{2}}{2}, \pm\tfrac{\sqrt{3}}{2}</math> and <math>\pm\tfrac{\sqrt{4}}{2} = \pm 1</math>, corresponding to angles of 90°, 60° or 120°, 45° or 135°, 30° or 150°, and 0° or 180°. Condition 2 says that no scalar multiples of ''α'' other than 1 and −1 can be roots, so 0 or 180°, which would correspond to 2''α'' or  −2''α'', are out. The diagram at right shows that an angle of 60° or 120° corresponds to roots of equal length, while an angle of 45° or 135° corresponds to a length ratio of <math>\sqrt{2}</math> and an angle of 30° or 150° corresponds to a length ratio of <math>\sqrt{3}</math>.

In summary, here are the only possibilities for each pair of roots.<ref>{{harvnb|Hall|2015}} Proposition 8.6</ref>
*Angle of 90 degrees; in that case, the length ratio is unrestricted.
*Angle of 60 or 120 degrees, with a length ratio of 1.
*Angle of 45 or 135 degrees, with a length ratio of <math>\sqrt 2</math>.
*Angle of 30 or 150 degrees, with a length ratio of <math>\sqrt 3</math>.