Introduction E9F5FC

Questions FFFFC0

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There are four choice templates:

• {$ \prod_{i}(\varnothing + \leftrightarrow_{i}) $} Simplexes.
• {$ \prod_{i}(\varnothing + (\leftarrow_{i} + \rightarrow_{i})) $} Cross-polytopes.
• {$ \prod_{i}((\leftarrow_{i} + \rightarrow_{i})+\leftrightarrow_{i}) $} Cubes.
• {$ \prod_{i}(\leftarrow_{i} + \rightarrow_{i}) $} Cube slices.

The root systems of the classical Lie algebras all include the roots of {$A_{n}$}: {$\pm (x_i-x_j)$}. The other root systems all include the additional roots of {$D_{n}$}: {$\pm (x_i+x_j), i\neq j$}. Finally, {$B_{n}$} includes {$\pm x_i$} and {$C_{n}$} includes {$\pm (x_i+x_i)$}.

The Weyl group of a root system is the group generated by the reflections across the hyperplanes of the roots. So the Weyl group is given by the directions of the roots but not their lengths.

There are three ways of defining the reflections {$e_1 \Leftrightarrow -e_1$}:

• {$B_n$}: By swapping with 0. By way of the hyperplane {$\pm(0-e_1)$}. Swapping {$e_1$} with {$0$} as with the cross polytopes. This takes place in the framework {$e_1 - 0$} so the minus sign is a feature of the pairing.
• {$D_n$}: By swapping with any root {$e_2$}. By composing the hyperplanes {$\pm(e_2-e_1)$} and {$\pm(e_2--e_1)$}. Swapping {$e_1$} with {$e_2$} and then {$e_2$} with {$-e_1$}
• {$C_n$}: By swapping directly with {$-e_1$}. By way of the hyperplane {$\pm(e_1--e_1)$}. Swapping {$e_1$} with {$-e_1$} directly. Then there is one minus sign in the pairing, and one carried by {$-e_1$}.

Pairing template

The sign can be attributed to the element or to the pairing template. So try to think that through.

Consider {$X-Y$} as a template where the minus sign imposes an ordering.

• {$A_n$}: {$X, Y$} are taken from {${e_1,e_2 \dots e_n}$} and {$X \neq Y$} is ruled out by {$X-Y=0$}.

Consider {${X,Y}$} as a template where the set indicates inclusion with no ordering.

• {$B_n$}: {$X, Y$} are taken from {${\dots -e_2,-e_1,e_1,e_2 \dots e_n}$} and {$X$} and {$Y$} may refer to the same number so long as they have the same sign, in which case they are understood to coincide.
• {$D_n$}: {$X, Y$} are taken from {${\dots -e_2,-e_1,e_1,e_2 \dots e_n}$} and {$X$} and {$Y$} must refer to different numbers.
• {$C_n$}: {$X, Y$} are taken from {${\dots -e_2,-e_1,e_1,e_2 \dots e_n}$} and {$X$} and {$Y$} may refer to the same number so long as they have the same sign, in which case they are understood to add.

Not satisfactory. :(

Orthogonalities

• Real (orthogonal), complex (unitary), quaternion (symplectic)

Readings

• Geometric shape for higher structures