F₄ (mathematics)

The Dynkin diagram for F₄ is: .

Its Weyl/Coxeter group ${\displaystyle G=W\left(\mathrm {F} _{4}\right)}$ is the symmetry group of the 24-cell: it is a solvable group of order 1152. It has minimal faithful degree ${\displaystyle \mu (G)=24}$[1] which is realized by the action on the 24-cell.

${\displaystyle \left[{\begin{array}{rrrr}2&-1&0&0\\-1&2&-2&0\\0&-1&2&-1\\0&0&-1&2\end{array}}\right]}$

The F₄ lattice is a four-dimensional body-centered cubic lattice (i.e. the union of two hypercubic lattices, each lying in the center of the other). They form a ring called the Hurwitz quaternion ring. The 24 Hurwitz quaternions of norm 1 form the vertices of a 24-cell centered at the origin.

The 24 vertices of 24-cell (red) and 24 vertices of its dual (yellow) represent the 48 root vectors of F₄ in this Coxeter plane projection

The 48 root vectors of F₄ can be found as the vertices of the 24-cell in two dual configurations, representing the vertices of a disphenoidal 288-cell if the edge lengths of the 24-cells are equal:

24-cell vertices:

• 24 roots by (±1,±1,0,0), permuting coordinate positions

Dual 24-cell vertices:

• 8 roots by (±1, 0, 0, 0), permuting coordinate positions
• 16 roots by (±½, ±½, ±½, ±½).

One choice of simple roots for F₄, , is given by the rows of the following matrix:

${\displaystyle {\begin{bmatrix}0&1&-1&0\\0&0&1&-1\\0&0&0&1\\{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}\\\end{bmatrix}}}$

Hasse diagram of F4 root poset with edge labels identifying added simple root position

Just as O(n) is the group of automorphisms which keep the quadratic polynomials x² + y² + ... invariant, F₄ is the group of automorphisms of the following set of 3 polynomials in 27 variables. (The first can easily be substituted into other two making 26 variables).

${\displaystyle C_{1}=x+y+z}$

${\displaystyle C_{2}=x^{2}+y^{2}+z^{2}+2X{\overline {X}}+2Y{\overline {Y}}+2Z{\overline {Z}}}$

${\displaystyle C_{3}=xyz-xX{\overline {X}}-yY{\overline {Y}}-zZ{\overline {Z}}+XYZ+{\overline {XYZ}}}$

Where x, y, z are real valued and X, Y, Z are octonion valued. Another way of writing these invariants is as (combinations of) Tr(M), Tr(M²) and Tr(M³) of the hermitian octonion matrix:

${\displaystyle M={\begin{bmatrix}x&{\overline {Z}}&Y\\Z&y&{\overline {X}}\\{\overline {Y}}&X&z\end{bmatrix}}}$

The set of polynomials defines a 24 dimensional compact surface.