F4 (mathematics)

The compact real form of F₄ is the isometry group of a 16-dimensional Riemannian manifold known as the octonionic projective plane OP². This can be seen systematically using a construction known as the magic square, due to Hans Freudenthal and Jacques Tits.

There are 3 real forms: a compact one, a split one, and a third one. They are the isometry groups of the three real Albert algebras.

The F₄ Lie algebra may be constructed by adding 16 generators transforming as a spinor to the 36-dimensional Lie algebra so(9), in analogy with the construction of E₈.

In older books and papers, F₄ is sometimes denoted by E₄.

Algebra

Dynkin diagram

The Dynkin diagram for F₄ is: .

Weyl/Coxeter group

Its Weyl/Coxeter group G = W(F₄) is the symmetry group of the 24-cell: it is a solvable group of order 1152. It has minimal faithful degree μ(G) = 24, which is realized by the action on the 24-cell. The group has ID (1152,157478) in the small groups library.

Cartan matrix

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

F4 lattice

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.

Roots of F4

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 (±1/2, ±1/2, ±1/2, ±1/2).

Simple roots

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

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

The Hasse diagram for the F₄ root poset is shown below right.

F4 polynomial invariant

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:

${\displaystyle M=\left[\begin{array}{ccc}x& \overline{Z}& Y\\ Z& y& \overline{X}\\ \overline{Y}& X& z\end{array}\right]}$

The set of polynomials defines a 24-dimensional compact surface (the 24-dimensional isoparametric hypersurface in the unit sphere ${\displaystyle C_2=1}$ with three distinct principal curvatures, E. Cartan, 1939).