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F₄ (mathematics)

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52-dimensional exceptional simple Lie group

For the field, seeFinite field § Field with four elements.

Algebraic structure →Group theory
Group theory

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simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable
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Classification of finite simple groups

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Integers ( $\mathbb {Z}$ )
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Inmathematics,F₄ is aLie group and also itsLie algebraf₄. It is one of the five exceptionalsimple Lie groups. F₄ has rank 4 and dimension 52. The compact form is simply connected and itsouter automorphism group is thetrivial group. Itsfundamental representation is 26-dimensional.

The compact real form of F₄ is theisometry group of a 16-dimensionalRiemannian manifold known as theoctonionic projective planeOP². This can be seen systematically using a construction known as themagic square, due toHans Freudenthal andJacques Tits.

There are3 real forms: a compact one, a split one, and a third one. They are the isometry groups of the three realAlbert algebras.

The F₄ Lie algebra may be constructed by adding 16 generators transforming as aspinor to the 36-dimensional Lie algebraso(9), in analogy with the construction ofE₈.

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

Algebra

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Dynkin diagram

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TheDynkin diagram for F₄ is:.

Weyl/Coxeter group

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ItsWeyl/Coxeter groupG =W(F₄) is thesymmetry group of the24-cell: it is asolvable group of order 1152. It has minimal faithful degreeμ(G) = 24,^[1] which is realized by the action on the24-cell. The group has ID (1152,157478) in thesmall groups library.

Cartan matrix

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\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

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The F₄lattice is a four-dimensionalbody-centered cubic lattice (i.e. the union of twohypercubic lattices, each lying in the center of the other). They form aring called theHurwitz quaternion ring. The 24 Hurwitz quaternions of norm 1 form the vertices of a24-cell centered at the origin.

Roots of F₄

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The 24 vertices of24-cell (red) and 24 vertices of its dual (yellow) represent the 48 root vectors of F₄ in thisCoxeter plane projection

The 48root vectors of F₄ can be found as the vertices of the24-cell in two dual configurations, representing the vertices of adisphenoidal 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

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

{\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}}

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

Hasse diagram of F₄root poset with edge labels identifying added simple root position

F₄ polynomial invariant

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Just as O(n) is the group of automorphisms which keep the quadratic polynomialsx² +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).

C_{1}=x+y+z

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

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

Wherex,y,z are real-valued andX,Y,Z are octonion valued. Another way of writing these invariants is as (combinations of) Tr(M), Tr(M²) and Tr(M³) of thehermitian octonion matrix:

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 (the 24-dimensional isoparametric hypersurface in the unit sphere $C_{2}=1$ with three distinct principal curvatures, E. Cartan, 1939).

Representations

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The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by theWeyl character formula. The dimensions of the smallest irreducible representations are (sequenceA121738 in theOEIS):

1, 26, 52, 273, 324, 1053 (twice), 1274, 2652, 4096, 8424, 10829, 12376, 16302, 17901, 19278, 19448, 29172, 34749, 76076, 81081, 100776, 106496, 107406, 119119, 160056 (twice), 184756, 205751, 212992, 226746, 340119, 342056, 379848, 412776, 420147, 627912...

The 52-dimensional representation is theadjoint representation, and the 26-dimensional one is the trace-free part of the action of F₄ on the exceptionalAlbert algebra of dimension 27.

There are two non-isomorphic irreducible representations of dimensions 1053, 160056, 4313088, etc. Thefundamental representations are those with dimensions 52, 1274, 273, 26 (corresponding to the four nodes in theDynkin diagram in the order such that the double arrow points from the second to the third).

Embeddings of the maximal subgroups of F₄ up to dimension 273 with associated projection matrix are shown below.

References

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^Saunders, Neil (2014). "Minimal Faithful Permutation Degrees for Irreducible Coxeter Groups and Binary Polyhedral Groups".arXiv:0812.0182 [math.GR].

Adams, J. Frank (1996).Lectures on exceptional Lie groups. Chicago Lectures in Mathematics.University of Chicago Press.ISBN 978-0-226-00526-3.MR 1428422.
John Baez,The Octonions, Section 4.2: F₄,Bull. Amer. Math. Soc.39 (2002), 145-205. Online HTML version athttp://math.ucr.edu/home/baez/octonions/node15.html.
Chevalley C, Schafer RD (February 1950)."The Exceptional Simple Lie Algebras F(4) and E(6)".Proc. Natl. Acad. Sci. U.S.A.36 (2):137–41.Bibcode:1950PNAS...36..137C.doi:10.1073/pnas.36.2.137.PMC 1063148.PMID 16588959.
Jacobson, Nathan (1971-06-01).Exceptional Lie Algebras (1st ed.). CRC Press.ISBN 0-8247-1326-5.

v t e Exceptional Lie groups
G₂ F₄ E₆ E₇ E₈

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