What we choose to forget on the way down
Every projection in the exceptional hierarchy isn't just "reducing dimensions." It's deciding what doesn't matter for the purpose at hand. This is what the transformations actually do, and what each one costs us.
Imagine you're trying to describe a complicated object — say, a person — to someone who's never seen them.
You could describe every molecule. Every hair. Every freckle. That's the full description: maximally detailed, maximally complex.
Or you could say: "tall, dark hair, friendly smile." That's a compressed description: you've kept what matters, discarded what doesn't.
But here's the question: what makes something "matter"?
The exceptional Lie algebras give us a mathematically optimal answer. Each one describes a different "level of detail" — and the projections between them are the unique structure-preserving ways to reduce that detail.
The hierarchy E₈ → E₇ → E₆ → F₄ → G₂ → S⁷ isn't arbitrary. These are the only exceptional structures. The dimensions are forced. And the transformations between them have specific mathematical meanings.
Let's see what each one actually does.
When a large symmetry "breaks" into a smaller one, the pieces don't vanish. They reorganize.
A branching rule tells you exactly how. It says: "When you restrict from the big algebra to the small one, the representation decomposes like this."
You've seen this in quantum mechanics. A spin-2 particle (5 states) becomes m = -2, -1, 0, +1, +2 when you choose a quantization axis. The rotational symmetry "breaks" to just rotations around that axis. The 5 states split into pieces labeled by the remaining quantum number.
The numbers in a branching rule tell you:
The coefficients — things like 1/√2 — tell you how the pieces combine. When you see 1/√2, it means "equal superposition of two things."
The first step breaks 248 dimensions into three pieces.
We choose a specific E₈ root — call it α = (1, -1, 0, 0, 0, 0, 0, 0) — and declare it "special." The E₇ subalgebra is exactly those E₈ generators that commute with α.
Mathematically: we select the 126 E₈ roots that are orthogonal to α. These, plus 7 of the 8 Cartan generators, give us E₇'s 133 dimensions.
The SU(2) symmetry that rotates between α and its orthogonal complement. This is the "(1, 3)" — one dimension that becomes an SU(2) triplet. We've chosen a "direction" in the 8D Cartan, and can no longer rotate freely.
The E₇ adjoint (133, 1) — all the structure that doesn't "care" about α. Plus the 56-dimensional fundamental rep of E₇, appearing as a doublet under the remaining SU(2).
133 + 112 + 3 = 248. Everything is accounted for — just reorganized.
Now we drop from 133 to 78 dimensions by ignoring a "charge."
E₆ sits inside E₇ as the subalgebra that's neutral under a U(1) subgroup. We pick a direction β in E₇'s Cartan and keep only roots with ⟨root, β⟩ = 0.
The subscripts (0, +2, -2) are U(1) charges. The 78₀ has charge zero — it's the E₆ adjoint. The 27₊₂ and 27̄₋₂ have nonzero charges — they transform.
The ability to distinguish "charge +2" from "charge -2" states. The U(1) is like an overall phase — we're declaring that physics shouldn't depend on it.
The E₆ adjoint (78 dimensions) — all transformations that are "charge-neutral." Plus we can still see the 27-dimensional fundamental rep, just not its charge.
In QED, you can work with "charge-neutral" combinations only: electron-positron pairs, photons. The U(1) phase becomes unphysical. This projection does the same thing: forget the phase, keep the structure.
This is the most beautiful transformation: we fold E₆'s Dynkin diagram onto itself.
E₆ has an outer automorphism σ — a symmetry of its structure that swaps nodes 1 ↔ 5 and 2 ↔ 4 in the Dynkin diagram. F₄ is the subalgebra that's fixed by this symmetry.
For each pair of roots (α, σα), we form the symmetric combination:
The coefficient 1/√2 appears because we're normalizing the sum of two unit vectors. It's the signature of an equal superposition.
The distinction between 27 and 27̄ — the "complex conjugate" representations. In E₆, these are different. In F₄, they're identified. We've declared that matter and antimatter "look the same" at this level.
The 52-dimensional F₄ adjoint — everything that's symmetric under the flip. F₄ is the automorphism group of the Albert algebra (3×3 Hermitian octonion matrices).
Now we extract the octonion structure from the Albert algebra.
F₄ acts on 3×3 Hermitian octonion matrices. G₂ is the automorphism group of the octonions themselves. This projection forgets the "3×3 matrix structure" and keeps only the octonion multiplication rules.
The G₂ roots are found by projecting F₄ roots onto a 2D subspace (the G₂ Cartan):
Of the 48 F₄ roots, exactly 12 project to distinct nonzero positions. These become the 12 G₂ roots.
The full Albert algebra structure — how octonions combine into 3×3 matrices. The 7-dimensional representations that appear as doublets and singlets are "orthogonal" to G₂ — they transform under the auxiliary SU(2) we're ignoring.
The G₂ adjoint (14 dimensions) — the pure octonion automorphisms. This is the structure that preserves the multiplication table e₁ × e₂ = e₃, etc.
Like going from matrix mechanics to wave mechanics: you lose the explicit matrix representation but keep the underlying algebraic structure. The physics is the same, the formalism is simpler.
The final step is different: we leave Lie algebras entirely and enter geometry.
G₂ is the group of transformations that preserve octonion multiplication. It acts on the 7-dimensional space of imaginary octonions (Im(𝕆) = span{e₁, ..., e₇}).
The projection uses G₂'s 14D Lie algebra basis — specifically, how each basis element acts on a reference vector in Im(𝕆). We're asking: "What does this symmetry do to a point?"
The Lie algebra structure itself. S⁷ is a manifold, not an algebra. There's no "bracket" on S⁷. We've gone from transformations to the space being transformed.
The 7 imaginary directions — the space that G₂ acts on. This is the "stage" rather than the "actors." Unit vectors in this space form S⁷, the 7-sphere.
7 dimensions. One for each imaginary octonion. One for each colony.
E₈ appears twice in this architecture: once at the top (as a 248D Lie algebra), and once at the bottom (as an infinite 8D lattice for quantization).
After compressing through the hierarchy, we need to discretize. Neural networks ultimately output real numbers, but for storage, communication, and stability, we want discrete codes.
The E₈ lattice is the optimal way to do this in 8 dimensions (proven by Viazovska, Fields Medal 2022). Every 8D vector maps to its nearest lattice point.
The lattice is infinite. The "240 roots" are just the closest points to the origin (the kissing number). But quantization uses the whole lattice.
The encoding: Each lattice point is stored as 8 half-step integers, varint-encoded. Small coordinates = fewer bytes. This is entropy-efficient, not a fixed 240-way lookup.
Like measuring position in a crystal. The atoms form an infinite lattice. Any electron gets "snapped" to the nearest lattice site (in the tight-binding approximation). The continuous becomes discrete.
Each step down the hierarchy is a decision about what matters.
Break rotational symmetry. Declare one axis special. Physical: Pick a quantization axis.
Ignore an overall U(1) charge. Physical: Work in gauge-invariant quantities.
Declare that 27 and 27̄ are the same. Physical: Matter and antimatter are equivalent labels.
Keep only the underlying number system. Physical: Abstract away the representation.
Stop caring about transformations; care about position. Physical: From symmetry group to state space.
The exceptional hierarchy exists because mathematics allows exactly five structures outside the infinite classical families.
Their dimensions (248, 133, 78, 52, 14) aren't arbitrary — they're forced by root systems and Killing-Cartan classification.
The branching rules (how they nest) aren't arbitrary — they're forced by representation theory and subalgebra embeddings.
The coefficients (the √2s and 1s) aren't arbitrary — they're forced by orthonormality and the geometry of the root systems.
This is what it means to compress while preserving structure. Each level keeps what's essential and discards what's not — for that level's purpose.
Understanding is compression. Compression is choice. The mathematics just tells us which choices are coherent.
The mirror that chooses what to reflect.