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Orthogonal Idempotents in Symmetric Tensor Powers of Composition Algebras

This paper explicitly constructs complete sets of primitive orthogonal idempotents within the $n$ -th symmetric powers of the Hamilton quaternion and Cayley octonion algebras, as well as their complexifications and specific associative subalgebras, providing exact counts based on the parity of $n$ .

Original authors: Aharon Razon

Published 2026-04-14

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CC BY 4.0

Original authors: Aharon Razon

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master architect trying to build a massive, complex cathedral out of a very strange, magical type of Lego brick. These aren't ordinary bricks; they are Quaternions (4-dimensional numbers) and Octonions (8-dimensional numbers). They are the building blocks of a universe that exists in more dimensions than we can easily visualize.

The paper you provided is a blueprint for taking these magical bricks, stacking them up in huge, symmetrical towers (called Symmetric Tensor Powers), and then figuring out exactly how to break those towers down into their smallest, indivisible, and perfectly unique pieces.

Here is the breakdown of the paper's journey, using simple analogies:

1. The Problem: The Giant Symmetrical Tower

Imagine you have a pile of $n$ magical bricks. You want to build a tower where the order of the bricks doesn't matter (if you swap two bricks, the tower looks the same). This is the Symmetric Tensor Power.

Mathematicians know that this giant tower isn't just one solid block. It's actually a collection of smaller, distinct rooms (called Simple Components) glued together. The paper starts by acknowledging that we already know where the walls between these rooms are. We have the "Central Idempotents"—think of these as the master keys that can unlock a specific room and close off the rest of the building.

2. The Goal: Finding the "Atoms" of the Rooms

Knowing where the rooms are is good, but the author wants to go deeper. Inside each room, there are even smaller, indivisible pieces called Primitive Orthogonal Idempotents.

Orthogonal: These pieces don't overlap. If you have one, you can't have the other at the same time. They are like distinct, non-touching islands.
Primitive: These are the smallest possible pieces. You can't break them down any further.
Complete Set: If you gather all these tiny islands together, they perfectly fill up the entire room (or the whole building).

The author's mission is to write down the exact recipe (the formula) for creating every single one of these tiny islands for any size tower ( $n$ ).

3. The Tools: The Magic Ingredients

To build these islands, the author uses a few special ingredients:

The Quaternions ( $H$ ): Think of these as a 4D version of complex numbers. They have a "real" part and three "imaginary" parts ( $i, j, k$ ).
The Octonions ( $O$ ): An even wilder 8D version.
The "Square" ( $\square$ ): A special combination of two bricks that acts like a filter or a lens.
Complex Numbers: The author often steps into the "Complex World" (adding $\sqrt{-1}$ ) to make the math easier, finds the pieces there, and then translates them back to the real world.

4. The Discovery: The Recipes

The paper provides three main "recipes" (Theorems A, B, and C) depending on the situation:

Recipe 1 (The Complex Version): If you are allowed to use complex numbers, the recipe is surprisingly elegant. You take a specific mix of "positive" and "negative" imaginary bricks, arrange them in a line, and symmetrize them. It's like making a perfect sandwich where the bread and filling are arranged in a specific pattern.
Recipe 2 (The Real Version, Odd $n$ ): If the tower has an odd number of layers and you must stay in the "Real World," the recipe involves a bit of a dance. You take your complex sandwiches, combine them with their "mirror images" (conjugates), and use a special formula (Waring's formula) to turn them back into real bricks.
Recipe 3 (The Real Version, Even $n$ ): If the tower has an even number of layers, it gets a bit more complicated. You have to split the tower down the middle and use a "splitting key" (involving a specific direction $e$ ) to separate the pieces into two distinct groups.

5. The Octonion Twist

The paper also looks at what happens when you take these Quaternion towers and place them inside the even bigger Octonion towers.

Imagine the Quaternion tower is a small house. The Octonion tower is a massive city.
The author shows that the small house fits perfectly inside a specific district of the city.
They then provide a new set of keys (idempotents) that work specifically for the part of the city where the house sits.

6. Why Does This Matter? (The "So What?")

You might ask, "Who cares about breaking down these mathematical towers?"

Think of it like cracking a safe.

In physics and engineering, complex systems (like quantum mechanics or signal processing) are often described by these algebras.
To solve a problem in such a system, you often need to "diagonalize" it—break it down into its simplest, independent parts so you can solve each part separately.
This paper gives you the exact list of keys to open every single lock in these specific types of mathematical safes. Without this list, you'd be guessing. With it, you can systematically dismantle the most complex structures built from Quaternions and Octonions.

Summary Analogy

Imagine you have a giant, glowing, multi-colored crystal (the Symmetric Tensor Power).

Previous work told us the crystal is made of different colored layers.
This paper tells us exactly how to cut each colored layer into tiny, perfect, non-overlapping diamonds (the primitive idempotents).
It gives you the laser cutter settings (the formulas) to do this for any size crystal, whether it's made of 4D bricks (Quaternions) or 8D bricks (Octonions), and whether you are working in the real world or the complex world.

The author, Aharon Razon, has essentially written the ultimate instruction manual for disassembling these high-dimensional mathematical structures into their most fundamental, usable parts.

1. Problem Statement

The paper addresses the structural decomposition of symmetric tensor powers ( $Sym^n C$ ) of composition algebras, specifically the Hamilton quaternion algebra ( $H$ ) and the Cayley octonion algebra ( $O$ ) over the real numbers $\mathbb{R}$ .

While previous work (notably by the author and others in references [3], [4], [5]) established that $Sym^n C$ decomposes into a direct sum of central simple algebras, the specific construction of a complete set of primitive orthogonal idempotents within these components remained an open problem.

The Goal: To explicitly construct a complete set of primitive orthogonal idempotents for:
1. The components $S^{(n)}_m H$ of the symmetric power of quaternions.
2. The complexified versions $S^{(n)}_m H \otimes_{\mathbb{R}} \mathbb{C}$ .
3. Specific associative subalgebras of the symmetric power of octonions, specifically $Sym^n H \cdot e^{(n)}_{m,O}$ , where $e^{(n)}_{m,O}$ are central idempotents derived from the octonion structure.

2. Methodology

The author employs a combination of representation theory, combinatorial algebra, and explicit tensor calculus.

Decomposition Framework: The paper relies on the known decomposition of $Sym^n C$ into simple components $S^{(n)}_m C$ indexed by integers $m$ from $\lceil n/2 \rceil$ to $n$ . These components are isomorphic to matrix algebras over $\mathbb{R}$ or $H$ (via the Wedderburn-Artin theorem).
Central Idempotents: The construction builds upon previously defined central idempotents $e^{(n)}_{m,C}$ which isolate the simple components. The author utilizes the recursive properties of these idempotents and the element $\Delta_C$ (a specific symmetric tensor related to the norm form).
Complexification Strategy:
- For the complexified case ( $Sym^n H \otimes \mathbb{C}$ ), the author introduces a basis of the complexified complex numbers $a = \frac{1}{2}(1 + i e_1)$ and its conjugate $a^c$ .
- A key technique involves symmetrizing tensor products of these elements: $(a^{\otimes k} \otimes (a^c)^{\otimes (n-k)})^\vee$ , where $\vee$ denotes the symmetrization operator (averaging over the symmetric group $S_n$ ).
Real Case Construction:
- For the real case ( $Sym^n H$ ), the author derives real idempotents by taking specific linear combinations of the complex idempotents to ensure they lie in the real algebra.
- This involves using the element $\square = \frac{1}{2}(1 \otimes 1 + e_1 \otimes e_1)$ and Waring's formula to express sums of powers in terms of elementary symmetric polynomials.
Embedding and Restriction: The paper exploits the embedding $H \subset O$ . It constructs idempotents in the subalgebra $Sym^n H \cdot e^{(n)}_{m,O}$ by restricting the idempotents of $Sym^n H$ using the central idempotents of the octonion algebra.
Combinatorial Tools: The proofs utilize combinatorial lemmas (including Noga Alon's polynomial lemma) to prove that certain constructed elements are non-zero and orthogonal.

3. Key Contributions and Results

A. Idempotents in Quaternion Symmetric Powers ( $Sym^n H$ )

The paper provides explicit formulas for primitive orthogonal idempotents in the components $S^{(n)}_m H$ and their complexifications.

Complex Case ( $Sym^n H \otimes \mathbb{C}$ ):
- The set $\left\{ \binom{n}{k} (a^{\otimes k} \otimes (a^c)^{\otimes (n-k)})^\vee e^{(n)}_{\ell, H} \mid k \in \{n-\ell, \dots, \ell\} \right\}$ forms a complete set of primitive orthogonal idempotents.
- Count: For a fixed $\ell$ , there are $2\ell - n + 1$ such idempotents.
- Total Count: Summing over all $\ell$ , the total number of primitive idempotents in $Sym^n H \otimes \mathbb{C}$ is $\frac{1}{4}(n+2)^2$ (if $n$ is even) or $\frac{1}{4}(n+1)(n+3)$ (if $n$ is odd).
Real Case ( $Sym^n H$ ):
- The author constructs real idempotents by combining the complex ones.
- If $n$ is odd: The idempotents are formed by summing conjugate pairs of the complex idempotents. The count for a component $S^{(n)}_\ell H$ is $\frac{2\ell - n + 1}{2}$ .
- If $n$ is even: The construction is more complex, involving a specific term for the middle index and splitting other terms using the element $e_2$ (a quaternion unit).
- Total Count: The total number of primitive idempotents in $Sym^n H$ is $\frac{1}{4}(n+2)^2$ (if $n$ is even) or $\frac{1}{8}(n+1)(n+3)$ (if $n$ is odd).

B. Idempotents in Octonion Subalgebras

The paper extends these results to the associative subalgebras of the octonion symmetric powers, specifically $Sym^n H \cdot e^{(n)}_{m,O}$ .

Structure: These subalgebras are isomorphic to the direct sum of quaternion components $S^{(n)}_\ell H$ for $\ell$ ranging from $\lceil n/2 \rceil$ to $m$ .
Explicit Construction: The idempotents are obtained by multiplying the quaternion idempotents by the octonion central idempotent $e^{(n)}_{m,O}$ .
Total Counts:
- For the complexified subalgebra: $\frac{1}{24}(n+2)(n+3)(n+4)$ (even $n$ ) or $\frac{1}{24}(n+1)(n+3)(n+5)$ (odd $n$ ).
- For the real subalgebra: $\frac{1}{24}(n+2)(n+3)(n+4)$ (even $n$ ) or $\frac{1}{48}(n+1)(n+3)(n+5)$ (odd $n$ ).

C. Example and Verification

The paper provides a concrete example for $n=2, m=2$ . It explicitly calculates the idempotents in $Sym^2 H \cdot e^{(2)}_{2,O}$ and demonstrates how they relate to a maximal set of idempotents in the octonion algebra $T_2 O$ , showing that the quaternion-derived set is a coarser decomposition (some octonion idempotents sum to a single quaternion idempotent).

4. Significance

Explicit Construction: Prior to this work, the existence of such idempotents was known via abstract structure theorems (Wedderburn-Artin), but explicit formulas were lacking. This paper provides the first explicit, constructive formulas for these idempotents in terms of tensor products of basis elements.
Bridge between Algebras: The work clarifies the structural relationship between the symmetric powers of quaternions and octonions, showing how the idempotents of the former embed into the latter.
Computational Utility: These explicit idempotents are crucial for computational representation theory, allowing for the decomposition of modules and the calculation of invariants in non-associative and non-commutative settings.
Generalization: The results are formulated in a way that suggests applicability to rational fields ( $\mathbb{Q}$ ) and other composition algebras, as noted in Remark 4.3.

In summary, Aharon Razon successfully resolves the problem of explicitly constructing primitive orthogonal idempotents for symmetric tensor powers of quaternions and specific octonion subalgebras, providing precise combinatorial formulas and counting the total number of such idempotents based on the parity of $n$ .