Orthogonal Idempotents in Symmetric Tensor Powers of Composition Algebras

Aharon Razon

arxiv: 2604.10319 · v2 · submitted 2026-04-11 · 🧮 math.RA

Orthogonal Idempotents in Symmetric Tensor Powers of Composition Algebras

Aharon Razon This is my paper

Pith reviewed 2026-05-10 14:59 UTC · model grok-4.3

classification 🧮 math.RA

keywords symmetric powerscomposition algebrasquaternionsoctonionsidempotentsorthogonal idempotentsprimitive idempotentsassociative subalgebras

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The pith

Explicit constructions give complete sets of primitive orthogonal idempotents in symmetric powers of quaternions and octonions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs explicit complete sets of primitive orthogonal idempotents in the nth symmetric power of the Hamilton quaternion algebra over the reals and complexes, with the number depending on whether n is even or odd. It also does analogous constructions for a subalgebra of the symmetric powers of the Cayley octonion algebra. A sympathetic reader would care because these idempotents allow the algebra to be decomposed into a direct sum of simpler ideals or components, facilitating the study of their structure and representations.

Core claim

We explicitly find a complete set of 1/4(n+2)^2 (resp. 1/4(n+1)(n+3)) primitive orthogonal idempotents in Sym^n H tensor_R C if n is even (resp. odd), where Sym^n H is the nth symmetric power of the Hamilton quaternion algebra H. We also give a complete set of 1/4(n+2)^2 (resp. 1/8(n+1)(n+3)) primitive orthogonal idempotents in Sym^n H if n is even (resp. odd). Moreover, we explicitly find a complete set of 1/24(n+2)(n+3)(n+4) (resp. 1/24(n+1)(n+3)(n+5)) primitive orthogonal idempotents in a certain associative subalgebra of Sym^n O tensor_R C if n is even (resp. odd).

What carries the argument

The explicitly listed elements in the symmetric powers that satisfy the idempotent, orthogonality, and primitivity conditions under the multiplication induced from the composition algebra.

If this is right

The symmetric powers admit a direct sum decomposition into the principal ideals generated by these idempotents.
The listed sets are complete, so they account for the full dimension of the algebra in each case.
The constructions extend from the quaternion case to an associative subalgebra in the octonion case while preserving the idempotent properties.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

These explicit bases could serve as a starting point for computing the Wedderburn decomposition or other structural invariants of the symmetric powers.
The parity-dependent counts of idempotents may reflect an underlying representation-theoretic alternation that could be tested by comparing dimensions for small n.
Similar listing techniques might generalize to other composition algebras or to symmetric powers over different base fields.

Load-bearing premise

The explicitly listed elements in the symmetric powers actually satisfy the idempotent, orthogonality, and primitivity conditions under the induced multiplication.

What would settle it

Direct computation for a specific small even n showing that the square of one listed element fails to equal itself or that two distinct listed elements have nonzero product.

read the original abstract

We explicitly find a complete set of ${1\over4}(n+2)^2$ (resp. ${1\over4}(n+1)(n+3)$) primitive orthogonal idempotents in ${\rm Sym}^n\mathbb{H}\otimes_\mathbb{R}\mathbb{C}$ if $n$ is even (resp. odd), where ${\rm Sym}^n\mathbb{H}$ is the $n^{\it th}$ symmetric power of the Hamilton quaternion algebra $\mathbb{H}$. We also give a complete set of ${1\over4}(n+2)^2$ (resp. ${1\over8}(n+1)(n+3)$) primitive orthogonal idempotents in ${\rm Sym}^n\mathbb{H}$ if $n$ is even (resp. odd). Moreover, we explicitly find a complete set of ${1\over24}(n+2)(n+3)(n+4)$ (resp. ${1\over24}(n+1)(n+3)(n+5)$) primitive orthogonal idempotents in a certain associative subalgebra of ${\rm Sym}^n\mathbb{O}\otimes_\mathbb{R}\mathbb{C}$ if $n$ is even (resp. odd), where ${\rm Sym}^n\mathbb{O}$ is the $n^{\it th}$ symmetric power of the Cayley octonion algebra $\mathbb{O}$. We also give a complete set of ${1\over24}(n+2)(n+3)(n+4)$ (resp. ${1\over48}(n+1)(n+3)(n+5)$) primitive orthogonal idempotents in a certain associative subalgebra of ${\rm Sym}^n\mathbb{O}$ if $n$ is even (resp. odd).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives explicit constructions of complete sets of primitive orthogonal idempotents in symmetric powers of quaternions and an associative subalgebra of octonions.

read the letter

This work focuses on finding explicit bases of orthogonal idempotents in symmetric tensor powers of composition algebras. For the quaternions, it gives sets of size roughly n squared over 4, split by parity of n, both over the reals and after complexification. For the octonions, it restricts to an associative subalgebra and provides similar explicit lists with cubic counts in n. What stands out is the concreteness. The abstract and the paper lay out the actual elements rather than just proving existence. That makes the result directly usable for someone wanting to compute in these algebras or decompose representations. The constructions appear to be new in this form. The literature on idempotents in composition algebras is standard, but applying it to symmetric powers with these specific counts and listings does not seem to have been done before. On the verification side, the paper must check that the listed elements square to themselves, multiply to zero when distinct, are primitive, and that their sum is the identity. The stress-test note indicates no dimension mismatch or obvious inconsistency, so if the calculations hold, the central claim is solid. A minor limitation is the restriction to quaternions and a subalgebra for octonions; it does not cover the full symmetric power of the octonions, which might be harder due to nonassociativity. This paper is aimed at researchers in non-associative algebra who need concrete tools for these specific structures. It is not going to change the broader field, but it fills a gap with explicit data. I think it deserves peer review. The explicit nature means referees can check the claims directly, and the result is self-contained.

Referee Report

2 major / 3 minor

Summary. The paper explicitly constructs complete sets of primitive orthogonal idempotents in the n-th symmetric power Sym^n H of the Hamilton quaternion algebra, both over R and after base change to C, giving 1/4(n+2)^2 such elements when n is even and 1/4(n+1)(n+3) when n is odd. Parallel explicit constructions are supplied for a certain associative subalgebra of Sym^n O (O the Cayley octonion algebra), with counts 1/24(n+2)(n+3)(n+4) (even n) and 1/24(n+1)(n+3)(n+5) (odd n) over C and adjusted counts over R. The constructions are given by direct formulas for the elements together with verification that they are idempotent, pairwise orthogonal, primitive, and sum to the unit.

Significance. If the listed elements satisfy the stated algebraic relations, the work supplies concrete, verifiable bases for the decomposition of these symmetric powers into direct sums of ideals. The explicit, parameter-free character of the listings (no fitted constants or recursive definitions) and the closed-form polynomial counts in n are genuine strengths that make the results immediately usable for further representation-theoretic or computational study of composition algebras and their symmetric powers.

major comments (2)

[§3] §3, the verification that the listed elements in Sym^n H are primitive: the argument that each corner ring e_i A e_i is a division ring (or local) relies on direct computation of the multiplication table; for general n this computation must be shown to hold uniformly rather than case-by-case, otherwise the primitivity claim for arbitrary n is not fully supported.
[§5] §5, definition of the associative subalgebra of Sym^n O: the subalgebra is described as the span of certain monomials closed under the induced multiplication, but the precise generators or the ideal-theoretic characterization used to guarantee associativity is not stated explicitly; without this, it is unclear whether the subalgebra is the largest possible or merely one convenient choice, affecting the completeness claim.

minor comments (3)

[§2] The notation for the symmetric power multiplication (induced from the composition algebra product) should be introduced once in §2 with a displayed formula rather than assumed from context.
[Table 1] Table 1 (counts for small n) contains a typographical error in the n=3 row for the real octonion case; the displayed number does not match the formula 1/48(n+1)(n+3)(n+5).
Several references to earlier work on idempotents in composition algebras are cited only by author name; full bibliographic details should be added.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation of the explicit constructions, and recommendation for minor revision. We address each major comment below and will update the manuscript accordingly.

read point-by-point responses

Referee: [§3] §3, the verification that the listed elements in Sym^n H are primitive: the argument that each corner ring e_i A e_i is a division ring (or local) relies on direct computation of the multiplication table; for general n this computation must be shown to hold uniformly rather than case-by-case, otherwise the primitivity claim for arbitrary n is not fully supported.

Authors: We appreciate the referee's comment on the primitivity argument in §3. The manuscript verifies idempotence, orthogonality, and primitivity via explicit formulas for the elements together with the standard multiplication rules in the symmetric power algebra. To strengthen the presentation for arbitrary n, we will add a uniform lemma in the revised §3 that computes the general form of products within each corner ring e_i (Sym^n H) e_i using the closed-form basis and the bilinear multiplication map. This will establish that each corner ring is a division algebra (over C) or local ring (over R) without case-by-case distinctions, confirming the primitivity claim uniformly. revision: yes
Referee: [§5] §5, definition of the associative subalgebra of Sym^n O: the subalgebra is described as the span of certain monomials closed under the induced multiplication, but the precise generators or the ideal-theoretic characterization used to guarantee associativity is not stated explicitly; without this, it is unclear whether the subalgebra is the largest possible or merely one convenient choice, affecting the completeness claim.

Authors: We thank the referee for noting the need for greater precision in the definition of the associative subalgebra in §5. The subalgebra is the linear span of a explicitly listed collection of monomials in Sym^n O that is closed under the induced multiplication; associativity follows from direct verification on this basis. In the revision we will state the generators explicitly (the symmetric powers of the unit element together with the images of a fixed imaginary basis of O satisfying the composition relations) and characterize the subalgebra as the associative subalgebra they generate inside Sym^n O. We do not claim it is maximal among associative subalgebras, but it is the natural one containing our constructed idempotents, so the completeness statement remains accurate for this subalgebra. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central claim is a direct constructive listing of explicit elements claimed to be primitive orthogonal idempotents in the symmetric powers, together with verification that they satisfy e_i^2 = e_i, e_i e_j = 0 (i≠j), primitivity, and completeness under the induced multiplication. No fitted parameters, self-referential definitions, or load-bearing self-citations appear in the stated derivation chain. The counts are presented as consequences of the explicit bases rather than inputs used to define them. The derivation is therefore self-contained against external benchmarks and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definition and multiplication rules of the quaternion and octonion algebras together with the construction of symmetric powers; no free parameters or new entities are introduced.

axioms (2)

standard math The Hamilton quaternion algebra H and Cayley octonion algebra O are the standard composition algebras over the reals with their usual multiplication tables.
Background definition of the objects whose symmetric powers are studied.
standard math Symmetric powers inherit a bilinear multiplication from the base algebra that is compatible with the symmetric product.
Standard algebraic construction used throughout.

pith-pipeline@v0.9.0 · 5616 in / 1369 out tokens · 60487 ms · 2026-05-10T14:59:37.174890+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

Alon,Combinatorial Nullstellensatz, Combinatorics, Probability and Computing8(1999), 7–29

N. Alon,Combinatorial Nullstellensatz, Combinatorics, Probability and Computing8(1999), 7–29

work page 1999
[2]

H. W. Gould,The Girard-Waring Power Sum Formulas for Symmetric Func- tions, and Fibonacci SequencesThe Fibonacci Quarterly,37(2), 13–140 (1999). DOI: 10.1080/00150517.1999.12428871

work page doi:10.1080/00150517.1999.12428871 1999
[3]

Razon,Symmetric tensor products of octonion algebras, In Jarden, Shaska (eds.), Abelian varieties and number theory, Contemporary Math.767, 161–179 (2021)

A. Razon,Symmetric tensor products of octonion algebras, In Jarden, Shaska (eds.), Abelian varieties and number theory, Contemporary Math.767, 161–179 (2021). DOI: 10.1090/conm/767

work page doi:10.1090/conm/767 2021
[4]

Razon,The structure of symmetric tensor powers of composition algebras, Journal of Algebra and its Applications, Vol

A. Razon,The structure of symmetric tensor powers of composition algebras, Journal of Algebra and its Applications, Vol. 23, No. 02 (2024). DOI: 10.1142/S0219498824500336

work page doi:10.1142/s0219498824500336 2024
[5]

Razon,The dual vector spaces of symmetric tensor powers of composition algebras, Journal of Algebra and its Applications, Vol

A. Razon,The dual vector spaces of symmetric tensor powers of composition algebras, Journal of Algebra and its Applications, Vol. 24, No. 12 (2025). DOI: 10.1142/S0219498825502779 Elta Systems Ltd., Ashdod, Israel Email address:razona@elta.co.il

work page doi:10.1142/s0219498825502779 2025

[1] [1]

Alon,Combinatorial Nullstellensatz, Combinatorics, Probability and Computing8(1999), 7–29

N. Alon,Combinatorial Nullstellensatz, Combinatorics, Probability and Computing8(1999), 7–29

work page 1999

[2] [2]

H. W. Gould,The Girard-Waring Power Sum Formulas for Symmetric Func- tions, and Fibonacci SequencesThe Fibonacci Quarterly,37(2), 13–140 (1999). DOI: 10.1080/00150517.1999.12428871

work page doi:10.1080/00150517.1999.12428871 1999

[3] [3]

Razon,Symmetric tensor products of octonion algebras, In Jarden, Shaska (eds.), Abelian varieties and number theory, Contemporary Math.767, 161–179 (2021)

A. Razon,Symmetric tensor products of octonion algebras, In Jarden, Shaska (eds.), Abelian varieties and number theory, Contemporary Math.767, 161–179 (2021). DOI: 10.1090/conm/767

work page doi:10.1090/conm/767 2021

[4] [4]

Razon,The structure of symmetric tensor powers of composition algebras, Journal of Algebra and its Applications, Vol

A. Razon,The structure of symmetric tensor powers of composition algebras, Journal of Algebra and its Applications, Vol. 23, No. 02 (2024). DOI: 10.1142/S0219498824500336

work page doi:10.1142/s0219498824500336 2024

[5] [5]

Razon,The dual vector spaces of symmetric tensor powers of composition algebras, Journal of Algebra and its Applications, Vol

A. Razon,The dual vector spaces of symmetric tensor powers of composition algebras, Journal of Algebra and its Applications, Vol. 24, No. 12 (2025). DOI: 10.1142/S0219498825502779 Elta Systems Ltd., Ashdod, Israel Email address:razona@elta.co.il

work page doi:10.1142/s0219498825502779 2025