[Paper Review] Structure theory for the group algebra of the symmetric group, with applications to polynomial identities for the octonions

Murray R. Bremner, Sara Madariaga+1 more|arXiv (Cornell University)|Jul 14, 2014

Advanced Topics in Algebra16 references9 citations

TL;DR

This paper develops a computational framework using the group algebra of the symmetric group $\mathbb{F}S_n$ to study multilinear polynomial identities for the octonions over a field of characteristic 0. By applying Clifton's algorithm and Wedderburn decomposition, the authors verify that all degree-7 identities are consequences of lower-degree identities, concluding that no new identities exist in degree 7 and conjecturing that the known identities of degree $\leq 6$ generate all octonion identities in characteristic 0.

ABSTRACT

In part 1, we review the structure theory of $\mathbb{F} S_n$, the group algebra of the symmetric group $S_n$ over a field of characteristic 0. We define the images $ψ(E^λ_{ij})$ of the matrix units $E^λ_{ij}$ ($1 \le i, j \le d_λ$), where $d_λ$ is the number of standard tableaux of shape $λ$, and obtain an explicit construction of Young's isomorphism $ψ\colon \bigoplus_λM_{d_λ}(\mathbb{F}) o \mathbb{F} S_n$. We then present Clifton's algorithm for the construction of the representation matrices $R^λ(p) \in M_{d_λ}(\mathbb{F})$ for all $p \in S_n$, and obtain the reverse isomorphism $ϕ\colon \mathbb{F} S_n o \bigoplus_λM_{d_λ}(\mathbb{F})$. In part 2, we apply the structure theory of $\mathbb{F} S_n$ to the study of multilinear polynomial identities of degree $n \le 7$ for the algebra $\mathbb{O}$ of octonions over a field of characteristic 0. We compare our results with earlier work of Racine, Hentzel & Peresi, and Shestakov & Zhukavets on the identities of degree $n \le 6$. We use computational linear algebra to verify that every identity in degree 7 is a consequence of the known identities of lower degrees: there are no new identities in degree 7. We conjecture that the known identities of degree $\le 6$ generate all octonion identities in characteristic 0.

Motivation & Objective

• To develop an efficient computational method for analyzing multilinear polynomial identities in nonassociative algebras using the group algebra $\mathbb{F}S_n$.
• To apply structure theory of $\mathbb{F}S_n$—including Young tableaux, matrix units, and Wedderburn decomposition—to study identities for the octonions.
• To determine whether new polynomial identities exist in degree 7 for the octonion algebra $\mathbb{O}$ beyond those implied by lower-degree identities.
• To compare results with prior work by Racine, Hentzel & Peresi, and Shestakov & Zhukavets on identities of degree $\leq 6$.
• To conjecture that the known identities of degree $\leq 6$ generate the full $T$-ideal of octonion identities in characteristic 0.

Proposed method

• Construct Young’s isomorphism $\psi: \bigoplus_{\lambda} M_{d_\lambda}(\mathbb{F}) \to \mathbb{F}S_n$ using matrix units $E^{\lambda}_{ij}$ and standard tableaux of shape $\lambda$.
• Implement Clifton’s algorithm to compute representation matrices $R^{\lambda}(p) \in M_{d_\lambda}(\mathbb{F})$ for all $p \in S_n$, enabling efficient computation of group algebra elements.
• Use the reverse isomorphism $\phi: \mathbb{F}S_n \to \bigoplus_{\lambda} M_{d_\lambda}(\mathbb{F})$ to decompose identities into irreducible $S_n$-modules.
• Apply computational linear algebra to compare row spaces of matrices representing all identities and consequences of the alternative laws, identifying new identities.
• Use rational arithmetic and matrix rank comparisons to determine whether identities in degree 7 are independent of lower-degree identities.
• Verify that the row space of the matrix for all identities in degree 7 has the same dimension as the row space of consequences of known identities, implying no new identities exist.

Experimental results

Research questions

• RQ1 Are there any multilinear polynomial identities of degree 7 satisfied by the octonion algebra $\mathbb{O}$ that are not consequences of identities of lower degree?
• RQ2 Can the known identities of degree $\leq 6$—including the alternative laws, (R2), (HP5), and (HP6) or (SZ)—generate all polynomial identities satisfied by $\mathbb{O}$ in characteristic 0?
• RQ3 What is the multiplicity of each irreducible $S_6$-module $[\lambda]$ in the space of all multilinear identities of degree 6 for $\mathbb{O}$?
• RQ4 Is there a new identity in degree 6 that is not of the form $[f(v,w,x,y,z), u] \equiv 0$ where $f$ is a central polynomial?
• RQ5 How do the computational techniques based on $\mathbb{F}S_n$ and matrix units in the group algebra enable the verification of identities in nonassociative algebras?

Key findings

• In degree 7, the row space of the matrix representing all multilinear identities has the same dimension as the row space of consequences of the alternative laws, (R2), (HP5), and either (HP6) or (SZ), indicating no new identities exist.
• For degree 6, the multiplicity of the trivial $S_6$-module $[111111]$ in the space of all identities is 40, while in the space of consequences of known identities it is 39, revealing a single new identity in this module.
• A new degree-6 identity was explicitly constructed: $\sum_{\sigma \in S_6} \epsilon(\sigma) \left(5x_1(x_2((x_3x_4)(x_5x_6))) - x_1(x_2(x_3(x_4(x_5x_6)))) \right) \equiv 0$, which is not of the form $[f, u] \equiv 0$.
• The identity (7) can replace (HP6) or (SZ) as the new generator in degree 6, as it generates the same $T$-ideal when combined with lower-degree identities.
• The matrix $\mathrm{allmat}(\lambda)$ for $\lambda = 11111$ in degree 5 has rank 11, while $\mathrm{oldmat}(\lambda)$ has rank 10, confirming a new identity not implied by the alternative laws.
• The authors conclude that the known identities of degree $\leq 6$ generate the full $T$-ideal of polynomial identities for $\mathbb{O}$, and conjecture this holds in general for characteristic 0.