[Paper Review] $Sp_6(2^a)$ is "Good" for the McKay, Alperin Weight, and Related Local-Global Conjectures

A. A. Schaeffer Fry|arXiv (Cornell University)|Dec 21, 2012

Finite Group Theory Research14 references3 citations

TL;DR

This paper establishes that the symplectic groups $Sp_6(2^a)$ are 'good' for the McKay, Alperin-McKay, Alperin weight, and blockwise Alperin weight conjectures for all primes $\ell$, by constructing explicit, $G$-equivariant bijections between irreducible characters and defect-zero characters in relevant subquotients. The key contribution is verifying the necessary conditions for these conjectures in a new family of finite classical groups, particularly for $\ell = 2$ and $\ell \neq 2$, using character table and decomposition number data.

ABSTRACT

The so-called "local-global" conjectures in the representation theory of finite groups relate the representation theory of $G$ to that of certain proper subgroups, such as the normalizers of particular $p$-groups. Recent results by several authors reduce some of these conjectures to showing that a certain collection of stronger conditions holds for all finite simple groups. Here, we show that $G=Sp_6(2^a)$ is "good" for these reductions for the McKay conjecture, the Alperin weight conjecture, and their blockwise versions.

Motivation & Objective

• To verify that $Sp_6(2^a)$ satisfies the conditions required to be 'good' for the McKay, Alperin-McKay, Alperin weight, and blockwise Alperin weight conjectures for all primes $\ell$.
• To extend the known list of finite simple groups that satisfy the reduction criteria for these local-global conjectures, particularly in cross-characteristic settings where $\ell \neq 2$.
• To provide a detailed verification using explicit character table and decomposition number data for $Sp_6(2^a)$ and lower-rank classical groups.
• To establish a framework for verifying the conjectures in higher-rank symplectic groups by analyzing $\ell$-radical subgroups and their associated character correspondences.

Proposed method

• Constructing $G$-equivariant bijections $\Omega_Q$ between height-zero irreducible characters of $G$ and those of $N_G(Q)/Q$ for each $\ell$-radical subgroup $Q$, ensuring compatibility with block induction.
• Using known character tables of $Sp_6(2^a)$ from [17] and decomposition numbers of unipotent blocks from [28], [27], [15], [21], [10] to analyze the structure of irreducible and defect-zero characters.
• Verifying that the induced block of each character under the bijection matches the block of the corresponding character in the normalizer, ensuring block compatibility.
• Analyzing the action of automorphisms on the character sets and ensuring the bijections commute with these automorphisms to satisfy the required equivariance conditions.
• Partitioning the set of irreducible Brauer characters $\mathrm{IBr}_\ell(G)$ according to $\ell$-radical subgroups $Q$, and constructing bijections $\ast_Q$ to defect-zero characters of $N_G(Q)/Q$.
• Confirming that the maps $\ast_Q$ and $\Omega_Q$ satisfy all required conditions for the reductions of the McKay, Alperin-McKay, and Alperin weight conjectures, including compatibility with block induction and automorphism actions.

Experimental results

Research questions

• RQ1 Does $Sp_6(2^a)$ satisfy the conditions required to be 'good' for the McKay conjecture for all primes $\ell$?
• RQ2 Can the Alperin-McKay conjecture be verified for $Sp_6(2^a)$ using explicit character bijections that commute with group automorphisms and preserve block structure?
• RQ3 For $\ell \neq 2$, does $Sp_6(2^a)$ admit a $G$-equivariant, block-preserving bijection between height-zero characters and defect-zero characters in the normalizers of $\ell$-radical subgroups?
• RQ4 Is the blockwise Alperin weight conjecture satisfied by $Sp_6(2^a)$ through a partition of $\mathrm{IBr}_\ell(G)$ indexed by $\ell$-radical subgroups and compatible bijections to defect-zero characters?
• RQ5 Does the proof strategy for $Sp_6(2^a)$ suggest a generalizable pattern for verifying the conjectures in higher-rank classical groups?

Key findings

• The group $Sp_6(2^a)$ is 'good' for the McKay conjecture for all primes $\ell$, including $\ell = 2$, by constructing $G$-equivariant, block-preserving bijections between height-zero characters and defect-zero characters in normalizers of $\ell$-radical subgroups.
• For $\ell \neq 2$, the paper confirms that $Sp_6(2^a)$ satisfies the reduction conditions for the Alperin-McKay conjecture, with explicit character correspondences verified via decomposition numbers and character table data.
• The blockwise Alperin weight conjecture holds for $Sp_6(2^a)$ because the set $\mathrm{IBr}_\ell(G)$ is partitioned over $\ell$-radical subgroups $Q$, and compatible bijections $\ast_Q$ to defect-zero characters of $N_G(Q)/Q$ are constructed.
• The group $Sp_6(2)$ is verified to be 'good' for the Alperin-McKay conjecture at $\ell = 2$, extending previous results to this exceptional case.
• The paper establishes that $Sp_4(2^a)$ with $q \geq 4$ even is 'good' for all conjectures for $\ell \neq 2$, and $Sp_4(2)' \cong A_6$ is 'good' for the blockwise Alperin weight conjecture.
• The results provide a foundation for extending the verification of local-global conjectures to higher-rank symplectic groups by demonstrating a consistent method using character tables and decomposition numbers.