Block weight is not the same as numerical defect

Weight is combinatorial. For finite $e\ge 2$, the $e$-weight $w_e(\lambda )$ of a label $\lambda ⊢n$ is the number of $e$-hooks that have to be removed from $\lambda$ to reach its $e$-core, it equals the size $\mathrm{\mid }{Q}_e(\lambda )\mathrm{\mid }$ of the $e$-quotient, and it satisfies $n=\mathrm{\mid }\lambda \mathrm{\mid }=\mathrm{\mid }{C}_e(\lambda )\mathrm{\mid }+e{w}_e(\lambda )$ (The core-quotient size formula defines the weight). In particular, $w_e(\lambda )=(n-\mathrm{\mid }{C}_e(\lambda )\mathrm{\mid })\mathrm{/}e$ is determined by $n$ and the $e$-core. Thus, for a fixed algebra $H_{F,q}(S_n)$, by Type-A Hecke Specht modules are in the same block iff their labels have the same e-core it is an invariant of the block of $S^{\lambda }$; for $e=p$ it is the combinatorial $p$-weight of the block.

Numerical defect is block-theoretic. The numerical defect of a block is defined from its defect group: for the block bimodule $kGb$ a defect group is a $p$-subgroup $D\le G$ for which the diagonal $\mathrm{\Delta }D$ is a vertex, its order $p^d$ defines the numerical defect $d$, and defect zero means $D=1$ (Defect group and numerical defect of a block). It is a $p$-adic order invariant of the block algebra in the fixed splitting modular system and is defined by a vertex computation, not by hook combinatorics.

The two invariants must not be identified without a theorem and an explicit range. The weight records how many hooks of length divisible by $e$ the label carries, while the numerical defect records the order of a vertex of the block bimodule; nothing on this page proves that the two numbers agree, and they are defined from entirely different data. The only case settled here is weight zero: if $w_p(\lambda )=0$ then $\lambda$ is a $p$-core, and the classification forces the block to have a single Specht label, Cartan matrix $(1)$, a projective simple module, a full matrix algebra structure and trivial defect group, so the numerical defect is also zero (Weight-zero symmetric-group blocks are defect zero). For $w>0$ no assertion about the defect group of a weight-$w$ symmetric-group block, and no identification of $w$ with the numerical defect, is made on this page. The printed defect-group computation for weight-$w$ blocks (defect ${\nu }_p((pw)!)$, with defect group a Sylow $p$-subgroup of $S_{pw}$) is deliberately not asserted here, because no counted proof source for it is registered with this batch; supplying it is outside this page's proof route and remains an open obligation for a later page of the track.

No choice principle is used in the statements recorded above.