Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 103 / 4 / iii

For every ${\chi }^{\mu }\in C$, level zero of the core tower of a partition is its core of a partition $\gamma$, of size $m$. The character degree coprime to p from the core tower criterion says that if $p\nmid {\chi }^{\mu }(1)$, this level must have size equal to the units digit ${\alpha }_0$ of $n$ in base $p$.

But $0\le {\alpha }_0<p$, whereas $m\ge p$. Hence every character in $C$ has degree divisible by $p$, andEquivalently, the core-tower size coefficient $m$ at position zero needs at least one carry in normalization to the base-$p$ digits. Each carry contributes one to the P-adic valuation of a symmetric-group character degree from the core tower.