Let $\mu=C_p(\lambda)$, put $m=|\mu|$, and let the first-level $p$-quotient partitions have sizes $n_0,\ldots,n_{p-1}$. Then
$$
n=m+p\sum_jn_j.
$$
Repeated <subadditivity of the base-p digit sum> gives
$$
d_p(n)-d_p(m)
\leq d_p\left(\sum_jn_j\right)
\leq\sum_jd_p(n_j).
$$
For any partition $\nu$ of size $s$, iterating the core-quotient relation $s=|C_p(\nu)|+p|Q_p(\nu)|$ and the same digit-sum inequality gives
$$
d_p(s)\leq\sum_{r\geq0}|TC_p(\nu)_r|.
$$
Apply this to every first-level quotient partition. Their core towers concatenate to levels $r\geq1$ of $TC_p(\lambda)$, so
$$
d_p(n)-d_p(m)
\leq\sum_{r\geq1}|TC_p(\lambda)_r|.
$$

Part a applied to $\lambda$ and to its $p$-core $\mu$, whose higher core-tower levels are empty, now gives
$$
\begin{aligned}
(p-1)\left(v_p(\chi^\lambda(1))-v_p(\chi^\mu(1))\right)
&=\sum_{r\geq1}|TC_p(\lambda)_r|-d_p(n)+d_p(m)\\
&\geq0.
\end{aligned}
$$
Therefore the <Character-degree valuation does not increase on taking the p-core>:
$$
\boxed{v_p(\chi^\lambda(1))\geq v_p(\chi^{C_p(\lambda)}(1))}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 160 / 4 / c / Solution

{c}

For every partition $\lambda$,
$$
v_p(\chi^\lambda(1))\geq v_p(\chi^{C_p(\lambda)}(1)).
$$
The core-tower formula reduces this to subadditivity of the base-$p$ digit sum across the sizes of the quotient partitions.


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