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

PutThe defining relation between the quotient tower of a partition and the core tower of a partition isSumming the resulting telescoping identities gives

By the Hook-length formula,The abacus divisible-hook correspondence says that the number of hooks divisible by $p^r$ is $q_r$, soIf $d_p(n)={\sum }_r{\alpha }_r$, the digit-sum form of the Legendre formula isCombining the three displayed identities proves the P-adic valuation of a symmetric-group character degree from the core tower formula

Add the base-$p$ expansions of $x$ and $y$ column by column. Before carrying, the sum of all displayed digits is $d_p(x)+d_p(y)$. Each carry removes $p$ units from one column and adds one unit to the next, decreasing the total digit sum by $p-1$. After all carries the digits are those of $x+y$, so the subadditivity of the base-p digit sum gives

Let $\mu =C_p(\lambda )$, put $m=|\mu |$, and let the first-level $p$-quotient partitions have sizes $n_0,\dots ,n_{p-1}$. ThenRepeated subadditivity of the base-p digit sum givesFor 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 givesApply this to every first-level quotient partition. Their core towers concatenate to levels $r\ge 1$ of $T{C}_p(\lambda )$, so

Part a applied to $\lambda$ and to its $p$-core $\mu$, whose higher core-tower levels are empty, now givesTherefore the Character-degree valuation does not increase on taking the p-core: