Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 103 / 3

The equivalence is true. Write $c_r(\eta )$ for the total size at level $r$ of the core tower of a partition $\eta$, and $q_r(\eta )$ for the corresponding total size in its quotient tower of a partition. We use the size relation and the P-adic valuation of a symmetric-group character degree from the core tower:Here $d_p$ is the base-p digit sum. These imply the character degree coprime to p from the core tower criterion: if $|\eta |={\sum }_r{\alpha }_r{p}^r$, then $p\nmid {\chi }^{\eta }(1)$ exactly when $c_r(\eta )={\alpha }_r$ at every level. Indeed carrying $p$ units from one position to the next decreases the sum of coefficients by $p-1$; the valuation vanishes precisely when there are no carries.

For $k\ge 1$, we also use power-core truncation of a prime-core tower: for $\gamma =C_{p^k}(\lambda )$, its core tower agrees with that of $\lambda$ below level $k$ and is empty at every level at least $k$. Moreover $q_k(\lambda )=w_{p^k}(\lambda )$, the weight of a partition for modulus $p^k$. To see the truncation, removal of a rim hook of length $p^k$ preserves the first $p$-core and, under the abacus divisible-hook correspondence, becomes removal of a $p^{k-1}$-hook in one quotient component. Iterating preserves all the lower cores; after all such removals, quotient level $k$ is empty.

If $p\nmid {\chi }^{\lambda }(1)$, the criterion gives $c_k(\lambda )=a$, $c_r(\lambda )=0$ for $r>k$, and the lower $c_r(\lambda )$ are the base-$p$ digits of $m$. It follows that $|\gamma |=m$, $p\nmid {\chi }^{\gamma }(1)$ andConversely, suppose $w_{p^k}(\lambda )=a$ and $p\nmid {\chi }^{\gamma }(1)$. Then $|\gamma |=m$ and its lower core-tower sizes are the digits of $m$. Quotient level $k$ of $\lambda$ has total size $a<p$. Each component there has size less than $p$, hence is already a $p$-core and has empty $p$-quotient. Thus $c_k(\lambda )=a$ and every higher level is empty. Combined with truncation, this gives exactly the digits of $n$, so $p\nmid {\chi }^{\lambda }(1)$.

ThereforeIf $k=0$, necessarily $m=0$ and $n=a<p$; the $1$-core is empty, its weight is $n=a$, and the Hook-length formula shows that every degree is coprime to $p$. Thus the equivalence holds in that boundary case too, with the empty partition having degree one.

The implication is false. ChooseThe inequalities $0\le 4<9$ and the relation $13=1\cdot 9+4$ hold. At cell $(1,4)$, the hook length is $12-4+1=9=a{p}^k$.

For the partition of an integer $(12,1)$, the Hook-length formula giveswhich is divisible by $3$. This also agrees with the degree of the standard representation of the symmetric group $S_{13}$.

The obstruction is visible in the core of a partition: removing the horizontal rim hook of length nine leaves $\gamma =(3,1)$, which has no hook length nine and hence is the $9$-core. Its degree is ${\chi }^{(3,1)}(1)=3$. The large hook ensures the required weight of a partition here, but the core degree is not coprime to $p$.

The implication is false. ChooseThe hook lengths areThus $3$ does not divide this irreducible character degree, but no hook has length $a{p}^k=6$.

There is no contradiction with part (i): removing two horizontal rim hooks of length three gives the empty core of a partition, so $w_3(\lambda )=2=a$ and the core degree is one. Having total weight of a partition two does not force a single hook of length twice the modulus.