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

We use the abacus divisible-hook correspondence in its multiset form: the hook lengths of $\lambda$ that are divisible by $p$, divided by $p$, form exactly the disjoint union of the multisets of hook lengths of the components of its quotient of a partition for modulus $p$.

Iterating this correspondence gives, for every $r\ge 1$,as multisets. Each Young diagram has one hook for every cell, so the cardinality of the right-hand side is $|T^Q(\lambda {)}_r|$, the sum of the component sizes.

For any positive integer $h$, its P-adic valuation is the number of positive $r$ for which $p^r\mid h$. Summing this identity over hooks and interchanging the finite sums provesThusThe sums are finite because hook lengths are bounded by $|\lambda |$, and the total size at each level of the quotient tower of a partition decreases by at least a factor $p$ until it reaches zero.

We use the core-quotient bijection for partitions: an ordered $p$-tuple $({\nu }_0,\dots ,{\nu }_{p-1})$ and a fixed $p$-core determine a unique partition of an integer, withThe irreducible characters in both $B$ and $C$ therefore correspond to the same ordered tuples of total size $w$. Replacing the empty core of a partition by $\gamma$ while keeping the quotient of a partition fixed gives a bijection ${\chi }^{\lambda }\mapsto {\chi }^{\mu }$ from $B$ to $C$. In particular $|B|=|C|$.

For such a pair, all levels of the core tower of a partition above level zero agree. At level zero, their sizes are $0$ and $m$, respectively. Put $D={\sum }_{r\ge 1}|T^C(\lambda {)}_r|={\sum }_{r\ge 1}|T^C(\mu {)}_r|$. When $0\le m<p$, the base-p digit sum satisfiesThe P-adic valuation of a symmetric-group character degree from the core tower then givesThus the same bijection preserves degree coprimality to $p$ and restricts to a bijection between the indicated sets of irreducible characters of degree coprime to p. ConsequentlyFor $m=0$, take $\gamma$ to be the empty partition; the construction becomes the identity.

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.