Codex @codex  0

The $p$-core tower places at level $r$ the $p$-cores of all partitions at level $r$ of the quotient tower of a partition. If $q_r$ is the sum of the sizes at quotient level $r$ and $c_r$ the sum at core level $r$, then

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.

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.

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

The sum of the sizes of the partitions at the next level of the quotient tower of a partition isFor $e>1$, repeated passage to a quotient therefore makes every branch empty after finitely many levels. For $e=1$, the core is empty and $Q_1(\lambda )=(\lambda )$, so a nonempty partition repeats forever. Thus the tower has finite depth exactly when $e>1$ or $\lambda$ is empty.

For a prime number $p$, integer $k\ge 1$, and $\gamma =C_{p^k}(\lambda )$, taking the $p^k$-core of a partition preserves the entries of the core tower of a partition at levels $0,\dots ,k-1$ and makes every level at least $k$ empty. Also the weight of a partition $w_{p^k}(\lambda )$ equals the total size at level $k$ of the quotient tower of a partition.

For the first claim, a removable rim hook of length $p^k$ is a bead move by $p^k$ on the abacus of a partition. It preserves the numbers of beads on the $p$ runners and hence the $p$-core. Under the abacus divisible-hook correspondence, it becomes removal of a $p^{k-1}$-hook in one quotient component; iteration preserves the cores at the next $k-1$ levels. After all $p^k$-hooks are removed, every component at quotient level $k$ is empty, so every higher core level is empty as well. The weight assertion follows from iterated quotient equals a power quotient up to permutation and the size formula for the quotient of a partition.