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Fix a positive integer $e$. A partition of an integer is uniquely specified by its core of a partition for modulus $e$ and its ordered quotient of a partition for that modulus. More precisely, for every $e$-core $\gamma$ and ordered tuple $({\nu }_0,\dots ,{\nu }_{e-1})$ of partitions, there is a unique partition $\lambda$ such thatThe ordered runners use the usual abacus of a partition convention with a number of beads divisible by $e$. Adding $e$ beads preserves both the tuple and its runner labels. The core fixes the relative runner offsets, and each component partition fixes the displacement of the beads on that runner, giving existence and uniqueness.

Let $\kappa$ be the core of a partition $\lambda$ at prime modulus $p$, with quotient component sizes $w_i$ and total weight $w$. Complete $p$-hook removal sequences number $N_p(\lambda )=\left(\genfrac{}{}{0ex}{}{w}{w_0,\dots ,w_{p-1}}\right){\prod }_i{f}^{{\lambda }^{(i)}}$. Their signs agree because abacus bead-crossing parities telescope to the permutation taking the initial labeled beads to the packed core order. Repeated Murnaghan–Nakayama rule gives the displayed formula for any permutation $\pi$ on the remaining letters.

At fixed degree $n$ in positive characteristic $p$, two Specht modules belong to the same block of a group algebra exactly when their labels have the same core of a partition at modulus $p$. The simple labels in that block are precisely the regular partitions with that core. The theorem follows by combining the residue-content criterion for symmetric-group blocks with the Jucys–Murphy description of the center of a symmetric-group algebra.

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.

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 $p$.

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.

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.

LetA direct comparison of the affected bead-gap pairs shows that removing a rim 2-hook preserves $D$: the pairs whose parities change cancel in odd-even pairs. Repeating this removal givesEvery 2-core is a staircaseAll hook lengths in this staircase are odd, and it has $1+2+\cdots +r=r(r+1)/2$ cells. Hence the odd-minus-even hook count of a partition isThus the requested integer is $m=r+1=\ell (C_2(\lambda ))+1$.

Moving a bead one step upward on an $e$-runner abacus of a partition removes an $e$-hook and decreases the size by $e$. Sliding all beads upward reaches the core of a partition after exactly $w_e(\lambda )$ moves. Therefore

True. A hook length divisible by $ef$ is divisible by $e$. If $\lambda$ is its own Core of a partition $C_e(\lambda )$, it has no such hook, so it is also its own $ef$-core.

Apply the Murnaghan–Nakayama rule successively to the $w$ disjoint $e$-cycles. A complete term requires a sequence of $w$ removable $e$-hooks. If $w>w_e(\lambda )$, no such sequence exists after the Weight of a partition is exhausted, so the character value is zero.

Suppose $w=w_e(\lambda )$. Every complete sequence ends at the Core of a partition $C_e(\lambda )$. Under the abacus divisible-hook correspondence, a removal chooses one cell from one component ${\lambda }^{(i)}$ of the quotient of a partition. The choices of which runner is used occur inorders. Within runner $i$, the signed complete removal sum is the degree ${\chi }^{{\lambda }^{(i)}}(1)$, and all inter-runner removal orders have the common Sign of an abacus hook-removal sequence $\epsilon \in \{1,-1\}$. The remaining permutation $\gamma$ acts on the core, giving

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.