# Frobenius form of the quotient of a partition

Let $\mu$ be a partition of $n \cdot l$ with a trivial $l$-core. To $\mu$ we can associate an $l$-multipartition of $n$ called the $l$-quotient of $\mu$. There are several equivalent descriptions of the $l$-quotient. It seems that the most popular one is the description in the book by James and Kerber on the representation theory of symmetric groups. To $\mu = (\mu_1, \mu_2, ...)$, where $\mu_i = 0$ for all sufficiently large $i$, we associate a set of $\beta$-numbers in the following way. We choose $r$ greater or equal to the number of rows in the Young diagram of $\mu$. The associated $\beta$-numbers are $\{\mu_1 + r - 1, \mu_2 + r - 2, ..., \mu_r - r + r \}$. The partition $\mu$ is uniquely determined by such a set. We can choose $r$ so that the set of $\beta$-numbers has cardinality which is a multiple of $l$. Define $X_i$ to be the subset of $\beta$-numbers which are congruent to $i$ mod $l$, for $i=0, ..., l-1$. Each set $X_i$ determines a partition, let's call it $\lambda^i$. The $l$-quotient of $\mu$ is then $quot_l(\mu) = (\lambda^0, ..., \lambda^{l-1})$. This process is often visualized by placing beads corresponding to the $\beta$-numbers on an abacus with $l$ runners.

The algorithm above yields a description of the $\beta$-numbers of $quot_l(\mu)$ in terms of the $\beta$-numbers of $\mu$. I have the following question. Is it possible to describe the Frobenius form of $quot_l(\mu)$ explicitly in terms of the Frobenius form of $\mu$? In other words, I consider $\mu$ as a sequence of nested one-hook partitions. The partition $\mu$ is then uniquely determined by the lengths of the arms and legs of these hooks. One writes $\mu = (a_1, ..., a_m | l_1, ..., l_m )$ if there are $m$ nested hooks with corresponding arm (leg) lengths $a_i$ ($l_i$).

The question is: can I describe $quot_l(\mu)$ in Frobenius form, i.e., describe the Frobenius form of each of the $l$ partitions in the multipartition $quot_l(\mu)$, given the data $\mu = (a_1, ..., a_m | l_1, ..., l_m )$?