I'll need some notation before I can phrase my question, so please bare with me for a little. I'll try to get there as fast as possible (it's also my first MO question...).
Let $\lambda$ be a partition of $n$, and let $S^{\lambda}$ be the corresponding Specht module. The Specht polynomials are a way of embedding $S^{\lambda}$, as a representation of $S_{n}$, into the ring $R$ of polynomials in $n$ variables $x_{i}$, $1 \leq i \leq n$ (over the complex field, say), or more precisely into its homogenous part $R_{d}$ of the appropriate degree $d$ (in the usual grading).
Explicitly, let $T$ be a Young tableau of shape $\lambda$, and let $C_{T}$ be the subgroup of $S_{n}$ that preserves the columns of $T$. Then in $S^{\lambda}$ there is a unique vector (up to scalars) on which $C_{T}$ acts via the sign representation. If the columns of $T$ decompose the set of integers between 1 and $n$ into sets, say $I_{r}$, $1 \leq r \leq l$ (with $l$ the length of the transpose partition $\lambda^{t}$) then the corresponding Specht polynomial is $$\Delta_{T}:=\prod_{r=1}^{l}\prod_{i,j \in I_{r},\ \ i<j}(x_{j}-x_{i}).$$ These polynomials are useful for understanding the action of $S_{n}$ on $R$. Note that like in the definition of the Schur symmetric functions as a quotient of determinants, any polynomial on which $C_{T}$ acts by the sign representation must be divisible by $\Delta_{T}$. By letting $T$ vary over the Young tableaux of shape $\lambda$, we get a spanning set for $S^{\lambda}$ (either abstractly or using the Specht polynomials).
However, the usual Specht polynomials, including their multiples by symmetric functions (the ring of which I'll denote by $R^{\mathrm{sym}}$, do not produce all of $R$. This is related to $R$ being the tensor product of $R^{\mathrm{sym}}$ with the group algebra of $S_{n}$, and the latter contains $S^{\lambda}$ with multiplicity that equals its dimension $f_{\lambda}$. More precisely, the quotient of $R$ by the ideal generated by the non-constant symmetric functions is isomorphic to the regular representation, thus contains $f_{\lambda}$ copies of $S^{\lambda}$. The higher Specht polynomials give explicit constructions for the other copies of $S^{\lambda}$ in $R$, that generate $R$ as an $R^{\mathrm{sym}}$-module. They are defined in the paper
Ariki, S., Terasoma, T., Yamada, H., Higher Specht polynomials, Hiroshima Math, vol 27 no. 1 ,177-188 (1997)
and are described very nicely in Maria Gillespie's blog (with a link to the original paper) here:
https://www.mathematicalgemstones.com/gemstones/diamond/higher-specht-polynomials/
Like the usual Specht polynomials, in every copy of $S^{\lambda}$ that is obtained in this way, there is unique explicit higher Specht polynomial for every Young tableau $T$ of shape $\lambda$, on which $C_{T}$ acts by the sign representation. This gives, for every such $\lambda$ and $T$, precisely $f_{\lambda}$ higher Specht polynomials (including the usual one).
I can now state my question. Fix $T$ as above, and let $P_{T}$ be a higher Specht polynomial that is associated with $T$. Since $C_{T}$ acts by the sign representation, we know that $P_{T}$ must be divisible by $\Delta_{T}$. This gives us $f^{\lambda}$ quotient polynomials. I played with them quite a bit, and saw some fascinating patterns, but I couldn't find anything on the web on it.
My question is:
Is there a reference dealing with the quotient polynomials? Are some of their basic or combinatorial properties known?
Here are the properties that they seem to have, by examining several families of partitions:
These quotients are invariant not only under $C_{T}$, but also under the larger group replacing every copy of $S_{m}^{k}$ inside it by the wreath product of $S_{k}$ acting on $k$ copies of $S_{m}$.
If the index $i$ shows up in the $r$th column, and the length of the $s$th column is $\lambda_{s}^{t}$ (using the conjugate partition), then the degree to which $x_{i}$ shows up in any such quotient is bounded by $\sum_{s \neq r}(\lambda_{s}^{t}-1)$.
Explicitly, if $\lambda$ is the hook partition $(n-k,1^{k})$ and $T$ is such that the number between 1 and $k$ appear in the longer column, then the quotients produce precisely the monomial symmetric functions $m_{\mu}(x_{k+1},\ldots,x_{n})$, where $\mu$ runs over all the partitions of length$\leq n-k$ and involving entries $\mu_{h} \leq k-1$.
For more general partitions, the quotients can involve several terms, sometimes with negative signs (which depend on the sign of certain partitions), and even multiplicities$>1$.
Any reference analyzing this question, or some insights into it, or even general interest in it, will be most welcome!
