Is the reduced plethysm (restricted to 2-columns in Young tableaux) of this Schur funtion known $\mathbb S_{3^1}(\mathbb S_{1^p})$?

Question

I am working on a physical problem, where I need to compute the "reduced plethysm" that is all the irreducibles characterised by the Young tableaux of 2 columns or less. The plethysm problem I want to solve is the following,

$$\mathbb s_{(3)}[\mathbb s_{1^p}] \text{ which can be written as } s_{(3)}[e_p],$$ where $e_p$ is the elementary symmetric polynomial of degree $p$, and we work in the permutation group $S_n$, so everything is in $n$ variables.

Edit: Since your partitions are special cases of "hooks", you can use this reference, which gives an expansion in terms of Schur functions. The coefficients are given as certain character values, and are, according to the paper, difficult to compute in general. However, in your special case, it might be easier.

Answer 1

This follows on from Dan Peterson's answer. For $r,s,a \in \mathbb{N}_0$, let $N_{r,s}(a)$ be the number of partitions of $a$ whose Young diagram fits into a box with $r$ rows and $s$ columns. By the Cayley–Sylvester formula, the multiplicity of $s_{(rs-a,a)}$ in $s_{(r)} \circ s_{(s)}$ is $N_{r,s}(a) - N_{r,s}(a-1)$ for $a$ such that $1 \le a \le rs/2$. See Corollary 2.12 in this paper of Giannelli for a short proof using the symmetric group, or this paper of Manivel for a generalization.

So the multiplicity of $s_{(3p-a,a)}$ in $s_{(3)} \circ s_{(p)}$ is $N_{(3,p)}(a) - N_{(3,p)}(a-1)$ for $a \le 3p/2$.

When $p$ is odd we need instead the multiplicity of $s_{(3p-a,a)}$ in $s_{(1^3)} \circ s_{(p)}$. By the Wronskian isomorphism (see here or Manivel's paper), we have

$$ \langle s_{(1^3)} \circ s_{(p)}, s_{(3p-a,a)} \rangle = \langle s_{(3)} \circ s_{(p-2)}, s_{(3p-a-3,a-3)} \rangle = N_{(3,p-2)}(a-3) - N_{(3,p-2)}(a-4) $$

for $p \ge 2$ and $4 \le a \le 3p/2$. If $a \le 2$ then the multiplicity is zero and if $a=3$ the multiplicity is $1$.

Answer 2

You are asking about $s_3 \circ s_{1^p}$. Recall that there is an involution $\omega$ on the ring of symmetric functions for which $\omega(s_\lambda) = s_{\lambda'}$ (conjugate partition). We use results from Macdonald's Symmetric functions and Hall polynomials: By Chapter 1, §8, Example 1, $ \omega( s_3 \circ s_{1^p})$ equals $s_3 \circ s_p$ if $p$ is even and $s_{1^3} \circ s_p$ if $p$ is odd. Both these plethysms are given completely explicitly in Example 9(b) in the same section, note that $s_p=h_p$ and $s_{1^p} = e_p$.

Your suspicion that there are no summands with only two columns (i.e. only two rows, after applying $\omega$) is incorrect for all $p>1$.

Answer 3

I am no expert in plethysm calculation, but a quick google on the involved terms leads to this paper, where combinatorial formulas for $s_{\lambda}[s_{\mu}]$ is given. Since your formula is a very special case of this, it is very likely that you can simplify the formula in the paper.

The formula in the paper expresses the plethysm as a sum over certain integer matrices. I think, with some combinatorial work, you can get an explicit formula.

EDIT: This paper give some coefficients in the Schur expansion in your case, since your partitions are hooks. However, it seems that even these coefficients are quite hard to compute in general, as they are certain character values.