I'm currently working on plethysm, i.e. the character of the composition $S^\lambda(S^\mu(V))$ of the Schur functors $S^\lambda$ and $S^\mu$ on a complex vector space $V$. We note this character $s_\lambda[s_\mu]$. One must know that the character of $S^\mu(V)$ is the Schur function $s_\mu$.

We want to find the irreductible representation of $S^\lambda(S^\mu(V))$, or equivalently, to write $s_\lambda[s_\mu]$ in a sum of Schur function. There is very little that is known about this, and this operation is rarely mentionned in the litterature I found.

I have worked with plethysm mostly in the context of a "math research traineeship", so I skipped some details of all this theory to emphasize on the use of the computer in pure maths.

I have 3 questions about all this and I don't find references:

1) I know how to compute plethysm in terms of power sum symmetric functions, but why is it defined this way?

2) What is known or conjectured about operations on plethysms that are Schur-positive (i.e. having only nonnegative coefficients when expressed in terms of Schur functions), like the Foulkes' conjecure, that said that $\forall a,b \in \mathbb{N}, \ a \leq b, \ h_b[h_a] - h_a[h_b]$ is Schur-positive, where $h_n = s_{(n)}$, the schur function indexed with only one part?

3) Where can I find clear versions (and with the use of modern notations) of the Thrall's proofs of $h_2[h_n]$, $h_n[h_2]$ and $h_3[h_n]$?

Thanks in advance for short explanations or for references!

• 1) Generally, many operations on symmetric functions are the easiest to represent in the power-sum basis. This holds particularly for operations whose combinatorial meaning is still mysterious. – darij grinberg Apr 2 '17 at 7:39

Of the three plethysms in 3), $h_n[h_2]$ is the simplest. A "modern" proof can be found for instance in Example A2.9 (page 449) of Enumerative Combinatorics, vol. 2. This was known to D. E. Littlewood well before Thrall. It may be even older though not stated in the language of plethysm.
For $h_2[h_n]$, here is one argument. Note that $h_2=\frac 12(p_1^2+p_2)$. Hence $$h_2[h_n] =\frac 12(h_n^2+h_n(x_1^2,x_2^2,\dots)).$$ Now \begin{eqnarray*} \sum_{n\geq 0}h_n(x_1^2,x_2^2,\dots) & = & \frac{1} {\prod(1-x_i^2)}\\ & = & \frac{1}{\prod(1-x_i)(1+x_i)}\\ & = & \left( \sum_{j\geq 0} h_j\right)\left( \sum_{k\geq 0} (-1)^k h_k\right), \end{eqnarray*} whence $$h_2[h_n] =\frac 12\left(h_n^2+\sum_{k=0}^{2n}(-1)^k h_k h_{2n-k}\right).$$ Expand $h_n^2$ and $h_kh_{2n-k}$ into Schur functions by Pieri's rule and collect terms to get $$h_2[h_n]=\sum_{k=0}^{\lfloor n/2\rfloor}s_{(2n-2k,2k)}.$$ A reference for $h_3[h_n]$ is S. P. O. Plunkett, Canad. J. Math. 24 (1972), 541--552.
• Is there a pdf version of this last article by Plunkett ? Simple curiosity : is there any expression for $h_k[h_m]$ ? – Synia Aug 25 '17 at 19:28
• Concerning $h_k[h_m]$, I found the talk www-math.mit.edu/~rstan/transparencies/plethysm.pdf where this is not mentioned, so I guess there is no such conjectural formula. – Synia Aug 26 '17 at 13:33