Scrapbook on plethysms

Wildon's Weblog 2022-10-06

I’m visiting OIST in Okinawa, Japan for 6 weeks, in which I plan to work on some of the problems in the (over)-extended research summary I wrote for the Oberwolfach meeting in September. The purpose of this post is to collect some bits and pieces maybe relevant to these problems or that come in my reading.

I was briefly tempted to title this post ‘tropical plethysms’, but then it occurred to me that perhaps the idea was perhaps not completely absurd: as a start, is there such a thing as a tropical symmetric function?

A generalized deflations recurrence

For this post, let us say that a skew partition $\lambda / \mu$ of $\ell m$ is a horizontal $\ell$ -border strip if there is a border-strip tableau $T$ of shape $\lambda / \mu$ comprised of $m$ disjoint $\ell$ -hooks, such that these hooks can be removed working right-to-left in the Young diagram. It is an exercise (see for instance the introduction to my paper on the plethystic Murnaghan–Nakayama rule) to show that at most one such $T$ exists. We define the $\ell$ -sign of the skew partition, denoted $\mathrm{sgn}_\ell(\lambda/\mu)$ , to be $0$ if $\lambda / \mu$ is not a horizontal $\ell$ -border strip, and otherwise to be $(-1)^k$ where $k$ is the sum of leg lengths in $T$ . Thus a skew partition is $1$ -decomposable if and only if it is a horizontal strip in the usual sense of Young’s rule, a skew partition of $\ell$ is $\ell$ -decomposable if and only if it is an $\ell$ -hook (and then its $\ell$ -sign is the normal sign), and

$\mathrm{sgn}_3\bigl((7,5,4)/(4,3)\bigr) = (-1)^{0+1+0} = -1,$

as shown by the tableau below:

Proposition 5.1 in this joint paper with Anton Evseev and Rowena Paget, restated in the language of symmetric functions is the following recurrence for the plethysm multiplicities relevant to Foulkes’ Conjecture:

$\begin{aligned}\langle s_{(n)}& \circ s_{(m)}, s_\lambda \rangle = \\ &\frac{1}{n} \sum_{\ell=1}^n \sum_{\alpha \in \mathrm{Par}((n-\ell) m))} \mathrm{sgn}_\ell(\lambda / \alpha)\langle s_{(n-\ell)} \circ s_{(m)}, s_\alpha \rangle.\end{aligned}$

The proof of Proposition 5.1 uses the theory of character deflations, as developed earlier in the paper, together with Frobenius reciprocity. My former Ph.D. student Jasdeep Kochhar used character deflations to prove a considerable generalization, in which $(n)$ is replaced with an arbitrary partition, and $(m)$ with an arbitrary hook partition. I think the only reason he stopped at hook partitions was because this was the only case where there was a convenient combinatorial interpretation (see the end of this subsection), because his argument easily generalizes still further to show that

$\begin{aligned}\langle s_{\nu}& \circ s_{\mu}, s_\lambda \rangle = \\ &\frac{1}{n} \sum_{\ell=1}^n \sum_{\alpha} \sum_\beta \langle p_\ell \circ s_\mu, s_{\lambda/\alpha} \rangle \mathrm{sgn}_\ell(\nu/\beta) \langle s_{\beta} \circ s_{\mu}, s_\alpha \rangle\end{aligned}$

where $\mu$ is any partition of $m$ , the first sum is over all $\alpha \in \mathrm{Par}\bigl( (n-\ell)m \bigr)$ (as before) and the second sum is over all $\beta \in \mathrm{Par}(n-\ell)$ . Since a one part partition has a unique $\ell$ -hook, if $\nu = (n)$ then the only relevant $\beta$ is $(n-\ell)$ . Hence a special case of Kochhar’s result, that generalizes the original result in only one direction, is

$\begin{aligned}\langle s_{(n)}& \circ s_{\mu}, s_\lambda \rangle = \\ &\frac{1}{n} \sum_{\ell=1}^n \sum_{\alpha} \sum_\beta \langle p_\ell \circ s_\mu, s_{\lambda/\alpha} \rangle \langle s_{n-\ell} \circ s_{\mu}, s_\alpha \rangle. \end{aligned}.$

In the special case where $\mu = (m)$ , the plethystic Murnaghan–Nakayama rule states that

$\langle p_\ell \circ s_{(m)}, s_{\lambda/\alpha} \rangle = \mathrm{sgn}_\ell(\lambda/\alpha)$

and substituting appropriately we recover the original result. More generally, Theorem 6.3 in the joint paper implies that $\langle p_\ell \circ s_\mu, s_{\lambda/\alpha} \rangle$ is the product of $\mathrm{sgn}_\ell(\lambda / \alpha)$ and the size of a certain set of $\lambda/\alpha$ border-strip tableaux in which all the border strips have length $\ell$ .