Trying to understand a notation $S_{\lambda} w S_{\mu}$.

Question

I am reading the paper (arXiv:1605.08545v5). There is a notation $S_{\lambda} w S_{\mu}$ on page 61 before the formula (27). It is said that $w$ is of maximal length in $S_{\lambda} w S_{\mu}$. Here $\lambda, \mu \in \mathbb{Z}^k$ are $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_k)$, $\lambda_1 \ge \cdots \ge \lambda_k$, $\mu = (\mu_1, \mu_2, \ldots, \mu_k)$, $\mu_1 \ge \cdots \ge \mu_k$. What are $S_{\lambda}$ and $S_{\lambda} w S_{\mu}$? How to compute maximal $w$ for given $\lambda, \mu$? Thank you very much.

Answer 1

On page 60, the authors say that if $\mu = \left(\mu_1, \mu_2, \ldots, \mu_k\right) \in \mathbb{Z}^k$ is any $k$-tuple, then $S_\mu$ denotes the stabilizer of the $k$-tuple $\mu$ in $S_k$. Presumably, the action of $S_k$ on the $k$-tuples being used here is the one that permutes the coordinates. Thus, $S_\mu$ is the set of all permutations $\sigma \in S_k$ such that all $i \in \left\{1,2,\ldots,k\right\}$ satisfy $\mu_{\sigma\left(i\right)} = \mu_i$.

This stabilizer $S_\mu$ can be characterized as the subgroup of $S_k$ consisting of all permutations that preserve a certain set partition (i.e., preserve each part of the set partition) of $\left\{1,2,\ldots,k\right\}$. If $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_k$, then this set partition is an interval partition (= set partition into intervals). Thus, in this case, $S_\mu$ is what is called a Young subgroup.

So you know that $S_\lambda$ and $S_\mu$ are Young subgroups in the case that you care about. Now, $S_\lambda w S_\mu$ hopefully means the double coset $\left\{x w y \mid x \in S_\lambda,\ y \in S_\mu\right\} \subseteq S_k$. These double cosets are fairly familiar objects -- see Mark Wildon's A model for the double cosets of Young subgroups. Note, however, that the definition of $S_\nu$ in Wildon's note is not literally equivalent to your definition of $S_\mu$ above -- more precisely, it is equivalent but not for $\mu = \nu$. To be more explicit: If $\nu = \left(\nu_1, \nu_2, \nu_3, \ldots\right)$ is a partition of $k$, then Wildon's $S_\nu$ is your $S_\mu$, where \begin{align} \mu = \left( \underbrace{-1,-1,\ldots,-1}_{\nu_1 \text{ times}}, \underbrace{-2,-2,\ldots,-2}_{\nu_2 \text{ times}}, \underbrace{-3,-3,\ldots,-3}_{\nu_3 \text{ times}}, \ldots\right) \in \mathbb{Z}^k . \end{align} You can easily check that, conversely, any $S_\mu$ in your sense can be rewritten as an $S_\nu$ in Wildon's sense for an appropriately chosen partition $\nu$ of $k$.

Also, keep in mind that Mark Wildon is British and writes composition of maps the "other" way round -- so the product $ab$ of two permutations $a,b\in S_k$ in Wildon's note will mean "apply $a$ first and $b$ second". Though I don't know whether the article you are reading doesn't do that as well.

I hope that was a reasonable Rosetta stone.