$\DeclareMathOperator{\Ind}{\operatorname{Ind}}$I'm reading *"The representation theory of the symmetric groups"* written by Gordon James.

I found the matrix $B$ in the chapter 6 (*"The character table of $S_{n}$"*) useful for my research.
So I want to find recent results on $B$ if there are any.
But as I don't know the name of the matrix I can't search for it directly. So I ask for the name of the matrix or any papers on $B$.

For those who have not read above book, let me explain the matrix $B$. It is a natural object.

The irreducible representations of $S_{n}$ over $\mathbb{Q}$ are represented by the Specht modules associated to the partitions of $n$. (In the book, for a partition $\lambda$, the Specht module associated to $\lambda$ is denoted $S_{\mathbb{Q}}^{\lambda}$). There is a partial order (dominance order) on the set of partitions of $n$.

$\lambda \lhd \mu$ if and only if for all $i \geq 1$, $\sum_{j=1}^{i} \lambda_{j} \leq \sum_{j=1}^{i}\mu_{j}$.

For a partition $\lambda$, there is a Young subgroup $G_{\lambda}$ and the induced character $\Ind_{G_{\lambda}}^{S_{n}}1$ is the regular representation on the $\lambda$-tabloids. It is denoted by $M_{\mathbb{Q}}^{\lambda}$ in the book.

In the book, we order the partitions of $n$ in the dictionary order(which is a total order) increasingly and define $B$ by

$B = ( b_{\lambda\mu} )$, where $b_{\lambda\mu}=|G_{\mu}| (\chi^{\lambda}, \Ind_{G_{\mu}}^{S_{n}}1).$ Here $\chi^{\lambda}$ is the character of $S_{\mathbb{Q}}^{\lambda}$.

It is a lower triangular matrix with 1's down the diagonal because there is a theorem that

$S_{\mathbb{Q}}^{\lambda}$ is a constituent of $M_{\mathbb{Q}}^{\mu}$ only if $\mu \lhd \lambda$. And the multiplicity of $S_{\mathbb{Q}}^{\lambda}$ in $M_{\mathbb{Q}}^{\lambda}$ is always 1.

Any references on $(\chi^{\lambda}, \Ind_{G_{\mu}}^{S_{n}}1)$ would be helpful to me.

Thank you in advance.