$\DeclareMathOperator{\Ind}{\operatorname{Ind}}$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.

The matrices in the book

  • $\begingroup$ @GeoffRobinson Thank you for the point. First of all, I've only read the first 8 chapters of the book yet so I don't know whether the above matrix is studied further in the book. Is it? I'm especially interested in the non-nullity of the entries. For example, in the book the penultimate column (2,6,6,6,4,2) has entries that are nonzero. Is it a general phenomenon for all n? $\endgroup$
    – gualterio
    Oct 7 '20 at 9:00
  • $\begingroup$ What is A? In the question I can only see definition of B. $\endgroup$
    – ArB
    Oct 7 '20 at 14:16
  • 1
    $\begingroup$ @ArB I added the picture, where B is included. Using A and B, one can get a matrix C, whose entries are the character values (Power-sums into Schur). $\endgroup$ Oct 7 '20 at 14:22
  • 1
    $\begingroup$ @gualterio as pointed out, the entires are (up to a multiple) the Kostka numbers $K_{\lambda\mu}$. They count the semistandard tableaux of shape $\lambda$ and weight $\mu$ (A good reference is Sagan's book on the symmetric groups). Thus, the penultimate column of your matrix is (twice) the number of semistandard tableaux with entries $1,1,2,3,4,\ldots, n-1$. For at least one to exist you need at least two boxes in the first row. Thus, all entries of that column, apart from the very last, will always be nonzero. More generally $K_{\lambda\mu} \neq 0$ if and only if $\lambda$ dominates $\mu$. $\endgroup$
    – Noah White
    Oct 8 '20 at 6:53

The $B$-matrix is the Kostka matrix, but where each entry has been multiplied by a certain factor. In Mathematica code, (using my symmetric functions package) the matrix is given by


which produces $$ \left( \begin{array}{ccccccc} 120 & 24 & 12 & 6 & 4 & 2 & 1 \\ 0 & 24 & 12 & 12 & 8 & 6 & 4 \\ 0 & 0 & 12 & 6 & 8 & 6 & 5 \\ 0 & 0 & 0 & 6 & 4 & 6 & 6 \\ 0 & 0 & 0 & 0 & 4 & 4 & 5 \\ 0 & 0 & 0 & 0 & 0 & 2 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) $$

That is, $B_{\lambda \mu} = K_{\lambda\mu}\cdot (\mu_1! \mu_2! \dotsb \mu_\ell !)$.

Similarly, the $A$-matrix in the question is given by

    , PowerSumSymbol@mu] Times @@ (lam!)
  , {lam, IntegerPartitions@5}
  , {mu, IntegerPartitions@5}] // MatrixForm

so $A_{\lambda\mu} = [p_\mu]h_\lambda \cdot (\lambda!)$ , where $[p_\mu]h_\lambda $ denotes the coefficient of $p_\mu$ in $h_\lambda$. The coefficient $[p_\mu]h_\lambda $ has a combinatorial formula, using ordered brick tabloids, see

Ömer Eğecioğlu and Jeffrey B. Remmel. Brick tabloids and the connection matrices between bases of symmetric functions. Discrete Applied Mathematics, 34(1-3):107–120, November 1991.

Since $K_{\lambda\mu} = [s_\mu] h_\lambda$, we can use the matrices above to compute $[s_\mu] p_\lambda$, which is the character value $\chi_{\lambda,\mu}$, the matrix $C$ in the same chapter.


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