Reading The Symmetric Group by Bruce Sagan and struggling with a couple of trivial definitions - where I've had trouble I've written (WHY).

If we have a tableau t with rows R_i and columns C_j, then we define the row- and column-stabilisers respectively to be

$\displaystyle R_t:=\Pi S_{R_i}, \quad C_t:=\Pi S_{C_j}.$

Apparently then equivalence classes of tableaux {t} (called tabloids, i.e. where the order of numbers in rows is immaterial) are given by $\displaystyle \{t\} = R_t t$ (WHY).

In addition, these groups are associated with certain elements of $\displaystyle \mathbb{C}[S_n]$. In general, given a subset $\displaystyle H\subseteq S_n$ we can form the group algebra sums $\displaystyle H^+:=\sum_{\pi\in H} \pi,\;\; H^-:=\sum_{\pi\in H}\textrm{sign}(\pi)\pi$.

For a tableau t, the element $\displaystyle R_t^+$ is already implicit in the corresponding tabloid by the remark above. (WHY. Don't understand this at all - isn't $\displaystyle R_t^+$ going to be an ordered tuple? What does it look like? This line makes no sense to me at all).

We also need to make use of

$\displaystyle \kappa_t:=C_t^-.$

Note that if t has columns $\displaystyle C_j$ then $\displaystyle \kappa_t$ factors as $\displaystyle \kappa_t=\Pi\kappa_{C_j}$ (WHY - how can a k-tuple (assuming there are k columns) be expressed as a product of single numbers? And what does the sum of permutations mean anyway? Very confused here).

Any light anyone could shed on any of the above would be great, I'm having a bit of trouble digesting it all. This is a great book though.

Cheers,

Atticus