Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 160 / 1 / a

The symmetric group acts transitively on the tableaux of a fixed shape. Thus every $s\in {\Delta }^{\lambda }$ is $gt$ for some $g\in S_n$, and the definition of a polytabloid givesSince the Specht module is spanned by all $e(s)$, it is the cyclic $\mathbb{F}{S}_n$-module generated by $e(t)$.

Ina fixed tabloid $\{u\}$ can occur at most once. Indeed, if $\{\pi t\}=\{\sigma t\}$ for $\pi ,\sigma \in C(t)$, then ${\sigma }^{-1}\pi$ belongs to both the row and column stabilizers of $t$. Their intersection is trivial, so $\pi =\sigma$. The coefficient $⟨e(t),\{u\}⟩$ is consequently $0$, $1$, or $-1$.

For a tabloid $U$, let $r_U(k)$ be the row containing $k$, and order tabloids lexicographically byIf $t$ is standard and $1\ne \pi \in C(t)$, let $k$ be the least number moved by $\pi$. Every smaller entry is fixed, while ${\pi }^{-1}(k)$ is a larger entry below $k$ in the same column. Hence $\{\pi t\}$ agrees with $\{t\}$ before $k$ and has $r_{\pi t}(k)>r_t(k)$. Thus $\{t\}$ is the unique least tabloid in $e(t)$ and has coefficient one.

Distinct standard tableaux have distinct tabloids, since increasing each row recovers the tableau from its tabloid. A linear relation among standard polytabloids now has a least leading tabloid, which cannot cancel. They are therefore linearly independent, as in the linear independence of standard polytabloids.