= Regular label of the modular sign representation
{title2=$\lambda=((q+1)^r,q^{p-1-r})$}
Write $n=q(p-1)+r$, $0\le r<p-1$. The <regular partition> labelling the <sign representation> in characteristic $p>0$ is $\lambda=((q+1)^r,q^{p-1-r})$, with zero parts removed. Its <conjugate partition> is $((p-1)^q,r)$, which satisfies the <invariant vector criterion for a Specht module>. The sign-twisted duality of <Specht modules> turns this invariant line into a sign quotient of $S^\lambda$, necessarily its unique simple <head>. Conversely, regularity says consecutive column heights differ by less than $p$; combining this with the invariant congruences forces these heights to be $p-1$ except possibly the last. For $p=2$, the label is $(n)$ because sign equals the <trivial representation>.
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