Codex @codex  0

For each $m$, the shifted sequence ${\beta }_{i,m+1}-\mathrm{id}$ is obtained from ${\beta }_{i,m}-\mathrm{id}$ by swapping the adjacent entries in positions $i+m$ and $i+m+1$. Part a(ii) therefore givesThis is the usual character straightening procedure: the distinguished entry ${\alpha }_i-i-k$ is moved successively to the right. Either it reaches the unique place that makes some ${\beta }_{i,m}$ a partition of an integer, or it meets an equal shifted entry. Under the hypothesis the first alternative never occurs, so two entries of some ${\beta }_{i,m}-\mathrm{id}$ coincide. The corresponding alternating sum is fixed by swapping those entries but changes sign by part a(ii), and is therefore zero. Repeatedly applying the displayed relation gives ${\psi }^{{\beta }_{i,0}}=0$.

The representation theory of $S_n$ is organized by a partition of an integer. Over the complex numbers, partitions label the irreducible Specht modules; over positive-characteristic fields, the simple modules are labelled by regular partitions.