= Dominance triangularity of modular symmetric-group decomposition
{title2=$d_{\mu\lambda}=0\text{ unless }\lambda\unrhd\mu,\quad d_{\lambda\lambda}=1$}
Rows of the symmetric-group <decomposition matrix (modular representation theory)> are all ordinary shapes, columns are <regular partitions>, and $d_{\mu\lambda}=[S^\mu:D^\lambda]$. A nonzero <column antisymmetrizer> of shape $\lambda$ on $M^\mu$ forces dominance: the first $r$ rows contain at most $\sum_j\min(r,\lambda'_j)$ column entries. For a regular shape its diagonal simple appears once, giving a rectangular triangular matrix and a unitriangular regular-row submatrix.
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