Decomposition matrix (modular representation theory, $D$)

The decomposition matrix is a specific invariant of modular representation theory.

The decomposition matrix records the multiplicities of simple modular representations in reductions of ordinary representations. If its rows are indexed by ordinary irreducible characters and its columns by irreducible Brauer characters, its entry $d_{\chi \varphi }$ is the decomposition number of $\varphi$ in the reduction of $\chi$.

Rows of the symmetric-group decomposition matrix 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 ${\sum }_j\min (r,{\lambda }_j^{\prime })$ column entries. For a regular shape its diagonal simple appears once, giving a rectangular triangular matrix and a unitriangular regular-row submatrix.

The Cartan matrix records composition-factor multiplicities in the projective indecomposable modules. For a split modular group algebra, it iswhere $D$ is the decomposition matrix.