Generalized decomposition numbers

Fix a splitting $p$-modular system $(K,\mathcal{O},k)$ for a finite group $G$, an ordinary irreducible character $\chi \in {\mathrm{Irr}}_K(G)$, and a $p$-element $u\in G$. Put $H=C_G(u)$. By Maschke's theorem the restricted character has a unique decomposition

${\mathrm{Res}}_H^G\chi ={\sum }_{\zeta \in {\mathrm{Irr}}_K(H)}{n}_{\chi ,\zeta }\zeta \mathrm{.}$

The element $u$ is central in $H$. On a simple $KH$-module affording $\zeta$, its action is therefore an $H$-endomorphism; Schur's lemma and the splitting-field condition make it a scalar ${\lambda }_{u,\zeta }\in K^{\times }$. For each irreducible Brauer character $\phi \in \mathrm{IBr}(H)$, define the generalized decomposition number

$d_{\chi ,\phi }^u:={\sum }_{\zeta \in {\mathrm{Irr}}_K(H)}{n}_{\chi ,\zeta }{\lambda }_{u,\zeta }{d}_{\zeta ,\phi },$

where $d_{\zeta ,\phi }$ is the ordinary decomposition number for $H$ from Decomposition numbers and the decomposition matrix.

The sum is finite and defines an element of $K$; unlike an ordinary decomposition number, it need not be a nonnegative integer. Since $u$ has $p$-power order, ${\lambda }_{u,\zeta }^{\mathrm{\mid }u\mathrm{\mid }}=1$. The next theorem identifies these scalars as the unique coefficients of the character expansion on the $p$-section defined in The p-section of a p-element. Brauer characters and their value convention are those of Brauer character of a finite-dimensional kG-module.