Variations of conjectures on counting irreducible characters of finite groups (Algebraic Combinatorics)
Katsuhiro Uno · Institutional Repositories DataBase (IRDB) · 2003
IntroductionLet $G$ be afinite group, $p$ aprime, $R$ the ring of algebraic integers in some finite Galois extension field $K$ of $\mathrm{Q}$ which contains enough roots of unity, $P$ aprime ideal of $R$ lying over $p\mathrm{Z}$ , $R_{P}$ the localization of $R$ at $P$ , and $k$ be the residue class field $R_{P}/PR_{P}$ of characteristic $p$ .For terminology used in modular representation theory, see [10].Let Irr(G) be the set of complex irreducible characters of $G$ , $e$ aprimitive idempotent of the center $Z(R_{P}G)$ of $R_{P}G$ , i.e., ablock idempotent of $G$ .Let $B$ be the-We say that $\chi\in \mathrm{I}\mathrm{r}\mathrm{r}(G)$ belongs to $B$ and write $\chi\in B$ , if $\chi(e) eq 0$ .-We also say that an indecomposable right $kG$ modular $M$ belongs to $B$ , if $M\overline{e} eq$ $0$ , where $\overline{e}$ is the image of $e$ via the canonical epimorphism from $R_{P}G$ to $kG$ .We also write $M\in B$ .Definition 1.1 For $\chi\in \mathrm{I}\mathrm{r}\mathrm{r}(G)$ , let $d(\chi)$ be the exponent of the highest power of $p$ in $|G|/\chi(1)$ .Let $d(B)= \max\{d(\chi)|\chi\in B\}$ .-For ablock $B$ , there exists a $p$-subgroup $D$ of $G$ such that every irreducible kG- module belonging to $B$ is isomorphic to adirect summand of a $kG$ -module induced from a $kD$ -module and that $|D|=p^{d(B)}$ .The above $D$ is unique up to $G$ -conjugate and called adefect group of $B$ .-For any $\chi\in \mathrm{I}\mathrm{r}\mathrm{r}(G)$ and aconjugacy classgives a $k$ -algebra homomorphism.It does not depend on the choice of $\chi\in B$ and is denoted by $\omega_{B}$ .Definition 1.2 Let $B$ be a block of $G$ and $H$ a subgroup of G.If a block $b$ of $H$ satisfies $\omega_{B}(\hat{C})=\omega_{b}(\overline{C\cap H})$ for all conjugacy class $C$ , of $G$ , th en we write $b^{G}=B$ .If this is the case, ate call $b^{G}$ the induced block and that the induced block can be