Some inequalities between invariants of blocks
The second author [Arch. Math. 73, No. 6, 407-413 (1999; Zbl 0959.20013)] showed if \(B\) is a \(p\)-block of a finite group \(G\) with Cartan matrix \(C=(c_{ij})\), \(k(B)\) ordinary irreducible characters, and \(\ell(B)\) irreducible Brauer characters, then \[ k(B)\leq\sum_{i=1}^{\ell(B)}c_{ii}-\sum_{i=1}^{\ell(B)-1}c_{i,i+1}. \] Here, the authors generalize this by showing that \[ k(B)\leq\sum_{1\leq i\leq j\leq\ell(B)}q_{ij}c_{ij},\text{ where }\sum_{1\leq i\leq j\leq\ell(B)}q_{ij}X_iX_j \] is any weakly positive integal quadratic form. Some conditions for equality to hold are also given.
scientific article; zbMATH DE number 1837269