Defect zero blocks for finite simple groups

Abstract:

We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a $p$-block with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero $p-$blocks remained unclassified were the alternating groups $A_{n}$. Here we show that these all have a $p$-block with defect 0 for every prime $p\geq 5$. This follows from proving the same result for every symmetric group $S_{n}$, which in turn follows as a consequence of the $t$-core partition conjecture, that every non-negative integer possesses at least one $t$-core partition, for any $t\geq 4$. For $t\geq 17$, we reduce this problem to Lagrange’s Theorem that every non-negative integer can be written as the sum of four squares. The only case with $t<17$, that was not covered in previous work, was the case $t=13$. This we prove with a very different argument, by interpreting the generating function for $t$-core partitions in terms of modular forms, and then controlling the size of the coefficients using Deligne’s Theorem (née the Weil Conjectures). We also consider congruences for the number of $p$-blocks of $S_{n}$, proving a conjecture of Garvan, that establishes certain multiplicative congruences when $5\leq p \leq 23$. By using a result of Serre concerning the divisibility of coefficients of modular forms, we show that for any given prime $p$ and positive integer $m$, the number of $p-$blocks with defect 0 in $S_n$ is a multiple of $m$ for almost all $n$. We also establish that any given prime $p$ divides the number of $p-$modularly irreducible representations of $S_{n}$, for almost all $n$.