Defect zero blocks for finite simple groups

Abstract

We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a p p -block with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero p − p- blocks remained unclassified were the alternating groups A n A_{n} . Here we show that these all have a p p -block with defect 0 for every prime p ≥ 5 p\geq 5 . This follows from proving the same result for every symmetric group S n S_{n} , which in turn follows as a consequence of the t t -core partition conjecture, that every non-negative integer possesses at least one t t -core partition, for any t ≥ 4 t\geq 4 . For t ≥ 17 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 t>17 , that was not covered in previous work, was the case t = 13 t=13 . This we prove with a very different argument, by interpreting the generating function for t 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 p -blocks of S n S_{n} , proving a conjecture of Garvan, that establishes certain multiplicative congruences when 5 ≤ p ≤ 23 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 p and positive integer m m , the number of p − p- blocks with defect 0 in S n S_n is a multiple of m m for almost all n n . We also establish that any given prime p p divides the number of p − p- modularly irreducible representations of S n S_{n} , for almost all n n .