Separating Characters by Blocks

Abstract

We investigate the problem of finding a set of prime divisors of the order of a finite group, such that no two irreducible characters are in the same p-block for all primes p in the set. Our main focus is on the simple and quasi-simple groups. For results on the alternating and symmetric groups and their double covers, some combinatorial results on the cores of partitions are proved, which may be of independent interest. We also study the problem for groups of Lie type. The sporadic groups (and their relatives) are checked using GAP.

Principally separated non-separated solvable groups

“…In [1], Bessenrodt, Malle and Olsson introduced the idea of separability of characters by blocks. If G is a finite group, IrrðGÞ is the set of irreducible complex characters of G, and p is a set of primes, we say that p separates G (by blocks) if for every two distinct w; c A IrrðGÞ there exists a prime p A p such that w and c lie in di¤erent pblocks.…”

“…The group G is separated if the set of all prime divisors of G separates G. If p ¼ fpg, then G is p-separated if and only if G is a p 0 -group, so one generally considers only sets p with more than one prime. It is shown in [1] that many simple and quasi-simple groups are separated by at most four of their prime divisors. It is shown in [4] that, for each n A N, there exist separated solvable finite groups which are not separated by any set of prime divisors of their order with fewer than n primes.…”

Bar weights of bar partitions and spin character zeros

“…Proof: For the spin characters ofS n , this follows from Theorem 1.1, after checking that the spin characters for the other exceptional partition labels for small n, i.e., (4,3,2), (4, 3, 2, 1), (7,3), indeed have zeros of odd prime order; in fact, they vanish on elements of cycle type (3 2 , 1 * ). For (3,2), (3, 2, 1) or (5, 2, 1), the corresponding spin characters do not have a zero of odd prime order.…”

Principal Blocks and the Steinberg Character

“…Then, if χ is an irreducible character of G, the value of the central character corresponding to χ on the class sum containing s equals (1) |G : C|χ(s)/χ(1).…”

“…Which are the finite simple groups of Lie type of characteristic p, such that the Steinberg character is in the principal -block for all primes = p dividing the group order? In the notation of [1], this asks for those groups G of this class for which the trivial character and the Steinberg character of G are not separated by π(G) \ {p}.…”