Green vertices and sources of simple modules of the symmetric group labelled by hook partitions

Abstract

We determine the Green vertices and sources of many of the simple modular representations of the finite symmetric group being parametrized by hook partitions. Mathematics Subject Classification (2000). 20C30, 20C20.

On vertices of exterior powers of the natural simple module for the symmetric group in odd characteristic

“…In the situation of Corollary 5.5, there is a positive answer to Conjecture (1.7) (a) in [11]. So far, the case n = pw with w ≡ 1 (mod p) and r = p − 1 still remains open.…”

“…As already mentioned, the vertices of D r with r < p − 1 have been determined in [11]. For this reason, we are now mainly concerned with the case r ≥ p − 1.…”

“…. , n) has to be con- With Theorem 5.3, we now obtain the following corollary which is a generalization of Theorem (1.2) in [11]. Proof.…”

“…p-regular partitions of shape (n − r, 1 r ). In [11] J. Müller and R. Zimmermann dealt with these modules and determined their vertices except for the case where p is odd, p 2 ≤ n ≡ 0 (mod p) and r = p − 1. They showed, that the vertices of D (n−r,1 r ) are precisely the defect groups of its block except for n = 4, p = 2 and r = 1, where the Sylow 2-subgroup of the alternating group A 4 is a vertex of D (3,1) .…”

“…Thus, in this situation, there is a positive answer to Conjecture (1.7) (a) in [11]. After all, in order to determine the vertices of all simple F S n -modules corresponding to hook partitions (n − r, 1 r ), it remains to treat the case where p is odd, n = wp for some w ≡ 1 (mod p) and r = p − 1.…”