Spread-invariant representations of symmetric groups (Q912207)

In his overview paper [in Proc. Symp. Pure Math. 47, 211-222 (1987; Zbl 0654.20005)] the second author has set up a program to study translation planes via faithful spread-invariant representations. This means modular representations whose representation space V possesses a partition into subspaces of half dimension, where the parts are permuted by the induced action of the group G. The result of the paper concerns the case G being the symmetric group of degree \(n\geq 4\). If, in addition, for each prime \(p\leq n\), p divides q-1, q the number of the elements in the field K, then V is a free KG-module and the non-trivial subspace of G-invariant elements of V has a spread which is induced from the spread of V. The main tool of the proof are consequences of the Murnaghan-Nakayama rule.