The quadratic type of certain irreducible modules for the symmetric group in characteristic two

Let λ be a 2-regular partition of n into two parts and let Dλ denote the corresponding irreducible -module. We say that Dλ is of quadratic type if there is a non-degenerate Σn-invariant quadratic form defined on this module. In this paper, we show that Dλ is not of quadratic type precisely when the smaller part of λ is a power of 2, say 2r, where r 0, and n≡kmod2r+2, where k is one of the 2r consecutive integers 2r+1+2r−1, …, 2r+2−2.