We prove that there exists a unique irreducible Brauer character of height zero in any 2-blocks of the symmetric group. This generalizes the theorem of P. Fong and G.D. James that the dimension of every non-trivial 2-modular simple module of the symmetric group is even. © 2012 Elsevier Inc.
Kiyota, M., Okuyama, T., & Wada, T. (2012). The heights of irreducible Brauer characters in 2-blocks of the symmetric groups. Journal of Algebra, 368, 329–344. https://doi.org/10.1016/j.jalgebra.2012.07.001