Let K be a field of characteristic two, and let λ be a two-part partition of some natural number r. Denote the permutation module corresponding to the (maximal) Young subgroup Σλ in Σr by Mλ. We construct a full set of orthogonal primitive idempotents of the centraliser subalgebra SK (λ) = 1λ SK (2, r) 1λ = EndK Σr (Mλ) of the Schur algebra SK (2, r). These idempotents are naturally in one-to-one correspondence with the 2-Kostka numbers. © 2006 Elsevier Inc. All rights reserved.
Doty, S., Erdmann, K., & Henke, A. (2007). Endomorphism rings of permutation modules over maximal Young subgroups. Journal of Algebra, 307(1), 377–396. https://doi.org/10.1016/j.jalgebra.2006.02.040