Endomorphism rings of permutation modules.
Let \(k\) be an algebraically closed field, let \(G\) be a finite group, and let \(P\) be a Sylow \(p\)-subgroup of \(G\). In this paper, the author studies the permutation module \(k_P^G=\mathrm{Ind}_P^G(k)\) and its endomorphism ring \(\mathfrak E=\mathrm{End}_{kG}(k_P^G)\). The paper is motivated by \textit{J. L. Alperin}'s suggestion to investigate his weight conjecture [Proc. Symp. Pure Math. 47, 369-379 (1987; Zbl 0657.20013)] through the endomorphism rings of certain permutation modules. The author focuses on the socle and head constituents of the indecomposable direct summands of \(k_P^G\) and of the projective indecomposable modules of \(\mathfrak E\). The computational methods utilize the computer algebra system GAP. The examples considered are: \(A_6\bmod 2\); \(S_6\bmod 3\); \(A_7\bmod 2,3\); \(S_7\bmod 2,3\); \(A_8\bmod 3\), \(A_9\bmod 2,3\); \(M_{11}\bmod 2,3,5,11\); \(M_{12}\bmod 2,3\); \(M_{22}\bmod 2\); \(U_3(3)\bmod 2\); \(J_2\bmod 2\); \(L_2(11)\bmod 2\); \(L_2(13)\bmod 2,3\); \(L_2(17)\bmod 2,3\); \(L_2(19)\bmod 2,3\).
scientific article; zbMATH DE number 5817242
scientific article