Parabolic coset algebras of W(H4), the Schur index of a character of degree 48, and non-commutative association schemes on the edges and cells of the 600-cell

We study the rational parabolic coset algebras of the Coxeter group W(H4), the symmetry group of the 600-cell. Of the 34 irreducible characters of W(H4), exactly one, the rational character of degree 48, has Schur index greater than 1 over Q; its index is 2 and the corresponding division algebra is the quaternion algebra (2,3) over Q. We determine the rational Wedderburn decompositions of the coset algebras of the edges and cells of the 600-cell, and show that this quaternion algebra occurs in both as the endomorphism algebra of a common simple summand. We also give explicit spectra of the vertex graph and edge line graph. This is a preliminary report; complete proofs will appear in a forthcoming paper. Let W = W(H4) be the symmetry group of the 600-cell, F = Q(sqrt(5)), and, for a standard parabolic subgroup W_J, let H_J = End_{QW}(Q[W/W_J]) be the rational coset algebra. We announce the following results, obtained by character theory and exact computation: (1) Of the 34 irreducible characters of W, exactly one has Schur index > 1 over Q, the rational character phi_48 of degree 48, whose index is 2; the other 33 indices are certified by odd multiplicities in seven rational modules induced from linear characters of parabolic subgroups together with the Brauer-Speiser theorem, without using the Benard-Bessis splitting field theorem. (2) The corresponding division algebra is D = (2,3)_Q, and every H_J has exactly one non-split simple factor M_{m_J/2}(D) unless J is in {Sigma, {s2, s3, s4}}; the vertex algebra of the 600-cell is split, H_{J_V} = Q^5 + F^2. (3) For the 720 edges and 600 cells, H_E and H_C have dimensions 62 and 45 (the numbers of orbitals), 32 and 27 self-paired orbitals, and Wedderburn decompositions that we determine over Q; D occurs in both as End_{QW}(S) for a simple module S of dimension 96 occurring exactly once in each permutation module, and dim Hom_{QW}(Q[cells], Q[edges]) = 49. (4) The adjacency matrices of the vertex graph and of the edge line graph have explicit spectra whose rank-one spectral idempotents certify four matrix blocks of H_E over Q. Complete proofs will appear in a forthcoming paper.`