Some decomposition numbers of Hecke algebras
We give some decomposition numbers of the Hecke algebra of type \(F_4\), \(H(F_4)\), and of Hecke algebras of type \(D_n\), \(H(D_n)\). The Hecke algebra of type \(F_4\) is defined here as a vector space over a field \(K\) of arbitrary characteristic and its definition depends on \(q\), \(Q\) where \(q\), \(Q\) are non-zero elements of \(K\). Geck and Lux have calculated the decomposition numbers of \(H(F_4)\) in the special case \(q=Q\) and Bremke has completed the same task for the cases \(q=Q^2\) and \(q=Q^4\). In this paper we give the composition factors of the ordinary irreducible representations of \(H(F_4)\) of dimension at most 4 for all choices of parameters of \(q\) and \(Q\). We also give a complete set of ordinary irreducible representations of \(H(F_4)\) of dimension at most 6. The representing matrices given here involve no denominators except in one of the cases where they involve denominators of the form \(Q^i\) (which are never zero). The representations of Hecke algebras of type \(D_n\), \(n\) odd, have been constructed on the restriction that a certain polynomial in \(q\) is non-zero. (The definition of \(H(D_n)\) depends on \(q\in K\).) Under the same restrictions on \(q\) we show here that, with a suitable ordering of the rows and columns, the decomposition matrices of Hecke algebras of type \(D_n\) for \(n\) odd are lower unitriangular and we give some examples of such matrices for small values of \(n\). (In the case where \(n\) is even part of the decomposition matrix is obtained).
scientific article; zbMATH DE number 1014021