The number of simple modules of a cellular algebra. (Q2496381)

Suppose that \(A\) is an indecomposable cellular algebra such that the spectrum of its Cartan matrix is of the form \(\{n,1,1,\dots,1\}\). The authors show that the number of non-isomorphic simple modules of \(A\) lies in a certain set [see \textit{J. J. Graham} and \textit{G. I. Lehrer}, Invent. Math. 123, No. 1, 1-34 (1996; Zbl 0853.20029) for the definition of cellular algebras]. The authors also give a partial answer to the question as to whether, for each number \(m\) in this set, there is such an algebra with \(m\) non-isomorphic simple modules. The proof involves the application of some number-theoretic results in terms of sums of squares and the consideration of special sets of partitions.