One of the deepest parts of the representation theory of finite groups is the theory of blocks with cyclic defect in which a vast array of detailed and useful information is obtained. These results are a model for much further research and open conjectures. A striking feature of this theory is how a graph, the Brauer tree, encapsulates almost all the results. This graph is a tree together with a planar embedding and a selected vertex, called the exceptional vertex, to which is attached a positive integer, the multiplicity. In the course of giving an exposition of the module-theoretic approach to this theory [l], a critical step was a combinatorial result about the Brauer tree. This observation is a special case of a more general and more symmetric result about trees with multiplicities which we shall present here.
To describe our result let T be a tree with multiplicities so each vertex has attached a positive integer, its multiplicity. Suppose that T has e edges and e + 1 vertices. Let C be the "Cartan matrix" (a name motivated by representation theory) of T so C is the square matrix with rows and columns indexed by the vertices with the i, j entry being the number of vertices, counting multiplicities, that the ith and jth edges have in common. Thus, if two distinct edges do not have a vertex in common then the corresponding entry is zero and otherwise it is the multiplicity of the one vertex these edges have in common, while the diagonal entry corresponding to an edge is the sum of the multiplicities of its two vertices. Theorem. If the multiplicities of the vertices of T are ml, m2, . . . , m, then the determinant of the Cartan matrix C is ? c. . m,,m,.-*rnj . . m,.
As usual, the caret denotes omission of the factor under it. We wish to indicate, as was shown to us, that the special case, where all the multiplicities are one, is a standard result. Indeed, under this assumption, if D is the e + 1 by e incidence matrix of T then C = D'D; but the characteristic polynomial of DD' is