The main combinatorial framework we will use in this book to study minuscule representations is that of heaps. A heap is a certain function from a partially ordered set to the set of vertices of a graph. We will often think of the heap as the Hasse diagram of the underlying poset, where the elements of the poset are labelled by vertices of the graph. For most purposes in this book, thinking of heaps pictorially as labelled partially ordered sets (as in Figure 2.7 below) will be adequate. However, for later applications, we will need concepts such as those of subheaps and quotient heaps, and it is unwieldy to describe these in terms of pictures. For such applications, it turns out to be concise and natural to define heaps in terms of categories, and in Section 2.1, we introduce the category of heaps and explain the connection between finite heaps and commutation monoids. A reader not comfortable with categories may skip most of Chapter 2 at a first reading, except Section 2.2, which introduces the key notion of “full heap”.

For the applications in this book, the most important kind of heap is those for which the associated graph is a Dynkin diagram, particularly one of finite or affine type. Dynkin diagrams are certain graphs that may have multiple and directed edges. The information in a Dynkin diagram may also be captured using amatrix with integer entries, called the generalized Cartan matrix of a Dynkin diagram.