No discussion of the combinatorics of minuscule representations and their applications would be complete without mentioning the topics surveyed in this final chapter. Unlike the exceptional structures discussed in Chapters 9 and 10, the concepts discussed in Chapter 11 make sense for minuscule representations of arbitrary type.

Section 11.1 develops the theory of minuscule elements in Weyl groups. Theorem 11.1.18, which is due to Proctor, Stembridge, and Pfeiffer & Röhrle, describes how minuscule representations induce relationships between distinguished coset representatives, lattices, and the weak and strong Bruhat orders. Section 11.2 discusses Proctor's work on the combinatorics of the underlying posets of principal subheaps. This gives some insight into the nature of Gaussian posets and their connections with the theory of plane partitions, which is the subject of Section 11.3. We mention some of the points of contact between Gaussian posets and the cyclic sieving phenomenon of Reiner, Stanton and White [66]. Finally, Section 11.4 shows how the combinatorics of heaps can be used to calculate Schubert intersection numbers in algebraic geometry, following the work of Thomas and Yong [84].

Minuscule elements of Weyl groups

Let Γ be a Dynkin diagram. Recall from Lemma 3.1.10 that the Weyl group W(Γ) is a quotient of the commutation monoid Co(Γ). Matsumoto's Theorem (Theorem 4.3.1) shows that any two reduced expressions for an element ω ∈ W(Γ) are equivalent via a finite sequence of braid relations.