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This chapter recalls the use of the Schützenberger involution on Gelfand-Tsetlin patterns to prove that Statement B implies Statement A. These statements will be discussed two more times in the later chapters of the book. Chapter 18 reinterprets both Statements A and B in terms of crystals, and directly proves that the reinterpreted Statement B implies the reinterpreted Statement A in Theorem 18.2. Then Chapter 19 again reinterprets Statements A and B in a different context, and yet again directly proves that the reinterpreted Statement B implies the reinterpreted Statement A in Theorem 19.10. The current chapter shows that the Schützenberger involution qᵣ can be formulated in terms of operations on short Gelfand-Tsetlin patterns. To facilitate the inductive proof, relevant equations are used.

This chapter presents the proof of the equivalence of the two definitions for the λ‎-parts in terms of Gelfand-Tsetlin patterns. The equivalence of these two descriptions is a deep fact that uses subtle combinatorial manipulations depending in an essential way on the properties of λ‎-th order Gauss sums. This equivalence is the key step in demonstrating functional equations. In outlining the proof, the chapter introduces many concepts and ideas as well as several equivalent forms of the result, called Statements A through G. Each statement is an intrinsically combinatorial identity involving products of Gauss sums, but with each statement the nature of the problem changes. The first reduction, Statement B, changes the focus from Gelfand-Tsetlin patterns to “short” Gelfand-Tsetlin patterns, consisting of just three rows.

This chapter translates Statements A and B into Statements A′ and B′ in the language of crystal bases, and explains in this language how Statement B′ implies Statement A′. It first introduces the relevant definition, which is provisional since it assumes that we can give an appropriate definition of boxing and circling for Ω‎. The crystal graph formulation in Statement A′ is somewhat simpler than its Gelfand-Tsetlin counterpart. In particular, in the formulation of Statement A, there were two different Gelfand-Tsetlin patterns that were related by the Schützenberger involution. In the crystal graph formulation, different decompositions of the long element simply result in different paths from the same vertex v to the lowest weight vector.

This chapter reinterprets Statements A and B in a different context, and yet again directly proves that the reinterpreted Statement B implies the reinterpreted Statement A in Theorem 19.10. The p-parts of Weyl group multiple Dirichlet series, with their deformed Weyl denominators, may be expressed as partition functions of exactly solved models in statistical mechanics. The transition to ice-type models represents a subtle shift in emphasis from the crystal basis representation, and suggests the introduction of a new tool, the Yang-Baxter equation. This tool was developed to prove the commutativity of the row transfer matrix for the six-vertex and similar models. This is significant because Statement B can be formulated in terms of the commutativity of two row transfer matrices. This chapter presents an alternate proof of Statement B using the Yang-Baxter equation.