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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 the definitions of the Weyl group multiple Dirichlet series into the language of crystal bases. It reinterprets the entries in these arrays and the accompanying boxing and circling rules in terms of the Kashiwara operators. Thus, what appeared as a pair of unmotivated functions on Gelfand-Tsetlin patterns in the previous chapter now takes on intrinsic representation theoretic meaning. The discussion is restricted to crystals of Cartan type Aᵣ. The Weyl vector, denoted by ρ‎, is considered as an element of the weight lattice, and the bijection between Gelfand-Tsetlin patterns and tableaux is described. The chapter also examines the λ‎-part of the multiple Dirichlet series in terms of crystal graphs.

This chapter describes Type A Weyl group multiple Dirichlet series. It begins by defining the basic shape of the class of Weyl group multiple Dirichlet series. To do so, the following parameters are introduced: Φ‎, a reduced root system; n, a positive integer; F, an algebraic number field containing the group μ‎₂ₙ of 2n-th roots of unity; S, a finite set of places of F containing all the archimedean places, all places ramified over a ℚ; and an r-tuple of nonzero S-integers. In the language of representation theory, the weight of the basis vector corresponding to the Gelfand-Tsetlin pattern can be read from differences of consecutive row sums in the pattern. The chapter considers in this case expressions of the weight of the pattern up to an affine linear transformation.

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 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 introduces a method of marking up a short Gelfand-Tsetlin pattern based on inequalities between its entries, that encodes the effect of the involution t 7 → t′ and the boxing and circling of its accordion. This will have another benefit: it will lead to the decomposition of the pattern into pieces called episodes that will ultimately lead to the reduction to the totally resonant case. The proof is easily checked using standard properties of Gauss sums. To define the cartoon, the chapter takes a slightly more formal approach to the short Gelfand-Tsetlin patterns. The vertices of the cartoon will be the elements of the substrate Θ‎, and the edges must be defined.

This chapter introduces the Knowability Lemma, which explains when products of Gauss sums associated to elements of a preaccordion are explicitly evaluable as polynomials in q, the order of the residue class field. It considers an episode in the cartoon associated to the short Gelfand-Tsetlin pattern and the three cases that apply according to the Knowability Lemma, two of which are maximality and knowability. Knowability is not important for the proof that Statement C implies Statement B. The chapter discusses the cases where ε‎ is Class II or Class I, leaving the remaining two cases to the reader. It also describes the variant of the argument for the case that ε‎ is of Class I, again leaving the two other cases to the reader.

This chapter introduces the Tokuyama's Theorem, first by writing the Weyl character formula and restating Schur polynomials, the values of the Whittaker function multiplied by the normalization constant. The λ‎-parts of Whittaker coefficients of Eisenstein series can be profitably regarded as a deformation of the numerator in the Weyl character formula. This leads to deformations of the Weyl character formula. Tokuyama gave such a deformation. It is an expression of ssubscript Greek small letter lamda(z) as a ratio of a numerator to a denominator. The denominator is a deformation of the Weyl denominator, and the numerator is a sum over Gelfand-Tsetlin patterns with top row λ‎ + ρ‎.

This chapter divides the prototypes into much smaller units called types. It fixes a top and bottom row, and therefore a cartoon. For each episode ε‎ of the cartoon, the chapter fixes an integer κ‎subscript Greek small letter epsilon. Then the set of all short Gelfand-Tsetlin patterns with the given top and bottom rows is called a type. Thus two patterns are in the same type if and only if they have the same top and bottom rows (and hence the same cartoon), and if the sum of the first (middle) row elements in each episode is the same for both patterns. The possible episodes may be grouped into four classes: Class I, II, III, and IV.

This chapter focuses on the language of resotopes and assumes that γ‎Lsubscript Greek small letter epsilon and γ‎Rsubscript Greek small letter epsilon are multiples of n for every totally resonant episode. It also recalls that s and d are the weights of the accordions under consideration. It begins with the proposition that Statement D is equivalent to Statement C, and that Statement D is true if n ≠ s. It then describes the case of a totally resonant short Gelfand-Tsetlin pattern before presenting the proof that Statement D implies Statement B. It shows that the reduction to Statement D is straightforward for Class I, but each of the remaining classes involves some nuances.