Chapter Eleven

Types

We now divide the prototypes into much smaller units that we call types. We fix a top and bottom row, and therefore a cartoon. For each episode ε of the cartoon, we fix an integer k_ε. Then the set of all short Gelfand-Tsetlin patterns (6.2) with the given top and bottom rows such that for each ε

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.

Let us choose Γ and Δ′ indexings as in Proposition 9.1. With notations as in that Proposition, and ε a fixed episode of the corresponding cartoon, there exist k and l such that (i) ∈ ε and (i) ∈ ε precisely when k ≤ i ≤ l. let

Then Proposition 9.1 implies that

for all elements of the type. We recall that our convention was that γ₀ = γ_2d+2 = 0. We take L_ε = 0 for the first (leftmost) cartoon and R_ε = 2d+2 for the last episode.

We may classify the possible episodes into four classes generalizing the classification in Table 9.1, and indicate in each case the locations of γ_Lε and γ_Rε in the Γ preaccordions, which may be checked by comparison with Table 9.1. Indeed, it must be remembered that in that proof, every panel of type R is replaced by one of type T or type B. Whichever choice is made, Table 9.1 gives the same location for L_ε and R_ε. The classification of the episode into one of four types is given in Table 11.1.

The location of and in the Δ preaccordion of may also be read off from Table 9.1. The classification of the episode into one of four classes is given in Table 11.2.

PROPOSITION 11.1 If is the episode that consecutively follows ε, then R_ε = L. The values γ_Lε() and γ_Rε() are constant on each type. Moreover G_Γ() = G_Δ() = 0 for all patterns in the type unless the γ_Lε are divisible by n. In the Γ and Δ′ preaccordions, γ_Lε and may be circled or not, but never boxed.

Table 11.1 The four classes of episodes in Γ.

Proof. From Tables 11.1 and 11.2, it is clear that R_ε = L for consecutive episodes.

If ε is of Class I or Class IV, then we see that

where the notation means that we sum over all episodes to the right of ε (including ε itself). If ε is of Class II or III, we have

In either case, these formulas imply that γ_Lε is constant on the patterns of the type.

Given their described locations, the fact that is never boxed in either the Γ or Δ′ preaccordions may be seen from the definitions.

We now show that G_Γ() = G_Δ() = 0 unless n|γ_Lε. Indeed, it follows from an examination of the locations of γ_Lε in the Γ preaccordions and in the Δ′ preaccordions (where the same value appears as ) that this entry is unboxed in both Γ and Δ′. If it is uncircled in the Γ preaccordion, then G_Γ() is divisible by h(γ_Lε), hence vanishes unless n|γ_Lε. If it is circled, then we apply the Circling Lemma (Lemma 9.6) to conclude that the same value appears somewhere else uncircled and unboxed, unless γ_Lε = 0 (which is divisible by n) which again forces G_Γ() = 0 if n γ_Lε; and similarly for G_Γ() = 0.

Table 11.2 The four classes of episodes in Δ.

• Due to this result, we may impose the assumption that n|γ_Lε for every episode. This assumption is in force for the rest of the book.

Now let ε₁, · · · , ε_N be the episodes of the cartoon arranged from left to right, and let k_i = k(ε_i). By a local pattern on ε_i subordinate to we mean an integer-valued function on ε_i that can occur as the restriction of an element of to ε_i. Its top and bottom rows are thus the restrictions of the given top rows, and it follows from the definition of the episode that if (0, t) and (2, t − 1) are both in ε_i then (0, t) = (2, t − 1); that is, if both an element of the top row and the element of the bottom row that is directly below it are in the same episode, then has the same value on both, and patterns in the type are resonant at t. The local pattern is subject to the same inequalities as a short pattern, and by (11.2) the sum of its first (middle) row elements must be k_i. Let _i be the set of local patterns subordinate to . We call _i a local type.

LEMMA 11.2 A pattern is in if and only if its restriction to ε_i is in _i for each i and so we have a bijection

Proof. This is obvious from the definitions, since the inequalities (11.1) for the various episodes are independent of each other.

Now if is a short pattern let us define for each episode ε

provided is locally strict at ε, by which we mean that if α, β ∈ ε ∩ Θ₁ and α is to the left of β then (α) > (β). If is not locally strict, then we define .

PROPOSITION 11.3 Suppose that n|γ_Lε for every episode. Assume also that for each _i we have

Then

This proposition is the bridge between types and local types. Two observations are implicit in the statement of (11.4).

• Since by its definition () depends only on the restriction _i of to _i, we may write (_i) instead of , and this is well-defined.
• The statement uses the fact that γ_Lε() and γ_Rε() are constant on the type, since otherwise would be inside the summation.

Proof. If _i ∈ _i is the restriction of ∈ , we have

By Proposition 11.1 we have R_εi = L_εi+1. Since our convention is that γ₀ = γ_2d+2 = 0, it follows that the factors cancel. Thus

This completes the proof.

In the rest of the chapter we will fix an episode ε = ε_i, and denote L = L_ε and R = R_ε to simplify the notation. The four remaining Propositions give relations between the Γ and Δ′ preaccordions within the episode ε.

PROPOSITION 11.4 Let be a short pattern whose cartoon contains the following Class II resonant episode ε of order d:

Then there exist integers s, μ₁, ν₁, μ₂, ν₂, · · · , μ_d, ν_d such that μ_i + ν_i = s (i = 1, · · · , d), and the Γ and Δ′ preaccordions are given in the following table.

The values s, γ_L and γ_R are constant on the type containing the pattern.

Note: If the episode ε occurs at the left edge of the cartoon, then our convention is that γ_L = γ₀ = 0, and if ε occurs at the right edge of the cartoon, then γ_R = γ_2d+2 = 0. We would modify the picture by omitting γ_L or γ_R in these cases, but the proof below is unchanged.

Proof. Let γ_L = γ_Lε and γ_R = γ_Rε in the notation of the previous chapter, and let s, μ_i, ν_i be defined by their locations in the Γ preaccordion. Let s = + γ_R + γ_L, and . It is immediate from the definitions that

From this it we see that μ_i + ν_i = s and .

In order to check the correctness of the Δ′ diagram, we observe that the resonance contains d panels of type R, each of which may be specialized to a panel of type T or B. We specialize these to panels of type T. We obtain the following canonical snakes, representing the Γ and Δ′ preaccordions.

Looking at the even-numbered locations in these indexings, starting with γ_L = , Proposition 9.1 asserts that the values ν₁, · · · , ν_d are as advertised in the Δ′ labeling. It also asserts that the value at the first odd-numbered location, which is the first spot in the bottom row of the episode, is (s − γ_L) + γ_L − ν₁ = μ₁; the second odd-numbered location gets the value μ₁ + ν₁ − ν₂ = μ₂, and so forth.

PROPOSITION 11.5 Let be a short pattern whose cartoon contains a Class I resonant episode ε of order d. There exist integers s, μ₁, ν₁, μ₂, ν₂, · · · , μ_d, ν_d such that μ_i + ν_i = s (i = 1, · · · , d), and the portions of in Γ and Δ′ preaccordions in ε are given in the following table.

The values s, γ_L, and γ_R are constant on the type containing the pattern.

Proof. We define s, μ_i and ν_i to be the quantities that make the Γ preaccordion correct. The correctness of the second diagram may be proved using snakes as in Proposition 11.4. The proof that μ_i + ν_i = s is also similar to Proposition 11.4.

PROPOSITION 11.6 Let be a short pattern whose cartoon contains a Class III resonant episode ε of order d. There exist integers s, μ₁, ν₁, μ₂, ν₂, · · · , μ_d, ν_d such that μ_i + ν_i = s (i = 1, · · · , d), and the portions of the Γ and Δ′ preaccordions in ε are given in the following table.

The values s, γ_L, γ_R and ξ are constant on the type containing the pattern.

Proof. This is similar and left to the reader.

PROPOSITION 11.7 Let be a short pattern whose cartoon contains a Class IV resonant episode ε of order d. There exist integers s, μ₁, ν₁, μ₂, ν₂, · · · , μ_d, ν_d such that μ_i + ν_i = s (i = 1, · · · , d), and the portions of the Γ and Δ′ preaccordions in ε are given in the following table.

The values s, γ_L, γ_R and ξ are constant on the type containing the pattern.

Proof. This is similar and left to the reader.