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The packets, we observe, can be identified empirically in any given case, but are difficult to characterize in general, and not even uniquely determined in some cases. See [9]. To proceed further, we introduce the notion of a short Gelfand-Tsetlin pattern or (for short) short pattern. By this we mean an array with just three rows   l1 l2 ··· ld+1   l0 a0 a1 ad t= . 2)   b0 ··· bd−1 where the rows are nonincreasing sequences of integers that interleave, that is, li ai li+1 , ai bi ai+1 . 3) We will refer to l0 , · · · , ld+1 as the top or zero-th row of t, a0 , · · · , ad as the first or middle row and b0 , · · · , bd−1 as the second or bottom row .

If T is a Gelfand-Tsetlin pattern, then in Γ(T) we use the right-hand rule in every row, and in ∆(T) we use the left-hand row in every row. But if t is a short Gelfand-Tsetlin pattern then in Γt and ∆t one row uses the right-hand rule, the other the left-hand rule. 39 Let us define  g(x)    x q GΓ (t) = h(x)  x∈Γt   0 if if if if x x x x is is is is boxed in Γt , but not circled; circled, but not boxed; neither boxed nor circled; both boxed and circled. Thus if t is not strict, then GΓ (t) = 0.

Proof Let Bµ be the Ar−1 crystal containing v which is obtained from Bλ by deleting the r-labeled edges. We make use of the word Ω∆,r−1 = (r − 1, r − 2, r − 1, r − 3, r − 2, r − 1, · · · , 1, 2, 3, · · · , r − 1) which represents the long element of Ar−1 and obtain a path v 0 0 0 ··· r − 1 r − 2 r − 1 ··· 0 b1 b2 b3 · · · r − 1 1 2 3 ··· br−1 r−1 where the initial string of 0’s is explained by the fact that fi v = 0 when 2 Thus we could equally well write v b1 b2 b3 · · · 1 2 3 ··· br r v i r −1.

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Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory by Ben Brubaker

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