By Ben Brubaker

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Additional resources for Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory

Sample text

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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