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Weyl Group Multiple Dirichlet Series by Ben Brubaker, Daniel Bump, Solomon Friedberg

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Chapter Two

Crystals and Gelfand-Tsetlin Patterns

We will translate the definitions of the Γ and Δ arrays in (1.11), and hence of the multiple Dirichlet series, into the language of crystal bases. The entries in these arrays and the accompanying boxing and circling rules will be reinterpreted 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. Despite the conceptual importance of this reformulation, the reader can skip this chapter and the subsequent chapters devoted to crystals with no loss of continuity. For further background information on crystals, we recommend Hong and Kang [42] and Kashiwara ...

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