广义指数的组合数学
Combinatorics of generalized exponents
摘要：
我们给一个纯粹的组合证明的稳定形式的广义指数的积极性与每个经典的根系。在有限型A_{n-1}中，我们重新推导了广义指数的晶体图描述，而不使用半标准表的组合数学或电荷统计。在有限型C_n中，我们得到了基于A_{2n-1}型晶体中所谓的区别点的广义指数的组合描述，并将其与辛King tableaux联系起来.这给出了辛李代数的不可约表示中与零权空间相关的Lusztig t-模拟的正性的组合证明。我们还提出了我们的组合公式的三个应用，并讨论了一些影响有关的两个C型分支规则。我们的方法有望扩展到正交类型。
英文摘要：
We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type A_{n-1}, we rederive the description of the generalized exponents in terms of crystal graphs without using the combinatorics of semistandard tableaux or the charge statistic. In finite type C_n, we obtain a combinatorial description of the generalized exponents based on the so-called distinguished vertices in crystals of type A_{2n-1}, which we also connect to symplectic King tableaux. This gives a combinatorial proof of the positivity of Lusztig t-analogues associated to zero weight spaces in the irreducible representations of symplectic Lie algebras. We also present three applications of our combinatorial formula, and discuss some implications to relating two type C branching rules. Our methods are expected to extend to the orthogonal types.
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