Puzzles and (equivariant) cohomology of Grassmannians
Puzzles and (equivariant) cohomology of Grassmannians
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DOI：
10.1215/s0012-7094-03-11922-5
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发表时间：
2001-12
影响因子：
2.5
通讯作者：
A. Knutson;T. Tao
中科院分区：
文献类型：
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作者：
A. Knutson;T. Tao
我们将Grassmannians上的普通Schubert微积分的谜题公式推广到T等变的Schubert微积分公式。要计算的结构常数是{y_{i+1}-y_i}中的多项式；它们在[Graham]Math.AG/9908172中(抽象地)显示为正系数。在这个意义上，我们的公式是第一个明显是积极的。特别是，通过从“最等变”的情况向后归纳法，这给出了普通谜题公式的一个新的、完备的证明。该公式的证明主要是组合的，但不需要事先的组合学，只需要少量的等变上同调(我们包括在内)。这个公式与[Molev-Sagan]q-ALG/9707028中关于三组变量阶乘Schur函数相乘的公式密切相关，尽管他们的规则没有给出[Graham]意义下的正公式。我们包括对这个问题的上同调解释，以及对它的谜题表述。
We generalize our puzzle formula for ordinary Schubert calculus on Grassmannians, to a formula for the T-equivariant Schubert calculus. The structure constants to be calculated are polynomials in {y_{i+1} - y_i}; they were shown (abstractly) to have positive coefficients in [Graham] math.AG/9908172. Our formula is the first to be manifestly positive in this sense. In particular this gives a new and self-contained proof of the ordinary puzzle formula, by an induction backwards from the "most equivariant" case. The proof of the formula is mostly combinatorial, but requires no prior combinatorics, and only a modicum of equivariant cohomology (which we include). This formula is closely related to the one in [Molev-Sagan] q-alg/9707028 for multiplying factorial Schur functions in three sets of variables, although their rule does not give a positive formula in the sense of [Graham]. We include a cohomological interpretation of this problem, and a puzzle formulation for it.