Crystal approach to affine Schubert calculus
Crystal approach to affine Schubert calculus
复制标题
仿射舒伯特微积分的晶体方法
DOI：
10.1093/imrn/rnv194
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发表时间：
2014
影响因子：
1
通讯作者：
A. Schilling
中科院分区：
文献类型：
--
作者：
J. Morse;A. Schilling
作者：莫尔斯，詹妮弗;席林，安妮|摘要：我们应用晶体理论仿射舒伯特演算，Gromov-Witten不变量的完整的旗流形，和正的格拉斯曼分层。我们引入操作员的类型$A$仿射Weyl群中的元素的分解，并产生一个晶体反映的内部结构的广义Young模块的Frobenius图像是由稳定的舒伯特多项式。我们应用晶体框架的产品Schur函数与$k$-Schur函数，从而证明了一个子类的3点Gromov-Witten不变量的完整的标志品种$\mathbb C^n$枚举的最高重量的元素下，这些运营商。这个类中包括舒伯特结构常数，它是舒伯特多项式与舒尔函数s_\lambda$的（量子）乘积，对于所有|\lambda^\vee| l n$。另一个副产品给出了一个最高的权重制定各种融合系数的Verlinde代数和舒伯特分解的某些positroid类。
Author(s): Morse, Jennifer; Schilling, Anne | Abstract: We apply crystal theory to affine Schubert calculus, Gromov-Witten invariants for the complete flag manifold, and the positroid stratification of the positive Grassmannian. We introduce operators on decompositions of elements in the type-$A$ affine Weyl group and produce a crystal reflecting the internal structure of the generalized Young modules whose Frobenius image is represented by stable Schubert polynomials. We apply the crystal framework to products of a Schur function with a $k$-Schur function, consequently proving that a subclass of 3-point Gromov-Witten invariants of complete flag varieties for $\mathbb C^n$ enumerate the highest weight elements under these operators. Included in this class are the Schubert structure constants in the (quantum) product of a Schubert polynomial with a Schur function $s_\lambda$ for all $|\lambda^\vee|l n$. Another by-product gives a highest weight formulation for various fusion coefficients of the Verlinde algebra and for the Schubert decomposition of certain positroid classes.