Peterson Isomorphism in $K$-theory and Relativistic Toda Lattice
Peterson Isomorphism in $K$-theory and Relativistic Toda Lattice
复制标题
DOI：
10.1093/imrn/rny051
复制
发表时间：
2017-03
期刊：
影响因子：
--
通讯作者：
T. Ikeda;Shinsuke Iwao;T. Maeno
中科院分区：
文献类型：
--
作者：
T. Ikeda;Shinsuke Iwao;T. Maeno
Lam、Schilling 和 Shimozono 研究了 $SL_n(C)$ 仿射格拉斯曼的 $K$ 同源环。它被实现为对称函数环的某个具体Hopf子环。另一方面，对于标志品种$Fl_n$的量子$K$理论，Kirillov和Maeno根据Givental和Lee获得的结果提供了推测性的表述。我们在这两个环的谱之间构造了一个显式的双有理态射。我们的方法依赖于 Ruijsenaars 具有单能初始条件的相对论 Toda 晶格。从这个结果中，我们获得了所谓的（共）同调的 Peterson 同构的 $K$ 理论类似物。我们提供了关于舒伯特基之间的详细关系的猜想，特别是，我们确定了与格拉斯曼排列相关的 Lenart-Maeno 量子 Grothendieck 多项式的图像。
The $K$-homology ring of the affine Grassmannian of $SL_n(C)$ was studied by Lam, Schilling, and Shimozono. It is realized as a certain concrete Hopf subring of the ring of symmetric functions. On the other hand, for the quantum $K$-theory of the flag variety $Fl_n$, Kirillov and Maeno provided a conjectural presentation based on the results obtained by Givental and Lee. We construct an explicit birational morphism between the spectrums of these two rings. Our method relies on Ruijsenaars's relativistic Toda lattice with unipotent initial condition. From this result, we obtain a $K$-theory analogue of the so-called Peterson isomorphism for (co)homology. We provide a conjecture on the detailed relationship between the Schubert bases, and, in particular, we determine the image of Lenart--Maeno's quantum Grothendieck polynomial associated with a Grassmannian permutation.