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.

中文翻译：

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K 理论中的彼得森同构和相对论 Toda 格

Lam、Schilling 和 Shimozono 研究了 $SL_n(C)$ 的仿射 Grassmannian 的 $K$-同源环。它被实现为对称函数环的某个具体的 Hopf 子环。另一方面，对于旗品种 $Fl_n$ 的量子 $K$ 理论，Kirillov 和 Maeno 提供了基于 Givental 和 Lee 获得的结果的推测性陈述。我们在这两个环的光谱之间构建了一个明确的双有理态射。我们的方法依赖于 Ruijsenaars 的具有单能初始条件的相对论 Toda 格。从这个结果中，我们获得了所谓的（共）同调的彼得森同构的 $K$ 理论模拟。我们提供了一个关于舒伯特基之间详细关系的猜想，特别是我们确定了 Lenart--Maeno' 的形象