论文标题
$ K $ - 理论加泰罗尼亚功能
$K$-theoretic Catalan functions
论文作者
论文摘要
我们证明，$ K $ - $ K $ -SCHUR功能是一个非构态对称函数家族的一部分，其最高均匀组件是加泰罗尼亚函数，是标志品种上某些向量捆绑包的欧拉特征。 Lam-Shilling-Shimozono确定了$ k $ -k $ -schur的功能，是舒伯特代表的$ k $ - affine grassmannian sl $ _ {k+1} $的代表。我们的观点表明，$ k $ - $ k $ -schur的功能满足了转移不变性的属性，我们从鲍德温和库马尔的积极性结果中推断出其分支系数的积极性。我们进一步表明，对$ K $ -K $ -SCHUR功能的配方进行稍微调整会产生第二个换档不变的基础，该基础具有阳性分支和矩形分解属性。在Ikeda-iwao-Maeno的作品的基础上，我们猜想这第二个基础为Lenart-Maeno量子Grothendieck多项式提供了$ k $ theeretic类似物的图像 - 彼得森同构的理论类似物。
We prove that the $K$-$k$-Schur functions are part of a family of inhomogenous symmetric functions whose top homogeneous components are Catalan functions, the Euler characteristics of certain vector bundles on the flag variety. Lam-Schilling-Shimozono identified the $K$-$k$-Schur functions as Schubert representatives for $K$-homology of the affine Grassmannian for SL$_{k+1}$. Our perspective reveals that the $K$-$k$-Schur functions satisfy a shift invariance property, and we deduce positivity of their branching coefficients from a positivity result of Baldwin and Kumar. We further show that a slight adjustment of our formulation for $K$-$k$-Schur functions produces a second shift-invariant basis which conjecturally has both positive branching and a rectangle factorization property. Building on work of Ikeda-Iwao-Maeno, we conjecture that this second basis gives the images of the Lenart-Maeno quantum Grothendieck polynomials under a $K$-theoretic analog of the Peterson isomorphism.
