Schubert Class and Cyclotomic NilHecke Algebras  

作　　者：Kai Zhou Jun Hu 

机构地区：[1]School of Mathematical Sciences,Zhejiang University,Hangzhou 310027,China [2]School of Mathematics and Statistics,Beijing Institute of Technology Beijing 100081,China

出　　处：《Algebra Colloquium》2021年第3期379-398,共20页代数集刊（英文版）

基　　金：The research was supported by the National Natural Science Foundation of China(No.11525102).

摘　　要：Let l and n be positive integers such that l≥n,and let Gn,l be the Grassmannian which consists of the set of n-dimensionsil subspaces of C^(l),There is a Z-graded algebra isomorphism between the cohomology H*(Gn,l,Z)of Gn,l and a natural Z-form B of the Z-graded basic algebra of the type A cyclotomic nilHecke algebraH^(0)l,n=<ψ1,…ψn-1,y1,…,yn>.We show that the isomorphism can be chosen such that the image of each(geometrically defined)Schubert class(a1,...,an)coincides with the basis element bλconstructed by Hu and Liang by purely algebraic method,where 0≤q1≤q2≤…≤an≤l-n with ai∈Z for each i,andλis the l-€-multipartition of n associated to(l+1-(an+n),l+1-(an-1+n-),...,l+1-(a1+1)).A similar correspondence between the Schubert class basis of the cohomology of the Grassmanni-an Gl-n,l and the bλ's basis(λis anl-multipartition of n with each component being either(1)or empty)of the natural Z-form B of the Z-graded basic algebra of H^(0)_(l,n)is also obtained.As an application,we obtain a second version of the Giambelli formula for Schubert classes.

关 键 词：cyclotomic nilHecke algebras GRASSMANNIANS Schur polynomials 

分 类 号：O15[理学—数学]