TY - JOUR

T1 - Schubert Class and Cyclotomic NilHecke Algebras

AU - Zhou, Kai

AU - Hu, Jun

N1 - Publisher Copyright:
© 2021 Academy of Mathematics and Systems Science, Chinese Academy of Sciences.

PY - 2021/9

Y1 - 2021/9

N2 - 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-dimensional subspaces of Cℓ. There is a Z-graded algebra isomorphism between the cohomology H(Gn,ℓ,M) of G n,ℓ and a natural Z-form B of the Z-graded basic algebra of the type A cyclotomic nilHecke algebra Hℓ,n(0)= ℓ ψ1,.., ψ n-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 bλ constructed by Hu and Liang by purely algebraic method, where 0≤a1≤ a2≤⋯≤an≤ℓ-n with ai Z for each i, and λ is the ℓ-multipartition of n associated to (ℓ+1-(an+n),ℓ+1-(an-1+n-1),..,ℓ+1-(a1+1)). A similar correspondence between the Schubert class basis of the cohomology of the Grassmannian Gℓ-n,ℓ and the bλ's basis (λ is an ℓ-multipartition of n with each component being either (1) or empty) of the natural Z-form B of the Z-graded basic algebra of Hℓ,n (0) is also obtained. As an application, we obtain a second version of the Giambelli formula for Schubert classes.

AB - 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-dimensional subspaces of Cℓ. There is a Z-graded algebra isomorphism between the cohomology H(Gn,ℓ,M) of G n,ℓ and a natural Z-form B of the Z-graded basic algebra of the type A cyclotomic nilHecke algebra Hℓ,n(0)= ℓ ψ1,.., ψ n-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 bλ constructed by Hu and Liang by purely algebraic method, where 0≤a1≤ a2≤⋯≤an≤ℓ-n with ai Z for each i, and λ is the ℓ-multipartition of n associated to (ℓ+1-(an+n),ℓ+1-(an-1+n-1),..,ℓ+1-(a1+1)). A similar correspondence between the Schubert class basis of the cohomology of the Grassmannian Gℓ-n,ℓ and the bλ's basis (λ is an ℓ-multipartition of n with each component being either (1) or empty) of the natural Z-form B of the Z-graded basic algebra of Hℓ,n (0) is also obtained. As an application, we obtain a second version of the Giambelli formula for Schubert classes.

KW - Grassmannians

KW - Schur polynomials

KW - cyclotomic nilHecke algebras

UR - http://www.scopus.com/inward/record.url?scp=85111463226&partnerID=8YFLogxK

U2 - 10.1142/S1005386721000304

DO - 10.1142/S1005386721000304

M3 - Article

AN - SCOPUS:85111463226

SN - 1005-3867

VL - 28

SP - 379

EP - 398

JO - Algebra Colloquium

JF - Algebra Colloquium

IS - 3

ER -