Schubert Class and Cyclotomic NilHecke Algebras

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Abstract: Let

e l l, n

be positive integers such that

e l l g e q n

. Let

m a t h b b G_{n, e l l}

be the Grassmannian which consists of the set of

n

-dimensional subspaces of

m a t h b b C^{e l l}

. There is a

m a t h b b Z

-graded algebra isomorphism between the cohomology

H^{*} (m a t h b b G_{n, e l l}, m a t h b b Z)

of

m a t h b b G_{n, e l l}

and a natural

m a t h b b Z

-form

B

of the

m a t h b b Z

-graded basic algebra of the type

A

cyclotomic nilHecke algebra

H_{e l l, n}^{(0)} = l a n g l e p s i_{1}, c d o t s, p s i_{n - 1}, y_{1}, c d o t s, y_{n} a n g l e

. In this paper, we show that the isomorphism can be chosen such that the image of each (geometrically defined) Schubert class

(a_{1}, c d o t s, a_{n})

coincides with the basis element

b_{m a t h b f l a m b d a}

constructed by Jun Hu and Xinfeng Liang by purely algebraic method, where

0 l e q a_{1} l e q a_{2} l e q c d o t s l e q a_{n} l e q e l l - n

with

a_{i} i n m a t h b b Z

for each

i

,

m a t h b f l a m b d a

is the

e l l

-multipartition of

n

associated to

(e l l + 1 - (a_{n} + n), e l l + 1 - (a_{n - 1} + n - 1), c d o t s, e l l + 1 - (a_{1} + 1))

. A similar correspondence between the Schubert class basis of the cohomology of the Grassmannian

m a t h b b G_{e l l - n, e l l}

and the

b_{l a m b d a}

's basis of the natural

m a t h b b Z

-form

B

of the

m a t h b b Z

-graded basic algebra of

H_{e l l, n}^{(0)}

is also obtained. As an application, we obtain a second version of Giambelli formula for Schubert classes.

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