On the Affine Schur Algebra of Type A

Abstract

Let n, r ∈ ℕ. The affine Schur algebra (of type A) over a field K is defined to be the endomorphism algebra of certain tensor space over the extended affine Weyl group of type A _r−1. By the affine Schur–Weyl duality it is isomorphic to the image of the representation map of the action on the tensor space when K is the field of complex numbers. We show that can be defined in another two equivalent ways. Namely, it is the image of the representation map of the semigroup algebra (defined in Section 3) action on the tensor space, and it equals to the “dual” of a certain formal coalgebra related to this semigroup. By these approaches, we can show many relations between different Schur algebras and affine Schur algebras and reprove one side of the affine Schur–Weyl duality.

Key Words:

• Affine Schur algebra
• Formal coalgebra
• Loop algebra
• Schur algebra

2000 Mathematics Subject Classification:

• 16D90
• 17B35

ACKNOWLEDGMENTS

The author acknowledges support by the AsiaLink network Algebras and Representations in China and Europe, ASI/B7-301/98/679-11.

The author thanks Steffen König for his inspiring and encouraging supervision, and he thanks the referee for much help in preparing the final version of this article.

Notes

Communicated by D. K. Nakano.