An elementary construction of monomial bases of modified quantum affine ${\mathfrak{gl}}_n$

Abstract

In 1990 Beilinson, Lusztig and MacPherson provided a geometric realization of modified quantum ${\mathfrak{gl}}_n$ and its canonical basis. A key step of their work is a construction of a monomial basis. Recently, Du and Fu provided an algebraic construction of the canonical basis for modified quantum affine ${\mathfrak{gl}}_n$, which among other results used an earlier construction of monomial bases using Ringel-Hall algebra of the cyclic quiver. In this paper, we give an elementary algebraic construction of a monomial basis for affine Schur algebras and modified quantum affine ${\mathfrak{gl}}_n$.

14 pages. Accepted by the Journal of London Mathematical Society. The introduction is reworded, and the references are updated. Some proofs are removed and now referred to the proper literature

References in corpus (4)

• The Integral quantum loop algebra of ${\mathfrak{gl}}_n$
• Positivity vs negativity of canonical bases
• Quantum affine ${\mathfrak{gl}}_n$ via Hecke algebras
• Positivity properties for canonical bases of modified quantum affine ${\mathfrak{sl}}_n$

Cited by in corpus (2)

• Affine flag varieties and quantum symmetric pairs
• Remarks on PBW bases of Ringel-Hall algebras of cyclic quivers