A geometric realisation of 0-Schur and 0-Hecke algebras

TL;DR

This paper introduces a geometric framework for understanding 0-Schur and 0-Hecke algebras via flag varieties, providing new bases, idempotents, and a presentation over non-invertible q.

Contribution

It constructs a geometric realisation of 0-Schur and 0-Hecke algebras, linking them to flag varieties and quiver representations, and explores their bases and algebraic structures.

Findings

Defined a new associative algebra G(n,r) from flag orbits

Established G(n,r) as a geometric realisation of 0-Schur algebra

Constructed bases and idempotents for 0-Schur algebras

Abstract

We define a new product on orbits of pairs of flags in a vector space, using open orbits in certain varieties of pairs of flags. This new product defines an associative $\mathbb{Z}$-algebra, denoted by $G(n,r)$. We show that $G(n,r)$ is a geometric realisation of the 0-Schur algebra $S_0(n, r)$ over $\mathbb{Z}$, which is the $q$-Schur algebra $S_q(n,r)$ at q=0. We view a pair of flags as a pair of projective resolutions for a quiver of type $\mathbb{A}$ with linear orientation, and study $q$-Schur algebras from this point of view. This allows us to understand the relation between $q$-Schur algebras and Hall algebras and construct bases of $q$-Schur algebras, which are used in the proof of the main results. Using the geometric realisation, we construct idempotents and multiplicative bases for 0-Schur algebras. We also give a geometric realisation of 0-Hecke algebras and a presentation of the $q$-Schur algebra over a base ring where $q$ is not invertible.