Abstract
We define a new product on orbits of pairs of flags in a vector space over a field k, using open orbits in certain varieties of pairs of flags. This new product defines an associative Z-algebra, denoted by G(n,r). We show that G(n,r) is a geometric realisation of the 0-Schur algebra S0(n,r) over Z, which is the q-Schur algebra Sq(n,r) at q=0. A pair of flags naturally determines a pair of projective resolutions for a quiver of type A with linear orientation, and we study q-Schur algebras from this point of view. This allows us to understand the relation between q-Schur algebras and Hall algebras and to construct bases of q-Schur algebras. 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 ground ring, where q is not invertible.
Highlights
Let k be a finite or an algebraically closed field and F the variety of partial n-step flags in an r-dimensional vector space V over k
In [1], using the double flag variety F ×F, Beilinson, Lusztig and MacPherson gave a geometric construction of some finite dimensional quotients of the quantised enveloping algebra Uq(gln)
The lemma shows that a pair of flags in F × F and a triple (α, [M ], [N ]), where α ∈ Λ(n, r) and [M ], [N ] are isomorphism classes of representations M and N of Λ with a surjection P (α) → M ⊕ N, mutually determine each other
Summary
Let k be a finite or an algebraically closed field and F the variety of partial n-step flags in an r-dimensional vector space V over k. In this paper the q-Schur algebras Sq(n, r) are defined as the quotients constructed in [1], which we recall below. We say that this q-Schur algebra is an R-algebra to emphasise the ground ring R.
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