Generalized negligible morphisms and their tensor ideals

We introduce a generalization of the notion of a negligible morphism and study the associated tensor ideals and thick ideals. These ideals are defined by considering deformations of a given monoidal category $${\mathcal {C}}$$ C over a local ring R. If the maximal ideal of R is generated by a single element, we show that any thick ideal of $${\mathcal {C}}$$ C admits an explicitly given modified trace function. As examples we consider various Deligne categories and the categories of tilting modules for a quantum group at a root of unity and for a semisimple, simply connected algebraic group in prime characteristic. We prove an elementary geometric description of the thick ideals in quantum type A and propose a similar one in the modular case.

Tilting modules, dominant dimensions and Brauer-Schur-Weyl duality

In this paper we use the dominant dimension with respect to a tilting module to study the double centraliser property. We prove that if A A is a quasi-hereditary algebra with a simple preserving duality and T T is a faithful tilting A A -module, then A A has the double centralizer property with respect to T T . This provides a simple and useful criterion which can be applied in many situations in algebraic Lie theory. We affirmatively answer a question of Mazorchuk and Stroppel by proving the existence of a unique minimal basic tilting module T T over A A for which A = E n d E n d A ( T ) ( T ) A=End_{End_A(T)}(T) . As an application, we establish a Schur-Weyl duality between the symplectic Schur algebra S K s y ( m , n ) S_K^{sy}(m,n) and the Brauer algebra B n ( − 2 m ) \mathfrak {B}_n(-2m) on the space of dual partially harmonic tensors under certain condition.