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.

On Quiver Grassmannians and Orbit Closures for Gen-Finite Modules

We show that endomorphism rings of cogenerators in the module category of a finite-dimensional algebra A admit a canonical tilting module, whose tilted algebra B is related to A by a recollement. Let M be a gen-finite A-module, meaning there are only finitely many indecomposable modules generated by M. Using the canonical tilts of endomorphism algebras of suitable cogenerators associated to M, and the resulting recollements, we construct desingularisations of the orbit closure and quiver Grassmannians of M, thus generalising all results from previous work of Crawley-Boevey and the second author in 2017. We provide dual versions of the key results, in order to also treat cogen-finite modules.

Silting Complexes and Partial Tilting Modules

It is well known that a partial tilting module may not be completed to a tilting module. However, it is still unknown whether a partial tilting module can be completed to a silting complex. The affirmative answer to this question will give an affirmative answer to the well-known rank question for tilting modules. In this paper, we prove that a partial tilting simple module can always be completed to a silting complex. More generally, we give the sufficient conditions for a partial tilting module to be completed to a silting complex.

Counterexamples to the tilting and (p,r)-filtration conjectures

In this paper the authors produce a projective indecomposable module for the Frobenius kernel of a simple algebraic group in characteristic p that is not the restriction of an indecomposable tilting module. This yields a counterexample to Donkin’s longstanding Tilting Module Conjecture. The authors also produce a Weyl module that does not admit a p-Weyl filtration. This answers an old question of Jantzen, and also provides a counterexample to the {(p,r)}-Filtration Conjecture.

Lefschetz Operators, Hodge–Riemann Forms, and Representations

For a field of characteristic $\ne 2$, we study vector spaces that are graded by the weight lattice of a root system and are endowed with linear operators in each simple root direction. We show that these data extend to a weight lattice graded semisimple representation of the corresponding Lie algebra, if and only if there exists a bilinear form that satisfies properties (roughly) analogous to those of the Hodge–Riemann forms in complex geometry. In the 2nd part of the article, we replace the field by the $p$-adic integers (with $p\ne 2$) and show that in this case the existence of a certain bilinear form is equivalent to the existence of a structure of a tilting module for the associated simply connected $p$-adic Chevalley group.

REPETITIVE EQUIVALENCES AND TILTING THEORY

Let $R$ be a ring and $T$ be a good Wakamatsu-tilting module with $S=\text{End}(T_{R})^{op}$ . We prove that $T$ induces an equivalence between stable repetitive categories of $R$ and $S$ (i.e., stable module categories of repetitive algebras $\hat{R}$ and ${\hat{S}}$ ). This shows that good Wakamatsu-tilting modules seem to behave in Morita theory of stable repetitive categories as that tilting modules of finite projective dimension behave in Morita theory of derived categories.

Wakamatsu tilting modules over ring extension

For a ring [Formula: see text], an extension ring [Formula: see text], and a fixed right [Formula: see text]-module [Formula: see text], we prove the induced left [Formula: see text]-module [Formula: see text] is a Wakamatsu tilting module when [Formula: see text] is a Wakamatsu tilting module.