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

2021 ◽  
Vol 8 (26) ◽  
pp. 823-848
Author(s):  
Jun Hu ◽  
Zhankui Xiao
Keyword(s):  
Lie Theory ◽  
Tilting Module ◽  
Tilting Modules ◽  
Brauer Algebra ◽  
Hereditary Algebra ◽  
Dominant Dimension ◽  
Content Type ◽  
Double Centralizer ◽  
Schur Algebra ◽  
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.

CLOSE NEIGHBORS IN THE QUIVER OF TILTING MODULES

2008 ◽  
Vol 07 (03) ◽  
pp. 379-392
Author(s):  
DIETER HAPPEL
Keyword(s):  
Representation Type ◽  
Tilting Module ◽  
Tilting Modules ◽  
Hereditary Algebra ◽  
Simple Modules ◽  
Finite Dimensional ◽  
Wild Representation Type ◽  
Local Properties

For a finite dimensional hereditary algebra Λ local properties of the quiver [Formula: see text] of tilting modules are investigated. The existence of special neighbors of a given tilting module is shown. If Λ has more than 3 simple modules it is shown as an application that Λ is of wild representation type if and only if [Formula: see text] is a subquiver of [Formula: see text].

scholarly journals Tilting modules over tame hereditary algebras

2012 ◽  
Vol 2013 (682) ◽  
pp. 1-48
Author(s):  
Lidia Angeleri Hügel ◽  
Javier Sánchez
Keyword(s):  
Complete Classification ◽  
Tilting Module ◽  
Tilting Modules ◽  
Hereditary Algebra ◽  
Infinite Dimensional ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Hereditary Algebras ◽  
Tame Hereditary Algebra

Abstract. We give a complete classification of the infinite dimensional tilting modules over a tame hereditary algebra R. We start our investigations by considering tilting modules of the form where is a union of tubes, and denotes the universal localization of R at in the sense of Schofield and Crawley-Boevey. Here is a direct sum of the Prüfer modules corresponding to the tubes in . Over the Kronecker algebra, large tilting modules are of this form in all but one case, the exception being the Lukas tilting module L whose tilting class consists of all modules without indecomposable preprojective summands. Over an arbitrary tame hereditary algebra, T can have finite dimensional summands, but the infinite dimensional part of T is still built up from universal localizations, Prüfer modules and (localizations of) the Lukas tilting module. We also recover the classification of the infinite dimensional cotilting R-modules due to Buan and Krause.

scholarly journals Double Centralizer Properties, Dominant Dimension, and Tilting Modules

2001 ◽  
Vol 240 (1) ◽  
pp. 393-412 ◽  
Author(s):  
Steffen König ◽  
Inger Heidi Slungård ◽  
Changchang Xi
Keyword(s):  
Tilting Modules ◽  
Dominant Dimension ◽  
Double Centralizer

scholarly journals Radically filtered quasi-hereditary algebras and rigidity of tilting modules

2017 ◽  
Vol 163 (2) ◽  
pp. 265-288
Author(s):  
AMIT HAZI
Keyword(s):  
Closed Field ◽  
Tilting Module ◽  
Tilting Modules ◽  
Hereditary Algebra ◽  
Algebraically Closed Field ◽  
Hereditary Algebras ◽  
Composition Factors

AbstractLetAbe a quasi-hereditary algebra. We prove that in many cases, a tilting module is rigid (i.e. has identical radical and socle series) if it does not have certain subquotients whose composition factors extend more than one layer in the radical series or the socle series. We apply this theorem to show that the restricted tilting modules forSL4(K) are rigid, whereKis an algebraically closed field of characteristicp≥ 5.

scholarly journals Dominant and global dimension of blocks of quantised Schur algebras

2021 ◽  
Author(s):  
Ming Fang ◽  
Wei Hu ◽  
Steffen Koenig
Keyword(s):  
Hecke Algebras ◽  
Symmetric Groups ◽  
Global Dimension ◽  
Group Algebras ◽  
Dominant Dimension ◽  
Schur Algebras ◽  
Homological Dimensions ◽  
Weyl Duality

AbstractGroup algebras of symmetric groups and their Hecke algebras are in Schur-Weyl duality with classical and quantised Schur algebras, respectively. Two homological dimensions, the dominant dimension and the global dimension, of the indecomposable summands (blocks) of these Schur algebras S(n, r) and $$S_q(n,r)$$ S q ( n , r ) with $$n \geqslant r$$ n ⩾ r are determined explicitly, using a result on derived invariance in Fang, Hu and Koenig (J Reine Angew Math 770:59–85, 2021).

scholarly journals PROPERLY STRATIFIED ALGEBRAS AND TILTING

2005 ◽  
Vol 92 (1) ◽  
pp. 29-61 ◽  
Author(s):  
ANDERS FRISK ◽  
VOLODYMYR MAZORCHUK
Keyword(s):  
Projective Dimension ◽  
Finitistic Dimension ◽  
Tilting Module ◽  
Tilting Modules ◽  
Category Of Modules ◽  
Cotilting Modules ◽  
Contravariantly Finite ◽  
Properly Stratified Algebras ◽  
Stratified Algebra ◽  
Generalized Tilting Module

We study the properties of tilting modules in the context of properly stratified algebras. In particular, we answer the question of when the Ringel dual of a properly stratified algebra is properly stratified itself, and show that the class of properly stratified algebras for which the characteristic tilting and cotilting modules coincide is closed under taking the Ringel dual. Studying stratified algebras whose Ringel dual is properly stratified, we discover a new Ringel-type duality for such algebras, which we call the two-step duality. This duality arises from the existence of a new (generalized) tilting module for stratified algebras with properly stratified Ringel dual. We show that this new tilting module has a lot of interesting properties; for instance, its projective dimension equals the projectively defined finitistic dimension of the original algebra, it guarantees that the category of modules of finite projective dimension is contravariantly finite, and, finally, it allows one to compute the finitistic dimension of the original algebra in terms of the projective dimension of the characteristic tilting module.

Wakamatsu tilting modules over ring extension

2019 ◽  
Vol 18 (10) ◽  
pp. 1950198
Author(s):  
Zhen Zhang ◽  
Jiaqun Wei
Keyword(s):  
Ring Extension ◽  
Tilting Module ◽  
Tilting Modules ◽  
Wakamatsu Tilting Module ◽  
Extension Ring

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.

TILTING MODULES OF FINITE PROJECTIVE DIMENSION: SEQUENTIALLY STATIC AND COSTATIC MODULES

2002 ◽  
Vol 01 (03) ◽  
pp. 295-305 ◽  
Author(s):  
ALBERTO TONOLO
Keyword(s):  
Projective Dimension ◽  
Tilting Module ◽  
Tilting Modules ◽  
Finite Projective Dimension

In [5], Miyashita introduced tilting modules of finite projective dimension. A tilting module AV of projective dimension less or equal than r furnishes r + 1 equivalences between subcategories of A-Mod and End V-Mod: we call static and costatic the modules in A-Mod and End V-Mod, respectively, involved in these equivalences. In this paper we characterize the modules in A-Mod and End V-Mod which have a filtration with static and costatic factors, respectively.

scholarly journals From Jantzen to Andersen filtration via tilting equivalence

2012 ◽  
Vol 110 (2) ◽  
pp. 161 ◽  
Author(s):  
Johannes Kübel
Keyword(s):  
Verma Module ◽  
The Other ◽  
Tilting Module ◽  
Tilting Modules ◽  
Projective Object ◽  
Other Hand

The space of homomorphisms between a projective object and a Verma module in category $\mathcal O$ inherits an induced filtration from the Jantzen filtration on the Verma module. On the other hand there is the Andersen filtration on the space of homomorphisms between a Verma module and a tilting module. Arkhipov's tilting functor, a contravariant self-equivalence of a certain subcategory of $\mathcal O$, which maps projective to tilting modules induces an isomorphism of these kinds of Hom-spaces. We show that this equivalence identifies both filtrations.

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