Let G be a semisimple, simply-connected algebraic group over an algebraically closed field of characteristic p > 0. We observe that the tensor product of the Steinberg module with a minuscule module is always indecomposable tilting. Although quite easy to prove, this fact does not seem to have been observed before. It has the following consequence: If p >= 2h-2 and a given tilting module has highest weight p-adically close to the r-th Steinberg weight, then the tilting module is isomorphic to a tensor product of two simple modules, usually in many ways.
We describe an algorithm, which - given the characters of tilting modules and assuming that Donkin's tilting conjecture is true - computes the characters of simple modules for an algebraic group in any characteristic.
We construct a representation of the blob algebra over a ring allowing base change to every interesting (i.e. non--semisimple) specialisation which, in quasihereditary specialisations, passes to a full tilting module.
In this paper the authors provide a complete answer to Donkin's Tilting Module Conjecture for all rank $2$ semisimple algebraic groups and $\text{SL}_{4}(k)$ where $k$ is an algebraically closed field of characteristic $p>0$. In the process, new techniques are introduced involving the existence of $(p,r)$-filtrations, Lusztig's character formula, and the $G_{r}$T-radical series for baby Verma modules.
Following the previous work, we determine exactly the highest weights for which a tensor product of two induced modules is a tilting module, for the algebraic group $SL_2$ over an algebraically closed field of positive characteristic.
We give a complete picture of when the tensor product of an induced module and a Weyl module is a tilting module for the algebraic group $SL_2$ over an algebraically closed field of characteristic $p$. Whilst the result is recursive by nature, we give an explicit statement in terms of the $p$-adic expansions of the highest weight of each module.
Let $G$ be a simple and simply connected algebraic group over an algebraically closed field $\Bbbk$ of characteristic $p>0$. We establish an isomorphism of $G$-modules between a direct sum of modules $\text{St} \otimes \text{St}$ and a direct sum of tensor products of simple modules of restricted highest weight with tilting modules that are projective over the Frobenius kernel of $G$. This isomorphism holds precisely when Donkin's Tilting Module Conjecture does, and thus can be seen as providing a $G$-module theoretic characterization of this conjecture.
In this article we study support $τ$-tilting modules, semibricks and more over blocks of group algebras. Let $k$ be an algebraically closed field of characteristic $p>0$, $\tilde{G}$ a finite group and $G$ a normal subgroup of $\tilde{G}$. Moreover, let $\tilde{B}$ be a block of $k\tilde{G}$ and $B$ a block of $kG$ covered by $\tilde{B}$. We show that, under certain conditions for the factor group $\tilde{G}/G$ and $B$, induced modules and extending modules of support $τ$-tilting modules and semibricks over $B$ are also the ones over $\tilde{B}$, respectively.
We show that the characters of tilting modules can be used, in a concrete and explicit way, to obtain the simple characters of a connected reductive algebraic group $G$ over an algebraically closed field $\Bbbk$ of characteristic $p$, for all $p$. Thus, once a formula for the characters of the indecomposable tilting $G$-modules has been found, a formula for the simple modules has been also. An immediate implication is that the work of Achar, Makisumi, Riche, and Williamson in \cite{AMRW} provides a character formula for simple $G$-modules when $p>h$, the Coxeter number of $G$.
Let G be a semisimple, simply connected linear algebraic group over an algebraically closed field k of characteristic p > 0. In a recent paper [4], Doty introduces the notion of r-minuscule weight and exhibits a tensor product factorization of a corresponding tilting module under the assumption p >= 2h-2, where h is the coxeter number. We remove this restriction and consider some variations involving the more general notion of (p,r)-minuscule weights.
Let G be a semisimple group over an algebraically closed field of characteristic p>0. We give a (partly conjectural) simple, closed formula for the character of many indecomposable tilting rational G-modules, assuming that p is large.
Linear differential algebraic groups (LDAGs) appear as Galois groups of systems of linear differential and difference equations with parameters. These groups measure differential-algebraic dependencies among solutions of the equations. LDAGs are now also used in factoring partial differential operators. In this paper, we study Zariski closures of LDAGs. In particular, we give a Tannakian characterization of algebraic groups that are Zariski closures of a given LDAG. Moreover, we show that the Zariski closures that correspond to representations of minimal dimension of a reductive LDAG are all i…
Let $G$ be a simple algebraic group over an algebraically closed field of prime characteristic. If $M$ is a finite dimensional $G$-module that is projective over the Frobenius kernel of $G$, then its character is divisible by the character of the Steinberg module. In this paper we study such quotients, showing that if $M$ is an indecomposable tilting module, then the multiplicities of the orbit sums appearing in its "Steinberg quotient" are well behaved.
We describe the semisimplification of the monoidal category of tilting modules for the algebraic group GL_n in characteristic p > 0. In particular, we compute the dimensions of the indecomposable tilting modules modulo p.
Let $A$ be a finite dimensional algebra over an algebraically closed field $k$. We investigate the structure properties of the endomorphism algebras of semi-tilting $A$-modules, and prove that the endomorphism algebras arising from the mutations of semi-tilting $A$-modules can be realized as the endomorphism algebras of BB-tilting modules.
We give a construction of Gorenstein projective $τ$-tilting modules in terms of tensor products of modules. As a consequence, we give a class of non-self-injective algebras admitting non-trivial Gorenstein projective $τ$-tilting modules. Moreover, we show that a finite dimensional algebra $Λ$ over an algebraically closed field is $CM$-$τ$-tilting finite if $T_n(Λ)$ is $CM$-$τ$-tilting finite which gives a partial answer to a question on $CM$-$τ$-tilting finite algebras posed by Xie and Zhang.
Let C be a connected noetherian hereditary abelian Ext-finite category with Serre functor over an algebraically closed field k, with finite dimensional homomorphism and extension spaces. Using the classification of such categories from math.RT/9911242, we prove that if C has some object of infinite length, then the Grothendieck group of C is finitely generated if and only if C has a tilting object.
Let $G$ be a simply-connected semisimple algebraic group over an algebraically closed field of characteristic $p$, assumed to be larger than the Coxeter number. The "support variety" of a $G$-module $M$ is a certain closed subvariety of the nilpotent cone of $G$, defined in terms of cohomology for the first Frobenius kernel $G_1$. In the 1990s, Humphreys proposed a conjectural description of the support varieties of tilting modules; this conjecture has been proved for $G = \mathrm{SL}_n$ in earlier work of the second author. In this paper, we show that for any $G$, the support variety of a til…
We use minimal tilting complexes to construct an explicit bijection between the set of thick tensor ideals with the two-out-of-three property in the category of finite-dimensional modules over a quantum group at a root of unity and the set of thick tensor ideals in the subcategory of tilting modules. We also explain why the analogous construction for rational representations of a reductive algebraic group over a field of positive characteristic does not give rise to a bijection.
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.
We use the category of linear complexes of tilting modules for the BGG category O, associated with a semi-simple complex finite-dimensional Lie algebra g, to reprove in purely algebraic way several known results about O obtained earlier by different authors using geometric methods. We also obtain several new results about the parabolic category O.
We prove that for $d \geq 2$, an algebraic $d$-Calabi-Yau triangulated category endowed with a $d$-cluster tilting subcategory is the stable category of a DG category which is perfectly $(d+1)$-Calabi-Yau and carries a non degenerate $t$-structure whose heart has enough projectives.
We study the quiver with relations of the endomorphism algebra of an APR tilting module. We give an explicit description of the quiver with relations by graded quivers with potential (QPs) and mutations. The result also implies that mutations of QPs with algebraic cuts induce tilting mutations between the associated truncated Jacobian algebras. Consequently, mutations of QPs provide a rich source of derived equivalence classes of algebras. As an application, we give a sufficient condition of QPs such that Derksen-Weyman-Zelevinsky's question has a positive answer.
For quantum groups at a root of unity, there is a web of theorems (due to Bezrukavnikov and Ostrik, and relying on work of Lusztig) connecting the following topics: (i) tilting modules; (ii) vector bundles on nilpotent orbits; and (iii) Kazhdan-Lusztig cells in the affine Weyl group. In this paper, we propose a (partly conjectural) analogous picture for reductive algebraic groups over fields of positive characteristic, inspired by a conjecture of Humphreys.
Let $\tilde{G}$ be a finite group, $G$ a normal subgroup of $\tilde{G}$ and $k$ an algebraically closed field of characteristic $p>0$. The first main result in this paper is to show that support $τ$-tilting $k\tilde{G}$-modules satisfying some properties are support $τ$-tilting modules as $kG$-modules too. As the second main result, we give equivalent conditions for support $τ$-tilting $k\tilde{G}$-modules to satisfy the above properties, and show that the set of the support $τ$-tilting $k\tilde{G}$-modules with the properties is isomorphic to the set of $\tilde{G}$-invariant support $τ$-tilti…
Let $A$ be a finite dimensional algebra over an algebraically closed field $k$. Let $T$ be a tilting $A$-module and $B={\rm End}_A\ T$ be the endomorphism algebra of $T$. In this paper, we consider the correspondence between the tilting $A$-modules and the tilting $B$-modules, and we prove that there is a one-one correspondence between the basic $T$-tilting $A$-modules in $T^{\perp}$ and the basic tilting $B$-modules in $^{\perp}(D_BT)$. Moreover, we show that there is a one-one correspondence between the $T$-contravariantly finite $T$-resolving subcategories of $T^{\perp}$ and the basic $T$-t…
Let $A$ be a hereditary algebra over an algebraically closed field $k$ and $A^{(m)}$ be the $m$-replicated algebra of $A$. Given an $A^{(m)}$-module $T$, we denote by $δ(T)$ the number of non isomorphic indecomposable summands of $T$. In this paper, we prove that a partial tilting $A^{(m)}$-module $T$ is a tilting $A^{(m)}$-module if and only if $δ(T)=δ(A^{(m)})$, and that every partial tilting $A^{(m)}$-module has complements. As an application, we deduce that the tilting quiver $\mathscr{K}_{A^{(m)}}$ of $A^{(m)}$ is connected. Moreover, we investigate the number of complements to almost til…
Let G be a reductive algebraic group over a field k. When k=C, R.K.Brylinski constructed a filtration of weight spaces of a G module, using the action of a principal nilpotent element of the Lie algebra, and proved that this filtration corresponds to Lusztig's q-analogue of the weight multiplicity. Later, Ginzburg discovered that this filtration has an interesting geometric interpretation via the geometric Satake correspondence. The goal of this article is to generalize these results to positive characteristics.
Let $p$ be a prime number, $k$ an algebraically closed field of characteristic $p$, $\tilde{G}$ a finite group, and $G$ a normal subgroup of $\tilde{G}$ having a $p$-power index in $\tilde{G}$. Moreover let $B$ be a block of $kG$ with a cyclic defect group and $\tilde{B}$ be the unique block of $k\tilde{G}$ covering $B$. We study tilting complexes over the block $\tilde{B}$ and show that the block $\tilde{B}$ is a tilting-discrete algebra. Moreover we show that the set of all tilting complexes over $\tilde{B}$ is isomorphic to that over $B$ as partially ordered sets.
This is the second paper of a series of papers on a version of categories $\mathcal{O}$ for root-reductive Lie algebras. Let $\mathfrak{g}$ be a root-reductive Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic $0$ with a splitting Borel subalgebra $\mathfrak{b}$ containing a splitting maximal toral subalgebra $\mathfrak{h}$. For some pairs of blocks $\overline{\mathcal{O}}[λ]$ and $\overline{\mathcal{O}}[μ]$, the subcategories whose objects have finite length are equivalence via functors obtained by the direct limits of translation functors. Tilting objects can also…
A finite directed category is a $k$-linear category with finitely many objects and an underlying poset structure, where $k$ is an algebraically closed field. This concept unifies structures such as $k$-linerizations of posets and finite EI categories, quotient algebras of finite-dimensional hereditary algebras, triangular matrix algebras, etc. In this paper we study representations of finite directed categories, discuss their stratification properties, and show the existence of generalized APR tilting modules for triangular matrix algebras under some assumptions.
Let $A$ be a finite dimensional hereditary algebra over an algebraically closed field $k$, $A^{(m)}$ be the $m$-replicated algebra of $A$ and $\mathscr{C}_{m}(A)$ be the $m$-cluster category of $ A$. We investigate properties of complements to a faithful almost complete tilting $A^{(m)}$-module and prove that the $m$-cluster mutation in $\mathscr{C}_{m}(A)$ can be realized in ${\rm mod} A^{(m)}$, which generalizes corresponding results on duplicated algebras established in [Z1].
We give an algorithm for working out the indecomposable direct summands in a Krull--Schmidt decomposition of a tensor product of two simple modules for G=SL_3 in characteristics 2 and 3. It is shown that there is a finite family of modules such that every such indecomposable summand is expressible as a twisted tensor product of members of that family. Along the way we obtain the submodule structure of various Weyl and tilting modules. Some of the tilting modules that turn up in characteristic 3 are not rigid; these seem to provide the first example of non-rigid tilting modules for algebraic gr…
Let $\Field$ be an algebraically closed field. For $n \in \mathbb{N}$ and $δ, δ_L, δ_R, κ_L, κ_R, κ\in \Field$, the symplectic blob algebra $\sba(δ, δ_L, δ_R, κ_L, κ_R, κ)$ is a finite dimensional non-commutative $\Field$-algebra that may be viewed as an extension of the Temperley-Lieb algebra. In a previous paper, we defined, for any $n \in \mathbb{N}$, a tensor space module $\tensor[_\sba]{\mathcal{V}(n)}{}$. In this paper we generalise an argument used by Martin and Ryom-Hansen in their study of the (ordinary) blob algebra to show that when $\sba$ is quasihereditary the module $\tensor[_\sb…
We apply tilting theory over preprojective algebras $Lambda$ to a study of moduli space of $Lambda$-modules. We define the categories of semistable modules and give an equivalence, so-called reflection functors, between them by using tilting modules over $Lambda$. Moreover we prove that the equivalence induces an isomorphism of algebraic varieties between moduli spaces. In particular, we study in the case when the moduli spaces related to the Kleinian singularity. We generalize a result of Crawley-Boevey which is known another proof of the McKay correspondence of Ito-Nakamura type.
We prove that some subquotient categories of exact categories are abelian. This generalizes a result by Koenig-Zhu in the case of (algebraic) triangulated categories. As a particular case, if an exact category B with enough projectives and injectives has a cluster tilting subcategory M, then B/M is abelian. More precisely, it is equivalent to the category of finitely presented modules over the stable category of M.
In this paper we propose a new approach to tilting modules for reductive algebraic groups in positive characteristic. We conjecture that translation functors give an action of the (diagrammatic) Hecke category of the affine Weyl group on the principal block. Our conjecture implies character formulas for the simple and tilting modules in terms of the p-canonical basis, as well as a description of the principal block as the anti-spherical quotient of the Hecke category. We prove our conjecture for GL_n using the theory of 2-Kac-Moody actions. Finally, we prove that the diagrammatic Hecke categor…
Let $G$ be a connected reductive algebraic group over an algebraically closed field $\Bbbk$ of characteristic $p \ge 0$, and let $\mathcal{N}$ be its nilpotent cone. Under mild hypotheses, we construct for each nilpotent $G$-orbit $C$ and each indecomposable tilting vector bundle $T$ on $C$ a certain complex $S(C,T)$ of $G \times \mathbb{G}_m$-equivariant coherent sheaves on $\mathcal{N}$. We prove that these objects are (up to shift) precisely the indecomposable objects in the coheart of a certain co-$t$-structure. We then show that if $p$ is larger than the Coxeter number, then the hypercoho…
Triangle presentations are combinatorial structures on finite projective geometries which characterize groups acting simply transitively on the vertices of a locally finite building of type $\tilde{\text{A}}_{n-1}$ ($n\ge3$). From a type $\tilde{\text{A}}_{n-1}$ triangle presentation on a geometry of order $q$, we construct a fiber functor on the diagrammatic monoidal category $\text{Web}(\text{SL}^{-}_{n})$ over any field $\mathbb{k}$ with characteristic $p\ge n-1$ such that $q \equiv 1$ mod $p$. When $\mathbb{k}$ is algebraically closed and $n$ odd, this gives new fiber functors on the categ…
Let $A$ be a finite dimensional algebra over an algebraically closed field $k$, and $M$ be a partial tilting $A$-module. We prove that the Bongartz $τ$-tilting complement of $M$ coincides with its Bongartz complement, and then we give a new proof of that every almost complete tilting $A$-module has at most two complements. Let $A=kQ$ be a path algebra. We prove that the support $τ$-tilting quiver $\overrightarrow{Q}({\rm s}τ$-${\rm tilt} A)$ of $A$ is connected. As an application, we investigate the conjecture of Happel and Unger in [9] which claims that each connected component of the tilting…
Let $\mathbf{G}$ be a connected reductive group over an algebraically closed field $\mathbb{F}$ of good characteristic, satisfying some mild conditions. In this paper we relate tilting objects in the heart of Bezrukavnikov's exotic t-structure on the derived category of equivariant coherent sheaves on the Springer resolution of $\mathbf{G}$, and Iwahori-constructible $\mathbb{F}$-parity sheaves on the affine Grassmannian of the Langlands dual group. As applications we deduce in particular the missing piece for the proof of the Mirkovic-Vilonen conjecture in full generality (i.e. for good chara…
Let k be an algebraically closed field of characteristic p>0 and let G be a symplectic or general linear group over k. We consider induced modules for G under the assumption that p is bigger than the greatest hook length in the partitions involved. We give explicit constructions of left resolutions of induced modules by tilting modules. Furthermore, we give injective resolutions for induced modules in certain truncated categories. We show that the multiplicities of the indecomposable tilting and injective modules in these resolutions are the coefficients of certain Kazhdan-Lusztig polynomials.…
Let A be a basic connected finite dimensional algebra over an algebraically closed field, let G be a group, let T be a basic tilting A-module and let B the endomorphism algebra of T. Under a hypothesis on T, we establish a correspondence between the Galois coverings with group G of A and the Galois coverings with group G of B. The hypothesis on T is expressed using the Hasse diagram of basic tilting A-modules and is always verified if A is of finite representation type. Then, we use the above correspondence to prove that A is simply connected if and only if B is simply connected, under the sam…
Let $V$ be a Weyl module either for a reductive algebraic group $G$ or for the corresponding quantum group $U_q$. If $G$ is defined over a field of positive characteristic $p$, respectively if $q$ is a primitive $l$'th root of unity (in an arbitrary field) then $V$ has a Jantzen filtration. The sum of the positive terms in this filtration satisfies a well known sum formula. If $T$ denotes a tilting module either for $G$ or $U_q$ then we can similarly filter the space $\Hom_G(V,T)$, respectively $\Hom_{U_q}(V,T)$ and there is a sum formula for the positive terms here as well. We give an easy an…
In this paper we study categories of tilting modules. Our starting point is the tilting modules for a reductive algebraic group G in positive characteristic. Here we extend the main result in [8] by proving that these tilting modules form a (strictly object-adapted) cellular category. We use this result to specify a subset of cellular basis elements, which generates all morphisms in this category. In a different direction we generalize the earlier results to the case where G is replaced by the infinitesimal thickenings G_rT of a maximal torus T in G by the Frobenius subgroup schemes G_r. Here …
We use tilting modules to study the structure of the tensor product of two simple modules for the algebraic group $\SL_2$, in positive characteristic, obtaining a twisted tensor product theorem for its indecomposable direct summands. Various other related results are obtained, and numerous examples are computed.
The category of level zero representations of current and affine Lie algebras shares many of the properties of other well-known categories which appear in Lie theory and in algebraic groups in characteristic p and in this paper we explore further similarities. The role of the standard and co-standard module is played by the finite-dimensional local Weyl module and the dual of the infinite-dimensional global Weyl module respectively. We define the canonical filtration of a graded module for the current algebra. In the case when $\mathfrak g$ is of type $\mathfrak{sl}_{n+1}$ we show that the wel…
We propose a theory of degenerations for derived module categories, analogous to degenerations in module varieties for module categories. In particular we define two types of degenerations, one algebraic and the other geometric. We show that these are equivalent, analogously to the Riemann-Zwara theorem for module varieties. Applications to tilting complexes are given, in particular that any two-term tilting complex is determined by its graded module structure.
Let $G$ be a simple algebraic group over an algebraically closed field $K$ of characteristic $p > 0$. We consider connected reductive subgroups $X$ of $G$ that contain a given distinguished unipotent element $u$ of $G$. A result of Testerman and Zalesski (Proc. Amer. Math. Soc., 2013) shows that if $u$ is a regular unipotent element, then $X$ cannot be contained in a proper parabolic subgroup of $G$. We generalize their result and show that if $u$ has order $p$, then except for two known examples which occur in the case $(G, p) = (C_2, 2)$, the subgroup $X$ cannot be contained in a proper para…
Dualities of resolving subcategories of finitely generated modules over Artin algebras are characterized as dualities with respect to Wakamatsu tilting bimodules. By restriction of these dualities to resolving subcategories of finitely generated modules with finite projective or Gorenstein-projective dimensions, Miyashita's duality and Huisgen-Zimmermann's correspondence on tilting modules as well as their Gorenstein version are obtained. Applications include constructing triangle equivalences of derived categories of finitely generated Gorenstein-projective modules and showing the invariance …
