Ring Theory and Representation Theory Seminar in 2017


This seminar is coorganized by Osamu Iyama and Ryo Takahashi.
Previous seminars 2015-16, 2014, 2013--.
25 January (Thursday), 10:30-12:00, 309 math.
Yuki Hirano (Kyoto University)
Relative singular locus and Balmer spectrum of matrix factorizations

Abstract: We classify certain thick subcategories of derived matrix factorization categories by using the notion of relative singular locus which we introduce in this work. As an application, we determine the Balmer spectrum of derived matrix factorization categories with respect to the natural tensor structures.
11 January (Thursday), 16:30-18:00, 409 math.
William Wong (Nagoya University)
Perverse equivalence

Abstract: In this talk I shall introduce perverse equivalence as defined by Chuang and Rouquier. As an attempt to combinatorically describe derived equivalences, it has very common properties with tilting and mutation. I will discuss some of my earlier work as an example of how to apply this notion.
30 November (Thursday), 16:30-18:00, 409 math.
Takuma Aihara (Tokyo Gakugei University)
Applying silting reduction

Abstract: Silting reduction had been introduced by Aihara-Iyama and was developed by Iyama-Yang. It is a powerful tool to obtain silting objects containing a given presilting one. In this talk, we discuss applying silting reduction to various cases from the viewpoint of homological conjectures.
24 November (Friday), 13:00-14:00, 452 math.
Sota Asai (Nagoya University)
Bricks over preprojective algebras

Abstract: Mizuno gave an isomorphism of lattices from a Coxeter group of Dynkin type to the set of torsion-free classes in the module category of the corresponding preprojective algebra. Combining it with my bijection on semibricks, we obtain a bijection from the Coxeter group to the set of semibricks over the preprojective algebra. My aim is to explicitly describe the semibrick associated to each element in the Coxeter group in this bijection. In this process, a combinatorial notion "canonical join representations" introduced by Reading, is very useful. I observed that the canonical join representation of an element in the Coxeter group gives the decomposition of the corresponding semibrick into a direct sum of bricks. I will mainly talk about such theoretic strategies to determine the semibrick in this seminar.

24 November (Friday), 14:45-15:45, 452 math.
Yuya Mizuno (Shizuoka University)
Preprojective algebra of Dynkin type and two-sided tilting complexes

Abstract: For the preprojective algebra of non-Dynkin type, there is a family of bimodules (two-sided tilting complexes), which admits auto-equivalences of the derived category by the derived tensor product. Moreover, one can give a map from the elements of the braid group to these functors. On the other hand, in the case of Dynkin case, the corresponding two-sided tilting complexes are missing and hence we can not apply the same method. In this talk, we discuss a construction of such two-sided tilting complexes and we explain these complexes induce mutation by derived tensor product.
11 November (Saturday), 16:30-18:00, 409 math.
Graham J. Leuschke (Syracuse University)
Auslander-Reiten sequences for Gorenstein rings of dimension one

Abstract: Let (R,m) be a complete local Gorenstein ring of dimension one. We show that there is a particular element of the total quotient ring of R which, when considered as an endomorphism of an indecomposable torsion-free R-module M satisfying a mild condition on the rank, induces the AR sequence for M. The rank condition is satisfied whenever M is an ideal, or R is a hypersurface ring, or R contains the rational numbers. This result can be applied to determine the shapes of some components of the stable AR quiver. This is a report on the PhD thesis work of Syracuse student Robert Roy.
24 October (Tuesday), 13:00-14:30, 309 math.
Emilie Bjørlo Arentz-Hansen (NTNU)
Classifying subcategories of the stable category of a Frobenius category

Abstract: We will discuss under which circumstances we have a one-to-one correspondence between appropriate subcategories of a Frobenius category and (thick) triangulated subcategories of the associated stable category. We will arrive at two main results, one for triangulated subcategories under some extra conditions on the Frobenius category, and one for thick triangulated subcategories.

24 October (Tuesday), 16:30-17:30, 309 math.
Toshiaki Shoji (Tongji University)
Generalized Springer correspondence for symmetric space associated to orthogonal groups

Abstract: The Springer correspondence gives a natural relationship between unipotent classes of a reductive group G and irreducible representations of its Weyl group W. If G is a general linear group, this gives a bijection , but in general not. As a generalization, Lusztig gave a natural bijection, for any G, between a certain set associated to unipotent classes and a union of irreducible representations of various reflection groups W_i, which is called the generalized Springer correspondence. We consider a symmetric space G/K, where G = GL(V) and K = SO(V). The set of K-orbits in the "unipotent part" of G/K is regarded as an analogue of unipotent classes for a reductive group. In this talk, we show that the generalized Springer correspondence holds for such K-orbits. Note that this seems to be a special phenomenon. For example, if K = Sp(V), the generalized Springer correspondence does not hold for G/K.
3 October (Tuesday), 13:00-14:30, 452 math.
Julia Sauter (Bielefeld University)
On tilting modules with projective splitting projectiles

Abstract: Let T be a tilting module of projective dimension at most one over a finite-dimensional algebra. We call T special if the splitting projective summand of T is projective. There is a bijection between special tilting modules and faithful projective modules (up to multiplicity of their summands). The faithful projectives (up to equivalence) give a finite very regular sublattice (with a unique minimum) of the poset of functorially-finite torsion classes. Our particular interest are special tilts of endomorphism rings. We recall the "generator correspondence" characterising endomorphism rings of generators (resp. cogenerators), this is a lesser known generalisation of the Morita-Tachikawa correspondence. Endomorphism rings of cogenerators have a canonical special tilt which we study further. This talk is based on joint work in progress with Matt Pressland.

3 October (Tuesday), 16:30-17:30, 452 math.
William Crawley-Boevey (Bielefeld University)
The Deligne Simpson Problem revisited

Abstract: The Deligne Simpson problem asks about the existence of irreducible representations of the fundamental group of a punctured sphere, where loops around the holes belong to prescribed conjugacy classes. Peter Shaw and I gave a sufficient condition for this in 2006, and although I announced a proof of necessity, it never appeared. I will discuss recent work with Andrew Hubery aimed at rectifying this.
21 September (Thursday), 13:00-14:30, 409 math.
Steffen Oppermann (NTNU)
Change of rings and singularity categories

Abstract: This talk is based on joint work with Chrysostomos Psaroudakis and Torkil Utvik Stai. The singularity category of a (finite dimensional) algebra is defined to be the localization of the bounded derived category modulo the subcategory of perfect complexes. The name "singularity category" is motivated by commutative algebra, where the singularity category contains information about the singularities of a ring while forgetting the regular parts. For (non-commutative) finite dimensional algebras the meaning is less clear. The aim of my talk is to investigate when ring-morphisms induce functors between singularity categories (and related cocomplete categories). One may hope that this gives some idea what information survives in the singularity category.

21 September (Thursday), 15:00-16:30, 409 math.
Martin Herschend (Uppsala University)
Wide subcategories of n-cluster tilting subcategories

Abstract: This talk is based on joint work with Peter Jorgensen and Laertis Vaso. In Iyama's higher dimensional Auslander-Reiten theory one shifts attention from module categories of algebras to so called n-cluster tilting subcategories for some fixed positive integer n. These are n-abelian (in the sense of Jasso) and so notions like kernel, cokernel and extension are replaced by their higher analogues: n-kernel, n-cokernel and n-extension, which are similar except that the exact sequences involved are longer. Thus it makes sense to consider subcategories of n-abelian categories that are closed under n-kernels, n-cokernels and n-extensions. We shall refer to these as wide subcategories. In my talk I will give a brief background on n-abelian categories and n-cluster tilting subcategories. I will then present a characterization of wide subcategories of n-cluster tilting subcategories using certain ring epimorphisms. This will then be used to classify thick subcategories on n-cluster tilting subacategories for Nakayama algebras of global dimension n.
19 July (Wednesday), 13:00-14:30, 452 math.
Yoshiyuki Kimura (Osaka Prefecture)
Twist automorphisms on quantum unipotent cells and the dual canonical bases

Abstract: Let $G$ be a connected simply-connected complex simple algebraic group with a fixed maximal torus $H$ , a pair of Borel subgroups $B_{\pm}$ such that $B_{+}\cap B_{-}=H$ and a Weyl group $W=\mathrm{Norm}_{G}\left(H\right)/H$ and the maximal unipotent subgroups $N_{\pm}\subset B_{\pm}$. For a Weyl group element $w\in W$, we consider the unipotent cell $N_{-}\cap B_{+}\dot{w}B_{+}$, where $\dot{w}$ is a lift of $w$ in $\mathrm{Norm}_{G}\left(H\right)$. Berenstein, Fomin and Zelevinsky introduced certain automorphism on the unipotent cell, called twist automorphism, for solving “the factorization problems” which describes the inverse of “toric chart” of the associated Schubert varieties.
The quantum unipotent cell is a quantum analogue of the coordinate ring of $N_{-}\cap B_{+}\dot{w}B_{+}$ which was introduced by De Concini and Procesi and they proved an isomorphism between it and a quantum analogue of the coordinate ring of $N_{-}\left(w\right)\cap\dot{w}G_{0}$, where $N_{-}\left(w\right)=N_{-}\cap\dot{w}N_{+}\dot{w}^{-1}$ and $\dot{w}G_{0}$ is the “Gauss” cell associated with $w$.
In this talk, we construct a quantum analogue of the twist automorphism, called a quantum twist automorphism, as a composite of the De Concini-Procesi isomorphism and the twist isomorphism between $N_{-}\cap B_{+}\dot{w}B_{+}$ and $N_{-}\left(w\right)\cap\dot{w}G_{0}$ which is defined by Gaussian decomposition and study its basic properties. In fact, we proved that the the quantum twist automorphism preserves the dual canonical basis of the quantum unipotent cell. This is a joint work with Hironori Oya.

19 July (Wednesday), 15:00-16:30, 452 math.
Hironori Oya (Tokyo)
Twist automorphisms and Chamber Ansatz formulae for quantum unipotent cells

Abstract: In the study of totally positive elements in unipotent cells, Berenstein, Fomin and Zelevinsky described the inverse rational maps of certain embeddings of tori into them because they provide coordinates behaving nicely to the totally positive elements. The resulting substitutions are called the Chamber Ansatz.
In this talk, we provide the quantum analogues of the Chamber Ansatz formulae, though we do not have the ``actual spaces'' but only have their ``coordinate algebras''. In terms of coordinate algebras, the torus embeddings above are the algebra homomorphisms from the coordinate algebras to the Laurent polynomial rings, whose quantum analogues are called Feigin homomorphisms. By the work of Geiss-Leclerc-Schr\"{o}er and Goodearl-Yakimov, it is known that there exist many embeddings of the quantum unipotent cells into skew Laurent polynomial rings as a consequence of their quantum cluster algebra structures. The Chamber Ansatz formulae give transition from the ``coordinate functions'' in the targets of the Feigin homomorphisms into the specific quantum clusters.
We also discuss the relation between the quantum twist automorphisms and Geiss-Leclerc-Schr\"{o}er's additive categorification of the quantum cluster algebra structures.
Some of the results in this talk are based on joint work with Yoshiyuki Kimura.
23 June (Friday), 16:30-18:00, 409 math.
Kengo Miyamoto (Osaka)
Classification of connected self-injective cellular algebras of polynomial growth representation type up to Morita equivalence

Abstract: An associative algebra is cellular if the algebra has a special basis called "cell basis". Cellular algebras which are introduced by Graham and Lehrer in 1996 often appear as finite dimensional algebras arising from various setting in Lie theory. For example, Iwahori Hecke algebras, Brauer algebras and Temperley-Lieb algebras are cellular algebras. In fact, the axiom of cellular algebras is a generalization of natural representations of some algebras arising from Lie theory. On the other hand, basic self-injective algebras over algebraically closed field of polynomial growth type have been classified several authors, and the property of being cellular algebras preserves under Morita equivalences if the base field has odd characteristic. In this talk, we classify Morita equivalence classes of connected self-injective cellular algebras of polynomial growth representation type when the characteristic of the base field (which is an algebraically closed) is odd.
This talk is based on a joint work with S. Ariki, R. Kase, K. Wada.
16 June (Friday), 13:00-14:30, 409 math.
Olgur Celikbas (West Virginia University)
Torsion in tensor products of modules

Abstract: In 1961 Auslander studied the torsion submodule of tensor products of finitely generated modules over unramified regular local rings. He proved that if the tensor product of nonzero finitely generated modules M and N is torsion-free, then M and N are both torsion-free.
In 1994 Huneke and R. Wiegand obtained a natural extension of Auslander’s result for hypersurface rings. Their result, referred to as the second rigidity theorem, establishes that if the tensor product of nonzero finitely generated modules M and N is reflexive over a hypersurface domain, then M and N are both reflexive.
In this talk I will briefly survey the history of these results. Then I will discuss my joint work with Greg Piepmeyer (Syzygies and tensor products, Math. Z., 276, 457–468, 2014). Our work was motivated by the second rigidity theorem of Huneke and Wiegand, and it relies upon an application of the new intersection theorem.
A special case of our result establishes that, if R is a local complete intersection ring of codimension c, M and N are nonzero finitely generated Tor-independent R-modules, and the tensor product of M and N satisfies Serre’s condition (S_{n+c}) for some nonnegative integer n, then both M and N satisfy (S_n).

16 June (Friday), 14:45-16:15, 409 math.
Ela Celikbas (West Virginia University)
Associated Graded Rings and Connected Sums

Abstract: In 2012 Ananthnarayan, Avramov and Moore defined a connected sum of two Gorenstein local rings as an appropriate quotient of their fiber product. This new construction of connected sums always produces Gorenstein rings.
In this talk, we will discuss some conditions on the associated graded ring of a Gorenstein Artin local ring Q that will make Q to be a connected sum. We will also give some results about short, and stretched, Gorenstein Artin rings, and talk about the rationality of the Poincare series of Q.
This presentation is based on a recent joint work with H. Ananthnarayan, Jai Laxmi, and Z. Yang.
19 May (Friday), 13:00-14:30, 409 math.
Yusuke Nakajima (IPMU)
Non-commutative crepant resolutions for some Hibi rings

Abstract: Conic divisorial ideals are special classes of divisorial ideals of toric rings. It is known that using conic divisorial ideals we can obtain a non-commutative resolution (which is a ring having the finite global dimension) of toric rings. In my talk, I especially discuss conic divisorial ideals of a Hibi ring, which is a toric ring arising from a partially ordered set. I will determine the precise description of conic divisorial ideals of Hibi rings using the associated partially ordered sets. Furthermore, I also discuss non-commutative crepant resolutions for some Hibi rings. This talk is based on a joint work with A. Higashitani.
28 April (Friday), 13:00-14:30, 409 math.
Haruhisa Enomoto (Nagoya)
Classification of exact structures and CM-finite Gorenstein algebras

Abstract: In this talk, I will give a classification of all possible exact structures on a given additive category, in terms of the module category over it. For a category of finite type, an exact structure corresponds to a set of dotted arrows of the AR quiver of it, or a set of simple modules satisfying the 2-regular condition. Using this, we reduce a classification of CM-finite Iwanaga-Gorenstein algebras to that of algebras with finite global dimension, which enables us to construct such algebras explicitly.
10 April (Monday), 13:00-14:30, 409 math.
Yann Palu (Amiens)
Non-kissing complex and tau-tilting over gentle algebras

Abstract: Gentle algebras form a class of algebras, described in terms of quivers and relations, whose representations are well understood and can be described combinatorially. In this talk, I will recall basic results on gentle algebras. I will then explain a bridge between the tau-tilting theory of gentle algebras and a combinatorial notion called "kissing".
14 March (Tuesday), 15:00-16:30, 409 math.
Yasuaki Ogawa (Nagoya)
Recollements for dualizing $k$-varieties

Abstract: A recollement of abelian categories is a special case of Serre quotients where both the inclusion and the quotient functor admit left and right adjoints. Given an algebra and its idempotent, we have a recollement of module categories. In this talk, we extend recollements of this type to functor categories over dualizing $k$-varieties. By a closer look at these recollements, we obtain a new proof of the (higher) Auslander-Reiten duality. If time allows, I would like to explain that under a certain condition these recollements are closely related to the Auslander-Bridger sequences.
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