The main result of this paper establishes a bijection between the set of equivalence classes of simple transitive 2-representations with a fixed apex ${\mathcal{J}}$ of a fiat 2-category $\mathscr{C}$ and the set of equivalence classes of faithful simple transitive 2-representations of the fiat 2-subquotient of $\mathscr{C}$ associated with a diagonal ${\mathcal{H}}$-cell in ${\mathcal{J}}$. As an application, we classify simple transitive 2-representations of various categories of Soergel bimodules, in particular, completing the classification in types $B_{3}$ and $B_{4}$.

We study the structure of the stable category $\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}(S/(f))$ of graded maximal Cohen–Macaulay module over $S/(f)$ where $S$ is a graded ($\pm 1$)-skew polynomial algebra in $n$ variables of degree 1, and $f=x_{1}^{2}+\cdots +x_{n}^{2}$. If $S$ is commutative, then the structure of $\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}(S/(f))$ is well known by Knörrer’s periodicity theorem. In this paper, we prove that if $n\leqslant 5$, then the structure of $\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}(S/(f))$ is determined by the number of irreducible components of the point scheme of $S$ which are isomorphic to $\mathbb{P}^{1}$.

In this paper we define almost gentle algebras, which are monomial special multiserial algebras generalizing gentle algebras. We show that the trivial extension of an almost gentle algebra by its minimal injective co-generator is a symmetric special multiserial algebra and hence a Brauer configuration algebra. Conversely, we show that any almost gentle algebra is an admissible cut of a unique Brauer configuration algebra and, as a consequence, we obtain that every Brauer configuration algebra with multiplicity function identically one is the trivial extension of an almost gentle algebra. We show that a hypergraph is associated with every almost gentle algebra A, and that this hypergraph induces the Brauer configuration of the trivial extension of A. Among other things, this gives a combinatorial criterion to decide when two almost gentle algebras have isomorphic trivial extensions.

We examine situations, where representations of a finite-dimensional F-algebra A defined over a separable extension field K/F, have a unique minimal field of definition. Here the base field F is assumed to be a field of dimension ≼1. In particular, F could be a finite field or k(t) or k((t)), where k is algebraically closed. We show that a unique minimal field of definition exists if (a) K/F is an algebraic extension or (b) A is of finite representation type. Moreover, in these situations the minimal field of definition is a finite extension of F. This is not the case if A is of infinite representation type or F fails to be of dimension ≼1. As a consequence, we compute the essential dimension of the functor of representations of a finite group, generalizing a theorem of Karpenko, Pevtsova and the second author.

We introduce the notion of a perfect path for a monomial algebra. We classify indecomposable non-projective Gorenstein-projective modules over the given monomial algebra via perfect paths. We apply the classification to a quadratic monomial algebra and describe explicitly the stable category of its Gorenstein-projective modules.

In this paper, we will consider derived equivalences for differential graded endomorphism algebras by Keller's approaches. First, we construct derived equivalences of differential graded algebras which are endomorphism algebras of the objects from a triangle in the homotopy category of differential graded algebras. We also obtain derived equivalences of differential graded endomorphism algebras from a standard derived equivalence of finite dimensional algebras. Moreover, under some conditions, the cohomology rings of these differential graded endomorphism algebras are also derived equivalent. Then we give an affirmative answer to a problem of Dugas (A construction of derived equivalent pairs of symmetric algebras, Proc. Amer. Math. Soc. 143 (2015), 2281–2300) in some special case.

For a finite quiver Q without sources, we consider the corresponding radical square zero algebra A. We construct an explicit compact generator for the homotopy category of acyclic complexes of projective A-modules. We call such a generator the projective Leavitt complex of Q. This terminology is justified by the following result: the opposite differential graded endomorphism algebra of the projective Leavitt complex of Q is quasi-isomorphic to the Leavitt path algebra of Qop. Here, Qop is the opposite quiver of Q, and the Leavitt path algebra of Qop is naturally ${\open Z}$-graded and viewed as a differential graded algebra with trivial differential.

This paper studies the combinatorics of lattice congruences of the weak order on a finite Weyl group $W$ , using representation theory of the corresponding preprojective algebra $\unicode[STIX]{x1D6F1}$ . Natural bijections are constructed between important objects including join-irreducible congruences, join-irreducible (respectively, meet-irreducible) elements of $W$ , indecomposable $\unicode[STIX]{x1D70F}$ -rigid (respectively, $\unicode[STIX]{x1D70F}^{-}$ -rigid) modules and layers of $\unicode[STIX]{x1D6F1}$ . The lattice-theoretically natural labelling of the Hasse quiver by join-irreducible elements of $W$ is shown to coincide with the algebraically natural labelling by layers of $\unicode[STIX]{x1D6F1}$ . We show that layers of $\unicode[STIX]{x1D6F1}$ are nothing but bricks (or equivalently stones, or 2-spherical modules). The forcing order on join-irreducible elements of $W$ (arising from the study of lattice congruences) is described algebraically in terms of the doubleton extension order. We give a combinatorial description of indecomposable $\unicode[STIX]{x1D70F}^{-}$ -rigid modules for type $A$ and $D$ .

The set of row reduced matrices (and of echelon form matrices) is closed under multiplication. We show that any system of representatives for the $\text{Gl}_{n}(\mathbb{K})$ action on the $n\times n$ matrices, which is closed under multiplication, is necessarily conjugate to one that is in simultaneous echelon form. We call such closed representative systems Grassmannian semigroups. We study internal properties of such Grassmannian semigroups and show that they are algebraic semigroups and admit gradings by the finite semigroup of partial order preserving permutations, with components that are naturally in one–one correspondence with the Schubert cells of the total Grassmannian. We show that there are infinitely many isomorphism types of such semigroups in general, and two such semigroups are isomorphic exactly when they are semiconjugate in $M_{n}(\mathbb{K})$ . We also investigate their representation theory over an arbitrary field, and other connections with multiplicative structures on Grassmannians and Young diagrams.

Ringel’s right-strongly quasi-hereditary algebras are a distinguished class of quasi-hereditary algebras of Cline–Parshall–Scott. We give characterizations of these algebras in terms of heredity chains and right rejective subcategories. We prove that any artin algebra of global dimension at most two is right-strongly quasi-hereditary. Moreover we show that the Auslander algebra of a representation-finite algebra $A$ is strongly quasi-hereditary if and only if $A$ is a Nakayama algebra.

Using that integrable derivations of symmetric algebras can be interpreted in terms of Bockstein homomorphisms in Hochschild cohomology, we show that integrable derivations are invariant under the transfer maps in Hochschild cohomology of symmetric algebras induced by stable equivalences of Morita type. With applications in block theory in mind, we allow complete discrete valuation rings of unequal characteristic.

Given an odd prime $p$, we investigate the position of simple modules in the stable Auslander–Reiten quiver of the principal block of a finite group with noncyclic abelian Sylow $p$-subgroups. In particular, we prove a reduction to finite simple groups. In the case that the characteristic is $3$, we prove that simple modules in the principal block all lie at the end of their components.

We prove a conjecture of Rouquier relating the decomposition numbers in category ${\mathcal{O}}$ for a cyclotomic rational Cherednik algebra to Uglov’s canonical basis of a higher level Fock space. Independent proofs of this conjecture have also recently been given by Rouquier, Shan, Varagnolo and Vasserot and by Losev, using different methods. Our approach is to develop two diagrammatic models for this category ${\mathcal{O}}$ ; while inspired by geometry, these are purely diagrammatic algebras, which we believe are of some intrinsic interest. In particular, we can quite explicitly describe the representations of the Hecke algebra that are hit by projectives under the $\mathsf{KZ}$ -functor from the Cherednik category ${\mathcal{O}}$ in this case, with an explicit basis. This algebra has a number of beautiful structures including categorifications of many aspects of Fock space. It can be understood quite explicitly using a homogeneous cellular basis which generalizes such a basis given by Hu and Mathas for cyclotomic KLR algebras. Thus, we can transfer results proven in this diagrammatic formalism to category ${\mathcal{O}}$ for a cyclotomic rational Cherednik algebra, including the connection of decomposition numbers to canonical bases mentioned above, and an action of the affine braid group by derived equivalences between different blocks.

This paper deals with the Green ring $\mathcal{G}(\mathcal{C})$ of a finite tensor category $\mathcal{C}$ with finitely many isomorphism classes of indecomposable objects over an algebraically closed field. The first part of this paper deals with the question of when the Green ring $\mathcal{G}(\mathcal{C})$ , or the Green algebra $\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$ K over a field K, is Jacobson semisimple (namely, has zero Jacobson radical). It turns out that $\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$ K is Jacobson semisimple if and only if the Casimir number of $\mathcal{C}$ is not zero in K. For the Green ring $\mathcal{G}(\mathcal{C})$ itself, $\mathcal{G}(\mathcal{C})$ is Jacobson semisimple if and only if the Casimir number of $\mathcal{C}$ is not zero. The second part of this paper focuses on the case where $\mathcal{C}=\text{Rep}(\mathbb {k}G)$ for a cyclic group G of order p over a field $\mathbb {k}$ of characteristic p. In this case, the Casimir number of $\mathcal{C}$ is computable and is shown to be 2p 2. This leads to a complete description of the Jacobson radical of the Green algebra $\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$ K over any field K.

In this paper we study categories ${\mathcal{O}}$ over quantizations of symplectic resolutions admitting Hamiltonian tori actions with finitely many fixed points. In this generality, these categories were introduced by Braden et al. We establish a family of standardly stratified structures (in the sense of the author and Webster) on these categories ${\mathcal{O}}$ . We use these structures to study shuffling functors of Braden et al. (called cross-walling functors in this paper). Most importantly, we prove that all cross-walling functors are derived equivalences that define an action of the Deligne groupoid of a suitable real hyperplane arrangement.

We study cluster categories arising from marked surfaces (with punctures and non-empty boundaries). By constructing skewed-gentle algebras, we show that there is a bijection between tagged curves and string objects. Applications include interpreting dimensions of $\operatorname{Ext}^{1}$ as intersection numbers of tagged curves and Auslander–Reiten translation as tagged rotation. An important consequence is that the cluster(-tilting) exchange graphs of such cluster categories are connected.

Let $\unicode[STIX]{x1D6EC}$ be a connected hereditary artin algebra. We show that the set of functorially finite torsion classes of $\unicode[STIX]{x1D6EC}$-modules is a lattice if and only if $\unicode[STIX]{x1D6EC}$ is either representation-finite (thus a Dynkin algebra) or $\unicode[STIX]{x1D6EC}$ has only two simple modules. For the case of $\unicode[STIX]{x1D6EC}$ being the path algebra of a quiver, this result has recently been established by Iyama–Reiten–Thomas–Todorov and our proof follows closely some of their considerations.

Let $(W,S)$ be a finite Coxeter group. Kazhdan and Lusztig introduced the concept of $W$ -graphs, and Gyoja proved that every irreducible representation of the Iwahori–Hecke algebra $H(W,S)$ can be realized as a $W$ -graph. Gyoja defined an auxiliary algebra for this purpose which—to the best of the author’s knowledge—was never explicitly mentioned again in the literature after Gyoja’s proof (although the underlying ideas were reused). The purpose of this paper is to resurrect this $W$ -graph algebra, and to study its structure and its modules. A new explicit description of it as a quotient of a certain path algebra is given. A general conjecture is proposed which would imply strong restrictions on the structure of $W$ -graphs. This conjecture is then proven for Coxeter groups of type $I_{2}(m)$ , $B_{3}$ and $A_{1}$ – $A_{4}$ .

Khovanov–Lauda–Rouquier (KLR) algebras of finite Lie type come with families of standard modules, which under the Khovanov–Lauda–Rouquier categorification correspond to PBW bases of the positive part of the corresponding quantized enveloping algebra. We show that there are no non-zero homomorphisms between distinct standard modules and that all non-zero endomorphisms of a standard module are injective. We present applications to the extensions between standard modules and modular representation theory of KLR algebras.