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We generalise a theorem of Tachikawa about reflexive Auslander-Reiten sequences. We apply this to give a new characterisation of the dominant dimension of gendo-symmetric algebras. We also generalise a formula due to Reiten about the…

In this paper we identify the cotangent to the derived stack of representations of a quiver $Q$ with the derived moduli stack of modules over the Ginzburg dg-algebra associated with $Q$. More generally, we extend this result to finite type…

This paper was written in 1994 and has attracted a large number of citations since then. The main result was a definition of what is called the affine Brauer algebra in the present version. It is now posted to affirm this terminology.

The rook monoid, also known as the symmetric inverse monoid, is the archetypal structure when it comes to extend the principle of symmetry. In this paper, we establish a Schur-Weyl duality between this monoid and an extension of the…

We construct a bijection between admissible representations for an affine Lie algebra $\mathfrak{g}$ at boundary admissible levels and $\mathbb{C}^\times$ fixed points in homogeneous elliptic affine Springer fibres for the Langlands dual…

This paper aims to use topological methods to compute $\mathrm{Ext}$ between an irreducible representation of a finite monoid inflated from its group completion and one inflated from its group of units, or more generally coinduced from a…

We construct a "Koszul duality" equivalence relating the (diagrammatic) Hecke category attached to a Coxeter system and a given realization to the Hecke category attached to the same Coxeter system and the dual realization. This extends a…

In this paper we give an explicit presentation of the centre of the restricted rational Cherednik algebra $\overline{H}_c(S_n\wr\mathbb{Z}/\ell\mathbb{Z})$ for a particular choice of parameter $c$. More precisely, we describe the centre of…

Given an oligomorphic group $G$ and a measure $\mu$ for $G$ (in a sense that we introduce), we define a rigid tensor category $\underline{\mathrm{Perm}}(G; \mu)$ of "permutation modules," and, in certain cases, an abelian envelope…

In this article, we derived some consequences to the symmetrization process developed in \cite{Deng23}. This consists a geometric derivation of part of the properties which uniquely determines the Kazhdan-Lusztig polynomials of type $A_n$…

We define a class of monoidal categories whose morphisms are diagrams, and which are enhancements and generalisations of the Brauer category obtained by adjoining infinitesimal braids, "coupons" and poles. Properties of these categories are…

We consider typical finite dimensional complex irreducible representations of a basic classical simple Lie superalgebra, and give a sufficient condition on when unique factorization of finite tensor products of such representations hold. We…

We completely determine the $\tau$-tilting finiteness of Borel-Schur algebras. To achieve this, we use two recently introduced techniques in silting theory: sign decomposition as introduced by Aoki, Higashitani, Iyama, Kase and Mizuno…

Let $U_{q}^{-}(\mathfrak g)$ be the negative half of a quantum Borcherds-Bozec algebra $U_{q}(\mathfrak g)$ and $V(\lambda)$ be the irreducible highest weight module with $\lambda \in P^{+}$. In this paper, we investigate the structures,…

Let $G$ be a reductive complex Lie group with Lie algebra $\mathfrak{g}$ and suppose that $V$ is a polar $G$-representation. We prove the existence of a radial parts map $\mathrm{rad}: \mathcal{D}(V)^G\to A_{\kappa}$ from the $G$-invariant…

Let $V$ be a symmetric space over a connected reductive Lie algebra $G$, with Lie algebra $\mathfrak{g}$ and discriminant $\delta\in \mathbb{C}[V]$. A fundamental object is the invariant holonomic system $\mathcal{G} =\mathcal{D}(V)\Big/…

We consider the Temperley-Lieb algebras $\textrm{TL}_n(\delta)$ at $\delta = 1$. Since $\delta = 1$, we can consider the multiplicative monoid structure and ask how this monoid acts on topological spaces. Given a monoid action on a…

We apply the local structure theorem and the polar decomposition to a real spherical space Z=G/H and control the volume growth on Z. We define the Harish-Chandra Schwartz space on Z. We give a geometric criterion to ensure…

We examine the relationship between the actions of two Weyl groups on the cohomology of a smooth quiver variety: the Maffei's action of the Weyl group associated to the quiver, and the symplectic Springer action of the Namikawa-Weyl group…

Given an action of a finite group on a triangulated category with a suitable strong exceptional collection, a construction of Elagin produces an associated strong exceptional collection on the equivariant category. We prove that the…