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We introduce the notion of extended admissible dissections of a marked surface, building upon the notion of an admissible dissection of a marked surface by Amiot--Plamondon--Schroll. For each extended admissible dissection we construct a…

Whereas exceptional sequences have a long history with many well-known connections to combinatorics, signed exceptional sequences are relatively recent. The authors introduced this concept in 2017 [19], although it was retroactively…

In this paper, we study the Sakellaridis-Venkatesh conjecture for the rank-1 spherical variety $X=\text{Spin}_9\backslash F_4$ using an exceptional theta correspondence. We establish the correct transfer map satisfying relative character…

We study the embeddings of the exceptional infinite-dimensional Lie superalgebra E(1,6) in the exceptional Lie superalgebras E(5,10) and E(4,4). These questions arose in the recent works on enhanced symmetries in some supersymmetric…

We study the action of Dynkin diagram automorphisms $\sigma$ on generalized Gaudin algebras, focusing in particular on the big Gaudin algebra $\mathcal{B}(\mathfrak{g}) \subset (U(\mathfrak{g}) \otimes S(\mathfrak{g}))^{\mathfrak{g}}$ and…

We construct a flat model structure on the category $_{\mathcal{Q},R}{\mathsf{Mod}}$ of additive functors from a small preadditive category $\mathcal{Q}$ satisfying certain conditions to the module category $_{R}{\mathsf{Mod}}$ over an…

We prove that Kaletha's toral supercuspidal L-packets satisfy the twisted endoscopic character relation in some cases, including the case of general linear groups equipped with an involution. Consequently, we verify that Kaletha's…

This paper provides the next step towards classification of algebras of generalized quaternion type. Previously algebras with 2-regular Gabriel quiver were classified (a quiver is 2-regular if at each vertex, two arrows start and two arrows…

Let $\bf T$ be the group of diagonal matrices in $SL_2(\bar{\mathbb{F}}_p)$, where $p$ is a prime number. Let $\Bbbk$ be an algebraically closed field of characteristic not equal to $2$ and $p$. We classify all the irreducible…

In this article, we study the representation theory of shifted super Yangians and finite $W$-superalgebras of type A. A criterion for the finite dimensionality of irreducible modules is obtained in the standard parity case. Furthermore, we…

We study tropicalisations of quasi-automorphisms of cluster algebras and show that their induced action on the g-vectors can be realized by tropicalising their action on the homogeneous $\hat{y}$ (or $\mathcal{X}$) variables of a chosen…

We establish an explicit formula for twisted Harish-Chandra characters of toral supercuspidal representations of p-adic reductive groups under several technical assumptions. Our setup especially includes the case of a quasi-split group…

We define the slant sum of quiver gauge theories, a gluing on the underlying quivers that identifies a gauge vertex with a framing vertex. Under some mild assumptions, we relate torus fixed points on the corresponding Higgs branches, which…

Let $n\geq 2$ be an integer, $S_n$ be the Lie algebra of vector fields on $\mathbb{C}^n$ with zero divergence, and $D_n$ be the Weyl algebra over the polynomial algebra $A_n=\mathbb{C}[t_1,t_2,\cdots,t_n]$. In this paper, we study the…

In this paper, we consider the big algebra recently introduced by Hausel for the $\mathrm{GL}_n$-action on the coordinate ring of the matrix space $\mathrm{Mat}(n,r)$. In particular, we obtain explicit formulas for the big algebra…

For an exact category we provide two constructions of an ambient category in which the initial category is resolving: In the derived category and in the Gabriel--Quillen embedding. For the first construction we describe a pre-aisle and its…

We construct a finite-dimensional algebra derived equivalent to the example of Kershaw--Rickard. For the Kershaw--Rickard example the delooping level and the sub-derived delooping level are both infinite, while for our algebra both…

Let $k$ be an algebraically closed field and $G$ a connected reductive group over $k((t))$ satisfying some conditions. We define a stratification by conjugacy classes of twisted Levi subgroups of $G$ on each Moy-Prasad quotient…

We study polynomial functors in the incompressible category $\text{Ver}_4^+$, which can be viewed as super polynomial functors in characteristic 2. Concretely, we classify additive, exact and simple polynomial functors, and describe how…

We introduce the notion of a Calabi--Yau quadruple as a generalization of Iyama--Yang's Calabi--Yau triple. For each $(d+1)$-Calabi--Yau quadruple, we show that the associated Higgs category is a $d$-Calabi--Yau Frobenius extriangulated…