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Lapid and M\'{i}nguez gave a criterion of the irreducibility of the parabolic induction $\sigma \times \pi$, where $\sigma$ is a ladder representation and $\pi$ is an arbitrary irreducible representation of the general linear group over a…

In this paper, we construct the permutation modules and Young modules for Brauer algebras of type $C$ by extending the representation theory of the group algebra of hyperoctahedral groups. Additionally, we develop a stratifying system for…

What might a combinatorial interpretation of the Kronecker coefficients even look like? We introduce a class of combinatorial objects called bitableaux, which we believe are a natural candidate, and we formulate a purely combinatorial…

To the symmetric space of the (positive half) of a real loop group, we attach a Borel--Serre type bordification and equip it with a Hausdorff topology. The attached boundary, indexed by certain rational parabolics of the loop group, is…

A tensor space is a vector space equipped with a finite collection of multi-linear forms. In recent years, a rich theory of infinite dimensional tensor spaces has emerged. In this note, we show that a large class of permutation groups can…

The classical Cayley transform is a birational map between a quadratic matrix group and its Lie algebra, which was first discovered by Cayley in 1846. Because of its essential role in both pure and applied mathematics, the classical Cayley…

Let $G$ be a Lie group, and $H\subset G$ a closed subgroup. Let $\pi$ be an irreducible unitary representation of $G$. In this paper, we briefly discuss the orbit method and its application to the branching problem $\pi|_{H}$. We use the…

As a sequel to [Duan-Shu-Yao], we introduce here a category $\mathscr{LC}$ arising from the BGG category $\mathcal{O}$ defined in [Duan-Shu-Yao] for Lie algebras of polynomial vector fields. The objects of $\mathscr{LC}$ are so-called…

Let $\mathfrak{g}=\mathfrak{g}_{\bar0}+\mathfrak{g}_{\bar1}$ be a basic classical Lie superalgebra over $\mathbb{C}$, and $e=e_{\theta}\in\mathfrak{g}_{\bar0}$ with $-\theta$ being a minimal root of $\mathfrak{g}$. Set $U(\mathfrak{g},e)$…

We present new techniques for removing arrows of bound quiver algebras, reducing thus the Finitistic Dimension Conjecture $\mathsf{(FDC)}$ for a given algebra to a smaller one. Unlike the classic arrow removal operation of…

The loop Hecke algebra is a generalization of the Hecke algebra to the loop braid group, introduced by Damiani, Martin and Rowell. We give a new presentation of the loop Hecke algebra provided a mild condition on the parameter and give a…

We continue the study of cactus flower moduli spaces $\overline{F}_n$ and Gaudin models started in arXiv:2308.06880, arXiv:2407.06424. We show that isomorphism classes of operadic coverings of the real form $\overline{F}_n(\mathbb{R})$ are…

We propose a framework for producing interesting subcategories of the category ${}_A\mathsf{Mod}$ of left $A$-modules, where $A$ is an associative algebra over a field $k$. The construction is based on the composition, $Y$, of the Yoneda…

Motivated by constructions from applied topology, there has been recent interest in the homological algebra of linear representations of posets, particularly in the context of homological algebra relative to non-standard exact structures. A…

Let $\mathfrak h$ be a Cartan subalgebra of a complex semisimple Lie algebra $\mathfrak g.$ We define a compactification $\bar {\mathfrak h}$ of $\mathfrak h$, which is analogous to the closure $\bar H$ of the corresponding maximal torus…

We study tensor categories that interpolate the representation categories of finite classical groups. There are (at least) two ways to approach these categories: via ultraproducts and via oligomorphic groups. Both have strengths and…

Let $G$ be a finite group and $k$ a field of characteristic $p > 0$. Balmer and Gallauer's recent result on finite $p$-permutation resolutions of $kG$-modules motivates the study of an intriguing new invariant; the $p$-permutation…

We study the validity of the local theta correspondence over a non-archimedean local field in the context of modular representation theory \textit{i.e.} for representations with coefficient fields of positive characteristic. For a…

We classify the irredible representations of $\mathrm{GL}_{2}(q)$ for which the induction to the product group $\mathrm{GL}_{2}(q)\times\mathrm{GL}_{2}(q)$, under the diagonal embedding, decomposes multiplicity free. It turns out that only…

In this paper, we introduce and study two cyclotomic level maps defined respectively on the set of nilpotent orbits $\underline{\mathcal{N}}$ in a complex semi-simple Lie algebra $\mathfrak{g}$ and the set of conjugacy classes…