Indecomposable Objects of $\underline{Rep}(GL_t)$ in Terms of Exterior Powers of the Tautological Object and its Dual

Idempotents in representation rings of quivers

For an acyclic quiver Q, we solve the Clebsch-Gordan problem for the projective representations by computing the multiplicity of a given indecomposable projective in the tensor product of two indecomposable projectives. Motivated by this problem for arbitrary representations, we study idempotents in the representation ring of Q (the free abelian group on the indecomposable representations, with multiplication given by tensor product). We give a general technique for constructing such idempotents and for decomposing the representation ring into a direct product of ideals, utilizing morphisms be…

张量幂中不可分解直和项数量的增长率

Growth rates of the number of indecomposable summands in tensor powers

该论文聚焦仿射（半）群（超）概形等相关对象的有限维表示张量积，研究其直和项数量的渐近行为。

Gelfand quiver的表示论与SL₂(R)的Harish-Chandra模的表示论

Representation theory of the Gelfand quiver and Harish-Chandra modules for $\mathsf{SL}_2(\mathbb{R})$

本文从导出范畴视角解决Gelfand问题，分类了Gelfand quiver的nilpotent表示的不可分解对象，并确定其在导出Auslander-Reiten翻译、符号反演和共轭对称下的像。

The Green rings of the generalized Taft Hopf algebras

In this paper, we investigate the Green ring $r(H_{n,d})$ of the generalized Taft algebra $H_{n,d}$, extending the results of Chen, Van Oystaeyen and Zhang in \cite{Coz}. We shall determine all nilpotent elements of the Green ring $r(H_{n,d})$. It turns out that each nilpotent element in $r(H_{n,d})$ can be written as a sum of indecomposable projective representations. The Jacobson radical $J(r(H_{n,d}))$ of $r(H_{n,d})$ is generated by one element, and its rank is $n-n/d$. Moreover, we will present all the finite dimensional indecomposable representations over the complexified Green ring $R(H…

奇特征下自然 $\mathrm{SL}_2$-模张量幂的不可分解分量数的渐近性

Asymptotics for number of indecomposable components of tensor powers of the natural $\mathrm{SL}_2$-module in odd characteristic

该研究在奇特征代数闭域上，证明了自然SL2-模张量幂的不可分解分量数渐近公式，并揭示了其与theta函数重标度卷积的极限密度联系。

Graded level zero integrable representations of affine Lie algebras

We study the structure of the category of integrable level zero representations with finite dimensional weight spaces of affine Lie algebras. We show that this category possesses a weaker version of the finite length property, namely that an indecomposable object has finitely many simple constituents which are non-trivial as modules over the corresponding loop algebra. Moreover, any object in this category is a direct sum of indecomposables only finitely many of which are non-trivial. We obtain a parametrization of blocks in this category.

Deligne's category Rep(GL_δ) and representations of general linear supergroups

We classify indecomposable summands of mixed tensor powers of the natural representation for the general linear supergroup up to isomorphism. We also give a formula for the characters of these summands in terms of composite supersymmetric Schur polynomials, and give a method for decomposing their tensor products. Along the way, we describe indecomposable objects in Rep(GL_δ) and explain how to decompose their tensor products.

Semisimplicity and Indecomposable Objects in Interpolating Partition Categories

We study Karoubian tensor categories which interpolate representation categories of families of so-called easy quantum groups in the same sense in which Deligne's interpolation categories $\mathrm{\underline{Rep}}(S_t)$ interpolate the representation categories of the symmetric groups. As such categories can be described using a graphical calculus of partitions, we call them interpolating partition categories. They include $\mathrm{\underline{Rep}}(S_t)$ as a special case and can generally be viewed as subcategories of the latter. Focusing on semisimplicity and descriptions of the indecomposab…

Injective objects in the category of finitely presented representations of an interval finite quiver

We characterize the indecomposable injective objects in the category of finitely presented representations of an interval finite quiver.

The Green Ring of Drinfeld Double $D(H_4)$

In this paper, we study the Green ring (or the representation ring) of Drinfeld quantum double $D(H_4)$ of Sweedler's 4-dimensional Hopf algebra $H_4$. We first give the decompositions of the tensor products of finite dimensional indecomposable modules into the direct sum of indecomposable modules over $D(H_4)$. Then we describe the structure of the Green ring $r(D(H_4))$ of $D(H_4)$ and show that $r(D(H_4))$ is generated, as a ring, by infinitely many elements subject to a family of relations.

关于0阶持续同调的加法像

On the additive image of 0th persistent homology

本文研究了0阶持续同调在有限范畴上的加法像，探讨了不可分解表示的性质，并开发了判断表示是否在加法像内的算法。

Thick Ideals in Deligne's category $\underline{\operatorname{Re}}\!\operatorname{p}(O_δ)$

We describe indecomposable objects in Deligne's category $\underline{\operatorname{Re}}\!\operatorname{p}(O_δ)$ and explain how to decompose their tensor products. We then classify thick ideals in $\underline{\operatorname{Re}}\!\operatorname{p}(O_δ)$. As an application we classify the indecomposable summands of tensor powers of the standard representation of the orthosymplectic supergroup up to isomorphism.

相对朗兰兹对偶对于 osp(2n + 1|2n)

Relative Langlands duality for $\mathfrak{osp}(2n + 1|2n)$

本文研究了 osp(2n + 1|2n) 的相对朗兰兹对偶，证明了 S-对偶关系并提出了相关全球猜想。