arXiv subjects

You-Hung Hsu

Publications and source records attributed to You-Hung Hsu.

6 recordsLinked to original sources

arXiv2025

$0$-affine quantum groups as K-theoretic Hall algebras

In this note, we show that the positive part of Arkhipov-Mazin's $0$-affine quantum group can be realized as the K-theoretic Hall algebra of the type $A$ Dynkin quiver. We then construct a categorical action of this positive part and demonstrate that such an action induces semiorthogonal decompositions on the corresponding weight categories. As a main example, we study the bounded derived category of coherent sheaves on $n$-step partial flag varieties.

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arXiv2024

On the Springer correspondence for wreath products

We establish a Bruhat decomposition indexed by the wreath product $\Sigma_m\wr \Sigma_d$ between two symmetric groups -- note that $\Sigma_m\wr \Sigma_d$ is not a Coxeter group in general. We show that such a decomposition affords a geometric variant in terms of the Bialynicki-Birula decomposition for varieties with $\mathbb{C}^*$-actions. Next, we construct a Steinberg variety whose top Borel-Moore homology realizes the group algebra $\mathbb{Q}[\Sigma_m\wr \Sigma_d]$ as a proper subalgebra. Such a geometric realization leads to a Springer-type correspondence which identifies the irreducible representations of $\Sigma_m\wr \Sigma_d$ with isotypic components of certain unconventional Springer fibers using type A geometry. In other words, we obtain a geometric counterpart of the (algebraic) Clifford theory, for the first time. Consequently, we obtain a new Springer correspondence of Weyl groups of type B/C/D using essentially type A geometry.

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arXiv2023

Categorical idempotents via shifted 0-affine algebras

We show that a categorical action of shifted 0-affine algebra naturally gives two families of pairs of complementary idempotents in the triangulated monoidal category of triangulated endofunctors for each weight category. Consequently, we obtain two families of pairs of complementary idempotents in the triangulated monoidal category $D^bCoh(G/P \times G/P)$. As an application, this provides examples where the projection functors of a semiorthogonal decomposition are kernel functors, and we determine the generators of the component categories in the Grassmannians case.

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arXiv2022

Semiorthogonal decomposition via categorical action

We show that the categorical action of the shifted $q=0$ affine algebra can be used to construct semiorthogonal decomposition on the weight categories. In particular, this construction recovers Kapranov's exceptional collection when the weight categories are the derived categories of coherent sheaves on Grassmannians and $n$-step partial flag varieties. Finally, as an application, we use this result to construct a semiorthogonal decomposition on the derived categories of coherent sheaves on Grassmannians of a coherent sheaf with homological dimension $\leq 1$ over a smooth projective variety $X$.

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arXiv2022

A categorical action of the shifted $q=0$ affine algebra

We introduce a new algebra called the shifted $q=0$ affine algebra, which arises naturally from the study of coherent sheaves on Grassmannians and n-step partial flag varieties via a natural correspondence. It has similar presentation as the shifted quantum affine algebra defined by Finkelberg-Tsymbaliuk arXiv:1708.01795v6. We then give a definition of its categorical action and prove that there is a categorical action on the bounded derived categories of coherent sheaves on n-step partial flag varieties. Finally, as an application, we use it to construct a categorical action of the $q=0$ affine Hecke algebra on the bounded derived category of coherent sheaves on the full flag variety.

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arXiv2021

Exceptional collections, t-structures, and categorical action of shifted $q=0$ affine algebra, $\mathfrak{sl}_{2}$ case

In this article, we show that the categorical action of the shifted $q=0$ affine algebra can be used to construct (or induce) t-structures on the weight categories. The main idea is to interpret the exceptional collection constructed by Kapranov as convolution of Fourier-Mukai kernels in the categorical action via using the Borel-Weil-Bott theorem. In particular, when the categories are the bounded derived category of coherent sheaves on Grassmannians. The t-structure we obtain is precisely the exotic t-structure defined by Bezrukavnikov of the exceptional collections given by Kapranov. As an application, we calculate the matrix coefficients for generators of the shifted $q=0$ affine algebra on the basis given by Kapranov exceptional collections.

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