arXiv subjects

Huang Lin

Publications and source records attributed to Huang Lin.

5 recordsLinked to original sources

arXiv2023

Representation type of cyclotomic quiver Hecke algebras of type $C^{(1)}_{\ell}$

We first investigate a connected quiver consisting of all dominant maximal weights for an integrable highest weight module in affine type C. This quiver provides an efficient method to obtain all dominant maximal weights. Then, we completely determine the representation type of cyclotomic Khovanov-Lauda-Rouquier algebras of arbitrary level in affine type C, by using the quiver we construct. We also determine the Morita equivalence classes and graded decomposition matrices of certain representation-finite and tame cyclotomic KLR algebras.

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arXiv2022

Trace forms on the cyclotomic Hecke algebras and cocenters of the cyclotomic Schur algebras

We define a unified trace form $\tau$ on the cyclotomic Hecke algebras $\mathscr{H}_{n,K}$ of type $A$, which generalize both Malle-Mathas' trace form on the non-degenerate version (with Hecke parameter $\xi\neq 1$) and Brundan-Kleshchev's trace form on the degenerate version. We use seminormal basis theory to construct a pair of dual bases for $\mathscr{H}_{n,K}$ with respect to the form. We also construct an explicit basis for the cocenter (i.e., the $0$th Hochschild homology) of the corresponding cyclotomic Schur algebra, which shows that the cocenter has dimension independent of the ground field $K$, the Hecke parameter $\xi$ and the cyclotomic parameters $Q_1,\cdots,Q_\ell$.

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arXiv2022

On the center conjecture for the cyclotomic KLR algebras

The center conjecture for the cyclotomic KLR algebras $R_\beta^\Lambda$ asserts that the center of $R_\beta^\Lambda$ consists of symmetric elements in its KLR $x$ and $e(\nu)$ generators. In this paper we show that this conjecture is equivalent to the injectivity of some natural map $\bar{\iota}_\beta^{\Lambda,i}$ from the cocenter of $R_\beta^\Lambda$ to the cocenter of $R_\beta^{\Lambda+\Lambda_i}$ for all $i\in I$ and $\Lambda\in P^+$. We prove that the map $\bar{\iota}_\beta^{\Lambda,i}$ is given by multiplication with a center element $z(i,\beta)\in R_\beta^{\Lambda+\Lambda_i}$ and we explicitly calculate the element $z(i,\beta)$ in terms of the KLR $x$ and $e(\nu)$ generators. We present an explicit monomial basis for certain bi-weight spaces of the defining ideal of $R_\beta^\Lambda$ and of $R_\beta^\Lambda$. For $\beta=\sum_{j=1}^n\alpha_{i_j}$ with $\alpha_{i_1},\cdots, \alpha_{i_n}$ pairwise distinct, we construct an explicit monomial basis of $R_\beta^\Lambda$, prove the map $\bar{\iota}_\beta^{\Lambda,i}$ is injective and thus verify the center conjecture for these $R_\beta^\Lambda$.

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arXiv2019

Crystal of affine $\widehat{\mathfrak{sl}}_{\ell}$ and Hecke algebras at a primitive $2\ell$th root of unity

Let $\ell\in\mathbb{N}$ with $\ell>2$ and $I:=\mathbb{Z}/2\ell\mathbb{Z}$. In this paper we give a new realization of the crystal of affine $\widehat{\mathfrak{sl}}_{\ell}$ using the modular representation theory of the affine Hecke algebras $H_n$ of type $A$ and their level two cyclotomic quotients with Hecke parameter being a primitive $2\ell$th root of unity. We realized the Kashiwara operators for the crystal as the functors of taking socle of certain two-steps restriction and of taking head of certain two-steps induction. For any finite dimensional irreducible $H_n$-module $M$, we prove that the irreducible submodules of $\rm{res}_{H_{n-2}}^{H_n}M$ which belong to $\widehat{B}(\infty)$ (Definition 6.1) occur with multiplicity two. The main results generalize the earlier work of Grojnowski and Vazirani on the relations between the crystal of affine $\widehat{\mathfrak{sl}}_{\ell}$ and the affine Hecke algebras of type $A$ at a primitive $\ell$th root of unity.

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arXiv2015

FairTest: Discovering Unwarranted Associations in Data-Driven Applications

In a world where traditional notions of privacy are increasingly challenged by the myriad companies that collect and analyze our data, it is important that decision-making entities are held accountable for unfair treatments arising from irresponsible data usage. Unfortunately, a lack of appropriate methodologies and tools means that even identifying unfair or discriminatory effects can be a challenge in practice. We introduce the unwarranted associations (UA) framework, a principled methodology for the discovery of unfair, discriminatory, or offensive user treatment in data-driven applications. The UA framework unifies and rationalizes a number of prior attempts at formalizing algorithmic fairness. It uniquely combines multiple investigative primitives and fairness metrics with broad applicability, granular exploration of unfair treatment in user subgroups, and incorporation of natural notions of utility that may account for observed disparities. We instantiate the UA framework in FairTest, the first comprehensive tool that helps developers check data-driven applications for unfair user treatment. It enables scalable and statistically rigorous investigation of associations between application outcomes (such as prices or premiums) and sensitive user attributes (such as race or gender). Furthermore, FairTest provides debugging capabilities that let programmers rule out potential confounders for observed unfair effects. We report on use of FairTest to investigate and in some cases address disparate impact, offensive labeling, and uneven rates of algorithmic error in four data-driven applications. As examples, our results reveal subtle biases against older populations in the distribution of error in a predictive health application and offensive racial labeling in an image tagger.

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