arXiv subjects

Jieru Zhu

Publications and source records attributed to Jieru Zhu.

11 recordsLinked to original sources

arXiv2026

SL2 tilting modules in the mixed case

Using the non-semisimple Temperley-Lieb calculus, we study the additive and monoidal structure of the category of tilting modules for $\mathrm{SL}_{2}$ in the mixed case. This simultaneously generalizes the semisimple situation, the case of the complex quantum group at a root of unity, and the algebraic group case in positive characteristic. We describe character formulas and give a presentation of the category of tilting modules as an additive category via a quiver with relations. Turning to the monoidal structure, we describe fusion rules and obtain an explicit recursive description of the appropriate analog of Jones-Wenzl projectors.

math.RT↗

arXiv2026

The conflated expression graph for an arbitrary permutation

We show that the conflated expression graph for an arbitrary permutation has a unique minimal element and a unique maximal element, and every reduced expression sits on a maximal chain from the source to the sink. This generalizes the work of Manin-Schechtman regarding higher Bruhat orders, and gives an independent and self-contained proof of certain results in Hothem. In addition, we give explicit algorithms for elements in the top and bottom commutation classes. Given any reduced expression $ρ$, we give an explicit method for producing a maximal chain containing $ρ$.

math.CO↗

arXiv2023

M-diagram basis of the Specht module for $(n,n,n)$

Motivated by the M-diagrams defined by Tymoczko, we show that these locally non-crossing $\mathfrak{sl}_3$-webs form a basis of the Specht module for the partition $(n,n,n)$. They further admit a unitriangular base change to both the polytabloid basis and the non-elliptic web basis.

math.RT↗

arXiv2023

Superalgebra deformations of web categories: finite webs

Let $\mathbb{k}$ be a characteristic zero domain. For a locally unital $\mathbb{k}$-superalgebra $A$ with distinguished idempotents $I$and even subalgebra $a \subseteq A_{\bar 0}$, we define and study an associated diagrammatic monoidal $\mathbb{k}$-linear supercategory $\mathbf{Web}^{A,a}_I$. This supercategory yields a diagrammatic description of the generalized Schur algebras $T^A_a(n,d)$. We also show there is an asymptotically faithful functor from $\mathbf{Web}^{A,a}_I$ to the monoidal supercategory of $\mathfrak{gl}_n(A)$-modules generated by symmetric powers of the natural module. When this functor is full, the single diagrammatic supercategory $\mathbf{Web}^{A,a}_I$ provides a combinatorial description of this module category for all $n \geq 1$. We also use these results to establish Howe dualities between $\mathfrak{gl}_{m}(A)$ and $\mathfrak{gl}_{n}(A)$ when $A$ is semisimple.

math.RT↗

arXiv2022

Tensor Representations for the Drinfeld Double of the Taft Algebra

Over an algebraically closed field $\mathbb k$ of characteristic zero, the Drinfeld double $D_n$ of the Taft algebra that is defined using a primitive $n$th root of unity $q \in \mathbb k$ for $n \geq 2$ is a quasitriangular Hopf algebra. Kauffman and Radford have shown that $D_n$ has a ribbon element if and only if $n$ is odd, and the ribbon element is unique; however there has been no explicit description of this element. In this work, we determine the ribbon element of $D_n$ explicitly. For any $n \geq 2$, we use the R-matrix of $D_n$ to construct an action of the Temperley-Lieb algebra $\mathsf{TL}_k(ξ)$ with $ξ= -(q^{\frac{1}{2}}+q^{-\frac{1}{2}})$ on the $k$-fold tensor power $V^{\otimes k}$ of any two-dimensional simple $D_n$-module $V$. This action is known to be faithful for arbitrary $k \geq 1$. We show that $\mathsf{TL}_k(ξ)$ is isomorphic to the centralizer algebra $\text{End}_{D_n}(V^{\otimes k})$ for $1 \le k \le 2n-2$.

math.RA↗

arXiv2021

McKay Matrices for Finite-dimensional Hopf Algebras

For a finite-dimensional Hopf algebra $A$, the McKay matrix $M_V$ of an $A$-module $V$ encodes the relations for tensoring the simple $A$-modules with $V$. We prove results about the eigenvalues and the right and left (generalized) eigenvectors of $M_V$ by relating them to characters. We show how the projective McKay matrix $Q_V$ obtained by tensoring the projective indecomposable modules of $A$ with $V$ is related to the McKay matrix of the dual module of $V$. We illustrate these results for the Drinfeld double $D_n$ of the Taft algebra by deriving expressions for the eigenvalues and eigenvectors of $M_V$ and $Q_V$ in terms of several kinds of Chebyshev polynomials. For the matrix $N_V$ that encodes the fusion rules for tensoring $V$ with a basis of projective indecomposable $D_n$-modules for the image of the Cartan map, we show that the eigenvalues and eigenvectors also have such Chebyshev expressions.

math.RA↗

arXiv2020

Transitioning between tableaux and spider bases for Specht modules

Regarding the Specht modules associated to the two-row partition $(n,n)$, we provide a combinatorial path model to study the transitioning matrix from the tableau basis to the $A_1$-web basis (i.e. cup diagrams), and prove that the entries in this matrix are positive in the upper-triangular portion with respect to a certain partial order.

math.RT↗

arXiv2020

Two Boundary Centralizer Algebras for $\mathfrak{q}(n)$

We define the degenerate two boundary affine Hecke-Clifford algebra $\mathcal{H}_d$, and show it admits a well-defined $\mathfrak{q}(n)$-linear action on the tensor space $M\otimes N\otimes V^{\otimes d}$, where $V$ is the natural module for $\mathfrak{q}(n)$, and $M, N$ are arbitrary modules for $\mathfrak{q}(n)$, the Lie superalgebra of Type Q. When $M$ and $N$ are irreducible highest weight modules parameterized by a staircase partition and a single row, respectively, this action factors through a quotient of $\mathcal{H}_d$. We then construct explicit modules for this quotient, $\mathcal{H}_{p,d}$, using combinatorial tools such as shifted tableaux and the Bratteli graph. These modules belong to a family of modules which we call calibrated. Using the relations in $\mathcal{H}_{p,d}$, we also classify a specific class of calibrated modules. The irreducible summands of $M\otimes N\otimes V^{\otimes d}$ coincide with the combinatorial construction, and provide a weak version of the Schur-Weyl type duality.

math.RT↗

arXiv2020

Quasi-split symmetric pairs of $U(\mathfrak{gl}_N)$ and their Schur algebras

We establish explicit isomorphisms of two seemingly-different algebras, and their Schur algebras, arising from the centralizers of two different type B Weyl group actions in Schur-like dualities. We provide a presentation of the geometric counterpart of the above Schur algebras in Bao-Kujawa-Li-Wang (2018) specialized at $q=1$.

math.RT↗

arXiv2018

Learning Symmetry Consistent Deep CNNs for Face Completion

Deep convolutional networks (CNNs) have achieved great success in face completion to generate plausible facial structures. These methods, however, are limited in maintaining global consistency among face components and recovering fine facial details. On the other hand, reflectional symmetry is a prominent property of face image and benefits face recognition and consistency modeling, yet remaining uninvestigated in deep face completion. In this work, we leverage two kinds of symmetry-enforcing subnets to form a symmetry-consistent CNN model (i.e., SymmFCNet) for effective face completion. For missing pixels on only one of the half-faces, an illumination-reweighted warping subnet is developed to guide the warping and illumination reweighting of the other half-face. As for missing pixels on both of half-faces, we present a generative reconstruction subnet together with a perceptual symmetry loss to enforce symmetry consistency of recovered structures. The SymmFCNet is constructed by stacking generative reconstruction subnet upon illumination-reweighted warping subnet, and can be end-to-end learned from training set of unaligned face images. Experiments show that SymmFCNet can generate high quality results on images with synthetic and real occlusion, and performs favorably against state-of-the-arts.

cs.CV↗

arXiv2018

Two Boundary Centralizer Algebras for $\mathfrak{gl}(n|m)$

We define an action of the degenerate two boundary braid algebra $\mathcal{G}_d$ on the $\mathbb{C}$-vector space $M\otimes N\otimes V^{\otimes d}$, where $M$ and $N$ are arbitrary modules for the general linear Lie superalgebra $\mathfrak{gl}(n|m)$, and $V$ is the natural representation. When $M$ and $N$ are parametrized by rectangular hook Young diagrams, this action factors through a quotient $\mathcal{H}^{\operatorname{ext}}_d$. The irreducible summands of $M\otimes N\otimes V^{\otimes d}$ for the centralizer of $\mathfrak{gl}(n|m)$, remain irreducible once regarded as modules for this quotient $\mathcal{H}^{\operatorname{ext}}_d$.

math.RT↗