arXiv subjects
Publications and source records attributed to Xin Fang.
We establish an explicit bijection between the toric degenerations of the Grassmannian $\textrm{Gr}(2,n)$ arising from maximal cones in tropical Grassmannians and the ones coming from plabic graphs corresponding to $\textrm{Gr}(2,n)$. We show that a similar statement does not hold for $\textrm{Gr}(3,6)$.
Linear degenerate flag varieties are degenerations of flag varieties as quiver Grassmannians. For type A flag varieties, we obtain characterizations of flatness, irreducibility and normality of these degenerations via rank tuples. Some of them are shown to be isomorphic to Schubert varieties and can be realized as highest weight orbits of partially degenerate Lie algebras, generalizing the corresponding results on degenerate flag varieties. To study normality, cell decompositions of quiver Grassmannians are constructed in a wider context of equioriented quivers of type A.
Following the historical track in pursuing $T$-equivariant flat toric degenerations of flag varieties and spherical varieties, we explain how powerful tools in algebraic geometry and representation theory, such as canonical bases, Newton-Okounkov bodies, PBW-filtrations and cluster algebras come to push the subject forward.
We introduce in this paper the marked chain-order polytopes associated to a marked poset, generalizing the marked chain polytopes and marked order polytopes by putting them as extremal cases in an Ehrhart equivalent family. Some combinatorial properties of these polytopes are studied. This work is motivated by the framework of PBW degenerations in representation theory of Lie algebras.
In this paper generalized Matsumoto-Tits sections lifting permutations to the algebra associated to a generalized virtual braid monoid are defined. They are then applied to study the defining relations of the quantum quasi-shuffle algebras via the total symmetrization operator.
We give a new proof for the fact that the number of torus fixed points for the degenerated flag variety is equal to the normalized median Genocchi number, using the identification with a certain Schubert variety. We further study the torus fixed points for the symplectic degenerated flag variety and develop a combinatorial model, symplectic Dellac configurations, so parametrize them. The number of these symplectic fixed points is conjectured to be the median Euler number.
We provide $\mathbb{N}$-filtrations on the negative part $U_q(\mathfrak{n}^-)$ of the quantum group associated to a finite-dimensional simple Lie algebra $\mathfrak{g}$, such that the associated graded algebra is a skew-polynomial algebra on $\mathfrak{n}^-$. The filtration is obtained by assigning degrees to Lusztig's quantum PBW root vectors. The possible degrees can be described as lattice points in certain polyhedral cones. In the classical limit, such a degree induces an $\mathbb{N}$-filtration on any finite dimensional simple $\mathfrak{g}$-module. We prove for type $\tt{A}_n$, $\tt{C}_n$, $\tt{B}_3$, $\tt{D}_4$ and $\tt{G}_2$ that a degree can be chosen such that the associated graded modules are defined by monomial ideals, and conjecture that this is true for any $\mathfrak{g}$.
We realize some powers of Dedekind $η$-function as traces on quantum coordinate algebras.
Two-dimensional (2D) layered materials with diverse properties have attracted significant interest in the past decade. The layered materials discovered so far have covered a wide, yet discontinuous electromagnetic spectral range from semimetallic graphene, insulating boron nitride, to semiconductors with bandgaps from middle infrared to visible light. Here, we introduce new layered semiconductors, black arsenic-phosphorus (b-AsP), with highly tunable chemical compositions and electronic and optical properties. Transport and infrared absorption studies demonstrate the semiconducting nature of b-AsP with tunable bandgaps, ranging from 0.3 to 0.15 eV. These bandgaps fall into long-wavelength infrared (LWIR) regime and cannot be readily reached by other layered materials. Moreover, polarization-resolved infrared absorption and Raman studies reveal in-plane anisotropic properties of b-AsP. This family of layered b-AsP materials extend the electromagnetic spectra covered by 2D layered materials to the LWIR regime, and may find unique applications for future all 2D layered material based devices.
We will introduce an $\mathbb{N}$-filtration on the negative part of a quantum group of type $A_n$, such that the associated graded algebra is a q-commutative polynomial algebra. This filtration is given in terms of the representation theory of quivers, by realizing the quantum group as the Hall algebra of a quiver. We show that the induced associated graded module of any simple finite-dimensional module (of type 1) is isomorphic to a quotient of this polynomial algebra by a monomial ideal, and we provide a monomial basis for this associated graded module. This construction can be viewed as a quantum analog of the classical PBW framework, and in fact, by considering the classical limit, this basis is the monomial basis provided by Feigin, Littelmann and the second author in the classical setup.
Two-dimensional (2D) layered tungsten diselenides (WSe2) material has recently drawn a lot of attention due to its unique optoelectronic properties and ambipolar transport behavior. However, direct chemical vapor deposition (CVD) synthesis of 2D WSe2 is not as straightforward as other 2D materials due to the low reactivity between reactants in WSe2 synthesis. In addition, the growth mechanism of WSe2 in such CVD process remains unclear. Here we report the observation of a screw-dislocation-driven (SDD) spiral growth of 2D WSe2 flakes and pyramid-like structures using a sulfur-assisted CVD method. Few-layer and pyramid-like WSe2 flakes instead of monolayer were synthesized by introducing a small amount of sulfur as a reducer to help the selenization of WO3, which is the precursor of tungsten. Clear observations of steps, helical fringes, and herring-bone contours under atomic force microscope characterization reveal the existence of screw dislocations in the as-grown WSe2. The generation and propagation mechanisms of screw dislocations during the growth of WSe2 were discussed. Back-gated field-effect transistors were made on these 2D WSe2 materials, which show on/off current ratios of 106 and mobility up to 44 cm2/Vs.
Two generating sets of the defining ideal of a Nichols algebra of diagonal type are proposed, which are then applied to study the bar involution and the specialization problem of quantum groups associated to non-symmetrizable generalized Cartan matrices.
We introduce in this paper the generalized virtual braid group on n strands GVB_n, generalizing simultaneously the braid groups and their virtual versions. A Mastumoto-Tits type section lifting shuffles in a symmetric group S_n to the monoid associated to GVB_n is constructed, which is then applied to characterize the quantum quasi-shuffle product. A family of representations of GVB_n is constructed using quantum groups.
We investigate a Hopf algebra structure on the cotensor coalgebra associated to a Hopf bimodule algebra which contains universal version of Clifford algebras and quantum groups as examples. It is shown to be the bosonization of the quantum quasi-shuffle algebra built on the space of its right coinvariants. The universal property and a Rota-Baxter algebra structure are established on this new algebra.
We construct multi-brace cotensor Hopf algebras with bosonizations of quantum multi-brace algebras as examples. Quantum quasi-symmetric algebras are then obtained by taking particular initial data; this allows us to realize the whole quantum group associated to a symmetrizable Kac-Moody Lie algebra as a quantum quasi-symmetric algebra and all highest weight irreducible representations can be constructed using this machinery. It also provides a systematic way to construct simple modules over the quantum double of a quantum group.
We prove in this paper a Borel-Weil-Bott type theorem for the coHochschild homology of a quantum shuffle algebra associated with quantum group datum taking coefficients in some well-chosen bicomodules, which can be looked as an analogue of equivariant line bundles over flag varieties in the non-commutative case.
This paper is devoted to understanding the defining ideal of a Nichols algebra from the decomposition of specific elements in the group algebra of braid groups. A family of primitive elements are found and algorithms are proposed. To prove the main result, the differential algebra of a Nichols algebra is constructed. Moreover, another point of view on Serre relations is provided.
We give a unified construction of quantum groups, q-Boson algebras and quantized Weyl algebras and an action of quantum groups on quantized Weyl algebras. This enables us to give a conceptual proof of the semi-simplicity of the category $\mathcal{O}(B_q)$ introduced by T.Nakashima and the classification of all simple objects in it.