arXiv subjects
Publications and source records attributed to Naihuan Jing.
In this paper, we study a class of non-weight modules over the generalized Heisenberg-Virasoro algebra of rank two $\widetilde{L}(p_1, p_2)$. We construct a family of irreducible $\widetilde{L}(p_1, p_2)$-modules, determine the isomorphism classes and show that these modules exhaust all the $\widetilde{L}(p_1, p_2)$-modules that are free modules of rank one over the Cartan subalgebra.
A $k$-ribbon tiling is a decomposition of a connected skew diagram into disjoint ribbons of size $k$. In this paper, we establish a connection between a subset of $k$-ribbon tilings and Petrie symmetric functions, thus providing a combinatorial interpretation for the coefficients in a Pieri-like rule for the Petrie symmetric functions due to Grinberg (Algebr. Comb. 5 (2022), no. 5, 947-1013). This also extends a result by Cheng, Chou and Eu et al. (Proc. Amer. Math. Soc. 151 (2023), no. 5, 1839-1854). As a bonus, our findings can be effectively utilized to derive certain specializations.
Manin matrices are quantum linear transformations of general quantum spaces. In this paper, we study the $q$-analogue of super Manin matrices and obtain several quantum versions of classical identities, such as Jacobi's ratio theorem, Schur's complement theorem, Cayley's complementary theorem, Muir's law, Sylvester's theorem, MacMahon Master Theorem and Newton's identities.
We first obtain a trace formula for immanants of generalized principal submatrix of any complex matrix based on any weight space for finite dimensional representations of the general linear group. Our trace formula contains Kostant's famous formula for immanants on $0$-weight spaces as special case. We then present a criterion for non-vanishing immanants for any generalized principal submatrix of positive definite Hermitian or nonsingular totally nonnegative matrices, which strengthened the well-known results of Schur and Stembridge. Furthermore, we present an inequality that contains Kostant, Schur and Stembridge's famous inequalities as special cases.
We introduce the quantum Berezinian for the quantum affine superalgebra $\mathrm{U}_q(\widehat{\mathfrak{gl}}_{M|N})$ and show that the coefficients of the quantum Berezinian belong to the center of $\mathrm{U}_q(\widehat{\gl}_{M|N})$. We also construct another family of central elements which can be expressed in the quantum Berezinian by a Liouville-type theorem. Moreover, we prove analogues of the Jacobi identities, the Schur complementary theorem, the Sylvester theorem and the MacMahon Master theorem for the generator matrices of $\mathrm{U}_q(\widehat{\gl}_{M|N})$.
In the 1940s Littlewood formulated three fundamental correspondences for the immanants and Schur symmetric functions on the general linear group, which establish deep connections between representation theory of the symmetric group and the general linear group parallel to the Schur-Weyl duality. In this paper, we introduce the notion of quantum immanants in the quantum coordinate algebra using primitive idempotents of the Hecke algebra. By employing $R$-matrix techniques, we establish the quantum analog of Littlewood correspondences between quantum immanants and Schur functions for the quantum coordinate algebra. In the setting of the Schur-Weyl-Jimbo duality, we construct an exact correspondence between the Gelfand-Tsetlin bases of the irreducible representations of the quantum enveloping algebra $U_q(\mathfrak{gl}(n))$ and Young's orthonormal basis of an irreducible representation of the Hecke algebra $\mathcal H_m$. This isomorphism leads to our trace formula for the quantum immanants, which settled the generalization problem of $q$-analog of Kostant's formular for $λ$-immanants. As applications, we also derive general $q$-Littlewood-Merris-Watkins identities and $q$-Goulden-Jackson identities as special cases of the quantum Littlewood correspondence III.
In this paper we present a modified version of the proof given Jing-Yang-Zhao's paper titled "Local Unitary Equivalence of Quantum States and Simultaneous Orthogonal Equivalence," which established the correspondance between local unitary equivalence and simultaneous orthogonal equivalence of $2$-qubits. Our modified proof utilizes a hypermatrix algebra framework, and through this framework we are able to generalize this correspondence to $3$-qubits. Finally, we apply a generalization of Specht's criterion (first proved in "Specht's Criterion for Systems of Linear Mappings" by V. Futorney, R. A. Horn, and V. V. Sergeichuk) to reduce the problem of local unitary equivalence of $3$-qubits to checking trace identities and a few other easy-to-check properties. We also note that all of these results can be extended to $2$ and $3$ qudits if we relax the notion of LU equivalence to quasi-LU equivalence, as defined in the aforementioned paper by Jing et. al.
In this paper we present a modified version of the proof given Jing-Yang-Zhao's paper "Local Unitary Equivalence of Quantum States and Simultaneous Orthogonal Equivalence," which established the correspondence between local unitary (LU) equivalence and simultaneous orthogonal equivalence of bipartite quantum states. Our modified proof utilizes a hypermatrix algebra framework, and with this framework we are able to generalize this correspondence to tripartite quantum states. Finally, we apply a generalization of Specht's criterion proved in Futorny-Horn-Sergeichuk' paper "Specht's Criterion for Systems of Linear Mappings" to \textit{essentially} reduce the problem of local unitary equivalence of tripartite quantum states to checking trace identities and a few other LU invariants. We also note that all of these results can be extended to arbitrary multipartite quantum states, however there are some practical limitations.
Based on the realignment moments of density matrix, we study parameterized entanglement criteria for bipartite and multipartite states. By adjusting the different parameter values, our criterion can detect not only bound entangled states, but also non-positive partial transpose entangled states for bipartite quantum systems. Moreover, we propose the definition of multipartite realignment moments and generalize the result of bipartite systems to obtain a sufficient criterion to detect entanglement for multipartite quantum states in arbitrary dimensions. And we further improve the conclusion to obtain another new entanglement criterion. The new method can detect more entangled states than previous methods as backed by detailed examples.
This paper studies irreducible characters of the $q$-rook monoid algebra $R_n(q)$ using the vertex algebraic method. Based on the Frobenius formula for $R_n(q)$, a new iterative character formula is derived with the help of the vertex operator realization of the Schur symmetric function. The same idea also leads to a simple proof of the Murnaghan-Nakayama rule for $R_n(q)$. We also introduce the bitrace for the $q$-rook monoid and derive its combinatorial formula as a generalization of the bitrace formula for the Iwahori-Hecke algebra. The character table of $R_n(q)$ with $|μ|=5$ is listed in the appendix.
Schur functions has been shown to satisfy certain plethysm stability properties and recurrence relations. In this paper, use vertex operator methods to study analogous stability properties of Schur's $Q$-functions. Although the two functions have similar stability properties, we find a special case where the plethysm of Schur's $Q$-functions exhibits linear increase.
We establish a parabolic presentation of the extended Yangian $\X(\mathfrak{g}_{N})$ associated with the Lie algebras $\mathfrak{g}_{N}$ of type $B$ and $C$, parameterized by a symmetric composition $ν$ of $N$. By formulating a block matrix version of the RTT presentation of $\X(\mathfrak{g}_{N})$, we systematically derive the generators and relations through the Gauss decomposition of the generator matrix in $ν$-block form. Furthermore, leveraging this parabolic presentation, we obtain a novel formula for the center of $\X(\mathfrak{g}_{N})$, offering new insights into its structure.
It is well known that any well-defined bipartite entanglement measure $\mathcal{E}$ obeys $γ$th-monogamy relations Eq. (1.1) and assisted measure $\mathcal{E}_{a}$ obeys $δ$th-polygamy relations Eq. (1.2). Recently, we presented a class of tighter parameterized monogamy relation for the $α$th $(α\geqγ)$ power based on Eq. (1.1). This study provides a family of tighter lower (resp. upper) bounds of the monogamy (resp. polygamy) relations in a unified manner. In the first part of the paper, the following three basic problems are focused: (i) tighter monogamy relation for the $α$th ($0\leq α\leq γ$) power of any bipartite entanglement measure $\mathcal{E}$ based on Eq. (1.1); (ii) tighter polygamy relation for the $β$th ($ β\geq δ$) power of any bipartite assisted entanglement measure $\mathcal{E}_{a}$ based on Eq. (1.2); (iii) tighter polygamy relation for the $ω$th ($0\leq ω\leq δ$) power of any bipartite assisted entanglement measure $\mathcal{E}_{a}$ based on Eq. (1.2). In the second part, using the tighter polygamy relation for the $ω$th ($0\leq ω\leq 2$) power of CoA, we obtain good estimates or bounds for the $ω$th ($0\leq ω\leq 2$) power of concurrence for any $N$-qubit pure states $|ψ\rangle_{AB_{1}\cdots B_{N-1}}$ under the partition $AB_{1}$ and $B_{2}\cdots B_{N-1}$. Detailed examples are given to illustrate that our findings exhibit greater strength across all the region.
Quantum discord, a key indicator of non-classical correlations in bipartite systems, has been recently extended to multipartite scenarios [Phys. Rev. Lett. 2020, 124:110401]. We present exact analytic formulas for the quantum discord of special families of N-qubit states, including generalized class of GHZ states. Our formulations span $2$, $3$, $4n$, $4n+1$, $4n+2$, and $4n+3$-qubit configurations where $n\in 1, 2, \ldots$, which refine the assessment of quantum correlations and provide an analytical tool in quantum computation. Moreover, we uncover a ``discord freezing'' in even-qubit systems under phase flip decoherence which provides a means for preserving quantum coherence in environmental perturbations.
We introduce two vertex operators to realize skew odd orthogonal characters $so_{λ/μ}(x^{\pm})$ and derive the Cauchy identity for the skew characters via Toeplitz-Hankel-type determinant similar to the Schur functions. The method also gives new proofs of the Jacobi--Trudi identity and Gelfand--Tsetlin patterns for $so_{λ/μ}(x^{\pm})$. Moreover, combining the vertex operators related to characters of types $C,D$ (\cite{Ba1996,JN2015}) and the new vertex operators related to $B$-type characters, we obtain three families of symmetric polynomials that interpolate among characters of $SO_{2n+1}(\mathbb{C})$, $SO_{2n}(\mathbb{C})$ and $Sp_{2n}(\mathbb{C})$, Their transition formulas are also explicitly given among symplectic and/or orthogonal characters and odd orthogonal characters.
$q$-Yangians can be viewed both as quantum deformations of the loop algebras of upper triangular Lie algebras and deformations of the Yangian algebras. In this paper, we study the quantum affine algebra as a product of two copies of the $q$-Yangian algebras. This viewpoint enables us to investigate the invariant theory of quantum affine algebras and their twisted versions. We introduce the twisted Sklyanin determinant for twisted quantum affine algebras and establish various identities for the Sklyanin determinants.
In this paper, we use vertex operator techniques to compute character values on unipotent classes of $\GL_n(\mathbb F_q)$. By realizing the Grothendieck ring $R_G=\bigoplus_{n\geq0}^\infty R(\GL_n(\mathbb F_q))$ as Fock spaces, we formulate the Murnanghan-Nakayama rule of $\GL_n(\mathbb F_q)$ between Schur functions colored by an orbit $ϕ$ of linear characters of $\overline{\mathbb F}_q$ under the Frobenius automorphism on and modified Hall-Littlewood functions colored by $f_1=t-1$, which provides detailed information on the character table of $\GL_n(\mathbb F_q)$. As applications, we use vertex algebraic methods to determine the Steinberg characters of $\GL_n(\mathbb F_q)$, which were previously determined by Curtis-Lehrer-Tits via geometry of homology groups of spherical buildings and Springer-Zelevinsky utilizing Hopf algebras.
We establish an explicit correspondence between the Drinfeld current algebra presentation for the two-parameter quantum affine algebra $U_{r, s}(\mathrm{C}_n^{(1)})$ and the $R$-matrix realization á la Faddeev, Reshetikhin and Takhtajan.