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In this paper, we establish product and coproduct rules for three bases of the Hopf algebra $\textsf{QSym}_F$ of quasisymmetric functions over $F$, with $F$ being either $\mathbb{C}(q,t)$ or $\mathbb{C}(q)$. These results are derived through the categorizations of $\textsf{QSym}_{\mathbb{C}}$ obtained by utilizing the normal lattice supercharacter theories. Firstly, we deal with a basis $\{\mathcal{D}_{\alpha}(q,t) \mid \alpha \in \textsf{Comp}\}$ of $\textsf{QSym}_{\mathbb{C}(q,t)}$, where $\textsf{Comp}$ denotes the set of all compositions. This basis is obtained from the direct sum of specific supercharacter function spaces and consists of superclass identifier functions. Upon appropriate specializations of $q$ and $t$, it yields notable bases of $\textsf{QSym}_{\mathbb{C}}$ and $\textsf{QSym}_{\mathbb{C}(q)}$, including enriched $q$-monomial quasisymmetric functions introduced by Grinberg and Vassilieva. Secondly, we deal with the basis $\{G_{\alpha}(q) \mid \alpha \in \textsf{Comp}\}$ of $\textsf{QSym}_{\mathbb{C}(q)}$, where $G_{\alpha}(q)$ represents the quasisymmetric Hall-Littlewood function introduced by Hivert. Our product rule is new, whereas our coproduct rule turns out to be equivalent to the existing coproduct rule of Hivert. Finally, we consider a basis $\{M_{\alpha}(q) \mid \alpha \in \textsf{Comp}\}$ of $\textsf{QSym}_{\mathbb{C}(q)}$, where $M_{\alpha}(q)$ is a $q$-analogue of the monomial quasisymmetric function., Comment: 51 pages. This paper is a revised version of arXiv:2207.11110, yet it is released independently due to substantial revisions and a significant shift in the research direction

The purpose of this paper is to study induction and restriction of two-variable Hecke algebras. First, we provide the explicit form of the Mackey decomposition formula. And then, we elucidate how (anti-)involutions interact with induction product and restriction., Comment: 16 pages

Layered LiMO2 (M = Ni, Co, Mn, and Al mixture) cathode materials used for Li-ion batteries are reputed to be highly reactive through their surface, where the chemistry changes rapidly when exposed to ambient air. However, conventional electron/spectroscopy-based techniques or thermogravimetric analysis fails to capture the underlying atom-scale chemistry of vulnerable Li species. To study the evolution of the surface composition at the atomic scale, here we use atom probe tomography and probed the surface species formed during exposure of a LiNi0.8Mn0.1Co0.1O2 (NMC811) cathode material to air. The compositional analysis evidences the formation of Li2CO3. Site specific examination from a cracked region of an NMC811 particle also suggests the predominant presence of Li2CO3. These insights will help to design improved protocols for cathode synthesis and cell assembly, as well as critical knowledge for cathode degradation

Let us fix a positive integer $\nu>1$. For each positive integer $n>1$, we consider a normal supercharacter theory $\mathcal{S}_n$ of $G_n$, where $G_n$ is the direct-product of $n-1$ copies of the cyclic group of order $\nu$. Then we endow $\bigoplus_{n \ge 0} \textsf{scf}(\mathcal{S}_n)$, the direct-product of supercharacter function spaces, with the Hopf algebra structure that is isomorphic to the Hopf algebra $\textsf{QSym}$ of quasisymmetric functions. Furthermore, we compute the structure constants of the Hopf algebra thus obtained for the basis consisting of superclass identifier functions. Using our categorifications, we study a new basis for the Hopf algebra $\textsf{NSym}_{\mathbb{C}(q,t)}$ of noncommutative symmetric functions over the rational function field $\mathbb{C}(q,t)$ in commuting variables $q$ and $t$, with an emphasis on the structure constants of $\textsf{NSym}_{\mathbb{C}(q,t)}$ for this basis. Some interesting applications are also obtained via the specializations of $q$ and $t$., Comment: 38 pages, typos corrected

The purpose of this paper is to provide a unified method for dealing with various 0-Hecke modules constructed using tableaux so far. To do this, we assign a $0$-Hecke module to each left weak Bruhat interval, called a weak Bruhat interval module. We prove that every indecomposable summand of the $0$-Hecke modules categorifying dual immaculate quasisymmetric functions, extended Schur functions, quasisymmetric Schur functions, and Young row-strict quasisymmetric Schur functions is a weak Bruhat interval module. We further study embedding into the regular representation, induction product, restriction, and (anti-)involution twists of weak Bruhat interval modules., Comment: 36 pages; v2: abstract, introduction, Subsection 3.1, and Subsection 3.3 are revised; reference [13] is added, v3: minor typos are revised; published in Math. Z

We extend the duality between acyclic orientations and totally cyclic orientations on planar graphs to dualities on graphs on orientable surfaces by introducing boundary acyclic orientations and totally bi-walkable orientations. In addition, we provide a reciprocity theorem connecting local tensions and boundary acyclic orientations. Furthermore, we define the balanced flow polynomial which is connected with tension polynomial by duality and with totally bi-walkable orientations by reciprocity., Comment: A revised version, 18 pages, 7 figures

We define the acyclic orientation polynomial of a graph to be the generating function for the sinks of its acyclic orientations. Stanley proved that the number of acyclic orientations is equal to the chromatic polynomial evaluated at $-1$ up to sign. Motivated by this link between acyclic orientations and the chromatic polynomial, we develop "acyclic orientation" analogues of theorems concerning the chromatic polynomial of Birkhoff, Whitney, and Greene-Zaslavsky. As an application, we provide a new proof for Stanley's sink theorem for chromatic symmetric functions $X_G$. This theorem gives a relation between the number of acyclic orientations with a fixed number of sinks and the coefficients in the expansion of $X_G$ with respect to elementary symmetric functions., Comment: 21 pages, 4 figures