We discuss a spectral property for the virial operator of the 2D Zakharov-Kuznetsov (ZK) equation. This is a crucial ingredient to establish blow-up or asymptotic stability of solitary waves in higher-dimensional problems. This model in 3D setting was originally introduced by Zakharov and Kuznetsov in plasma physics, and is also a higher-dimensional generalization of the well-known Korteweg-de Vries (KdV) equation. The problem of stability of solitary waves in ZK equation or stable blow-up in modified ZK (or KdV-type) equation is an important physical question, for which virial operators and their spectral properties are the essential elements of the analysis. In this paper we investigate this problem analytically and reduce it to verifying numerically only some signs of inner products and certain eigenvalues.

In many operational contexts, solutions to NP-hard combinatorial optimization problems, modeled by means of Mixed-Integer Linear Programming (MILP), may become infeasible due to unpredictable disruptions. Typically, reoptimizing by solving the MILP formulation on the perturbed instance is not possible as new solutions must be obtained in a very short computing time, while simple repairing heuristics may result in low-quality solutions. To bridge this gap, we propose a learning-to-reoptimize framework, and apply it to the Lot Sizing Problem (LSP) under machine breakdown disruptions. We design a fix-and-optimize strategy aided by a Graph Neural Network (GNN) that efficiently computes a new solution within the neighborhood of a repaired solution. By representing the instance, the original solution and the disruption as a feature graph, we train a GNN to predict the likelihood that specific binary variables require to be modified. These predictions guide the selection of a small subset of variables to be reoptimized by an MILP solver, while the other variables are hard-fixed. Numerical experiments on a large dataset demonstrate that our approach handles effectively different problem sizes, and that it significantly outperforms a baseline alternative approach, yielding larger cost reductions within the same limited time budget.

We prove a mixed-characteristic analogue of Kunz's theorem in terms of perfectoid towers: a Noetherian local ring of residue characteristic $p$ is regular if and only if it admits a flat map to a Noetherian ring that extends to a perfectoid tower. This result is deduced from another mixed-characteristic analogue due to O. Gabber and J. Lurie. We also characterize regularity for perfectoid towers via vanishing of single higher $\mathrm{Tor}$-module of the residue field with a perfectoid algebra.

We characterize positivity preserving maps $T: B(\mathcal{H})_h \otimes \mathbb{R}[x_1, \dots, x_n] \to B(\mathcal{H})_h \otimes \mathbb{R}[x_1, \dots, x_n]$ on $\mathbb{R}^n$ and on compact sets $K \subseteq \mathbb{R}^n$. This also characterizes local operator moment sequences and general operator moment sequences via positivity preserving maps.

A spectral collocation operator with the summation-by-parts property was introduced by Chan to develop entropy-stable discontinuous Galerkin (DG) semi-discretizations (https://doi.org/10.1016/j.jcp.2018.02.033). The present work shows that semi-discretizations based on this collocation operator produce solutions that are equivalent to solutions of a DG semi-discretization using the same underlying quadrature. The equivalence holds regardless of the number of degrees of freedom in the collocation scheme and when the quadrature is not strictly positive. Extraneous degrees of freedom in the collocation scheme are associated with the nullspace of the operator and remain zero throughout an unsteady simulation. If necessary, nullspace consistency can be recovered by introducing projection-based numerical dissipation that targets only the extraneous modes. The equivalence between collocation and DG solutions is verified for the constant-coefficient advection equation and Burgers' equation on triangular meshes. The numerical results show that equivalence breaks down for entropy-stable semi-discretizations of Burgers' equation based on a skew-symmetric splitting, but that equivalence can be recovered by projecting the collocation scheme's residual onto the relevant polynomial space. In addition to investigating equivalence, the results demonstrate that the collocation operator produces semi-discretizations with favorable spectral radii compared with a commonly used summation-by-parts operator construction.

Probabilistic smoothing is a standard tool for global optimization, but existing methods rely on Gaussian kernels and specific transforms, often resulting in strong hyperparameter sensitivity and limited robustness. We propose a general smoothing framework that combines flexible symmetric unimodal kernels with monotonic ratio-based transformations. Under mild conditions, we show that the smoothed objective preserves the global maximizer and that all stationary points concentrate near the true optimum for sufficiently large amplification, without requiring a decreasing smoothing schedule. We further provide explicit complexity bounds for stochastic gradient ascent and show that a leave-one-out baseline provably reduces variance. Experiments on high-dimensional benchmarks and black-box adversarial attacks demonstrate improved robustness and competitive performance.

The automorphism group of a code is the group of permutations that map a code to itself. Berman codes are a class of binary linear codes characterized by two integer parameters $n\geq 2$ and $m\geq 1$, and this class includes the Reed-Muller codes as well. The class of Berman codes and their duals were recently shown to achieve the capacity of the binary erasure channel. A number of abelian codes that arise from the intersection and subspace sums of Berman and Dual Berman codes were also identified recently, for odd $n\geq 3$. A subclass of these abelian codes was shown to have good short block-length performance for AWGN channels, with efficient decoding algorithms. In this work, we identify the exact automorphism group for Berman codes and their duals. Further, we find the exact automorphism group for the above mentioned abelian codes, when $n\geq 5$. In the case of such abelian codes with $n=3$, we present partial characterizations of the automorphism groups for a large collection of parameter choices, and complete characterizations for a few.

Partial differential equations (PDEs) on surfaces are fundamental to scientific computing and geometry processing. A popular approach to solving PDEs on surfaces is the finite element method (FEM), where the surface is divided into discrete geometric elements (usually triangles). Recently, physics-informed neural networks (PINNs) have emerged as a continuous, mesh-free alternative that does not suffer from FEM's sensitivity to mesh quality or geometric discretization errors. We present PINNSur, a simple framework for using PINNs on curved surfaces: we train a neural field to approximate the surface's normals, and then we express surface differential operators using their projection from $\mathbb{R}^3$ onto the surface. Since every orientable manifold has well-defined normals, our method is suitable for all such surfaces, regardless of curvature or topology, enabling many geometry processing applications. Moreover, despite their empirical success in solving PDEs in flat Euclidean domains, PINNs lack convergence guarantees to the true solution of the underlying PDE, and there is limited systematic experimental evidence demonstrating such convergence. This gap restricts their adoption as reliable solvers compared to established methods like FEM, where convergence to the true solution is well understood and theoretically grounded. These surface PDEs are particularly challenging to solve convergently, as one must not only deal with the convergence of the function approximation, but also with the convergence of the geometric approximation of the surface itself. In this work, we empirically investigate the convergence behavior of PINNs for solving surface PDEs by introducing a simple empirical convergence test.

Lie algebra $\mathfrak{sl}(2)$ can be realised by vector fields on $\mathbb{R}^1\ni x$ with polynomial coefficients $1$, $-2x$, $-x^2$; their Wronskian determinants yield the Lie bracket. Likewise, the monomials $1$, $\ldots$, $x^k/k!$, $\ldots$, $x^N/N!$ span finite-dimensional strong homotopy (SH) Lie algebras with the Wronskians $\mathbf{1} \wedge \partial_x \wedge \ldots \wedge \partial_x^{N-1}$ as the $N$-ary brackets. Over dimension $d=2$ with $\mathbb{R}^2\ni(x,y)$ and for the generalised complete Wronskian $W^{d=2}_{k=1}=\mathbf{1}\wedge \partial_x \wedge \partial_y$ of differential order $k=1$ as the ternary bracket, the finite-dimensional polynomial SH-Lie algebras are spanned by $\langle 1$, $x$, $y$, $p\rangle$ with $p\in\{x^2$, $xy$, $y^2\}$. We explicitly describe all finite-dimensional polynomial SH-Lie algebras $\Bbbk_k[{\boldsymbol{x}}]\subseteq \mathcal{A} \subseteq \Bbbk[x^1,\ldots,x^d]$ (over $\Bbbk=\mathbb{R}$ or $\mathbb{C}$) with the complete generalised Wronskians $W^{d\geqslant 1}_{k\geqslant 1}$ of order $k$ as $N$-ary bracket: $N=\binom{d+k}{d}$. We obtain a factorisation formula for the generalised Vandermonde determinants which show up in the structure constants of the polynomial algebras $\mathcal{A}$.

Sequence models, and particularly Linear Recurrent Neural Networks (LRNNs) of the form $\mathbf{h}_{k+1} = \mathbf{W} \mathbf{h}_{k} + \mathbf{y}_k + \mathbf{b}$, are widely applicable in time-series analysis for dynamical systems, yet, as black-box algorithms, much is unknown about why they perform well. In this work, we leverage Takens' embedding theorem, which provides conditions under which partially observed time series organized into delay-coordinate vectors can faithfully represent the original system's dynamics, as a theoretical framework for explaining how and why sequence models preserve and reconstruct dynamical systems. For LRNNs, concatenating output states into delay-coordinate vectors gives rise to a ``delay" matrix $\mathbb{M}_{n,m}\in \mathbb{C}^{(nm) \times (n+1)m}$: a block matrix consisting of identity matrices $\mathbf{I} \in \mathbb{R}^{m \times m}$ repeated $n$ times along the main diagonal and weight matrices $\mathbf{W} \in \mathbb{C}^{m \times m}$ featured $n$ times along the super-diagonal. $\mathbb{M}_{n,m}$ relates the delay-coordinates of the input time series to those of the LRNN output states, and, for $\mathbb{M}_{n,m}$ to be an embedding, it must be full row-rank. We provide explicit conditions for $\mathbb{M}_{n,m}$ to be full row-rank and prove the condition number of $\mathbb{M}_{n,m}$ and determinant of $\mathbb{M}_{n,m} \mathbb{M}_{n,m}^*$--measures of embedding stability--are bounded independent of $n$, at least for certain ranges of $\mathbf{W}$'s singular values: namely, when $σ_{\max}(\mathbf{W}) \le 1$. This result explains why the spectrum of $\mathbf{W}$ for trained LRNNs tends to converge to within the unit circle.

We prove that every perfectoid tower can be decomposed into a fiber product of perfectoid towers that are either $p$-torsion free or perfect of characteristic $p$. As an application, we show that separated perfectoid towers are reduced. We also establish the tilting invariance of étale cohomology and Koszul homology for perfectoid towers.

The first part of this article is devoted to characterizing the cocycles $α$ of a finite group $G$ that give rise to faithful projective representations of $G$. We prove that a $p$-group $G$ admits a faithful irreducible projective representation if and only if the cohomology class $[α]$ does not lie in the image of the inflation map $\operatorname{inf}: \mathrm{H}^2\!\left(G / N, \mathbb{C}^{\times}\right) \longrightarrow \mathrm{H}^2\!\left(G, \mathbb{C}^{\times}\right)$ for any non-trivial central subgroup $N$ of $G$. In the case where $[α] \in \operatorname{Im}(\operatorname{inf})$, we determine a criterion such that a direct sum of irreducible $α$-representations is faithful. We conclude this part by describing the behaviour of cocycles $α$ that yield faithful irreducible representations for direct products of groups. In the second part, we introduce the notion of the projective embedding degree of a finite group $G$, defined as the smallest integer $n$ such that $G$ embeds into $\mathrm{PGL}_n(\mathbb{C})$; equivalently, it is the smallest $n$ such that $G$ has a faithful complex projective representation of degree $n$. We also define the analogous notion of the irreducible projective embedding degree of $G$. These invariants have been investigated for several classes of groups, including direct products of groups, finite abelian groups, extra-special $p$-groups, Heisenberg groups, and groups of order $p^3$, $p^4$ (for primes $p$), and $p^5$ (for $p \geq 5$).

We characterize perfectoid towers in terms of conormal cones rather than torsion parts. This result is deduced from a refined study of the relationship between torsion with respect to a principal ideal and the associated conormal cone, building on the work of O. Gabber and L. Ramero.

Let $T_R(M)$ be a tensor ring, where $R$ is a ring and $M$ is an $N$-nilpotent $R$-bimodule. Under certain conditions, we characterize the Gorenstein flat-cotorsion modules over $T_R(M)$, showing that a $T_R(M)$-module $(X, u)$ is Gorenstein flat-cotorsion if and only if $u$ is monomorphic and $\Coker u$ is a Gorenstein flat-cotorsion $R$-module. As applications, we describe the Gorenstein flat-cotorsion modules over some trivial extension rings and Morita context rings.

In this article we provide a classification and description of compact Riemann surfaces admitting a triangular action of a group of order $2p^2,$ where $p$ is an odd prime number. We obtain that all such Riemann surfaces are isomorphic to curves defined over the rational numbers. As a by-product, we derive a classification of orientably-regular hypermaps whose orientation-preserving automorphism group has order $2p^2.$

We consider three categories arising from the higher Auslander algebras of type $A$ in relation to $d$-dimensional cluster combinatorics: $d$-exact subcategory of the module category of $A^d_{n+1}$ generated by the $d$-cluster-tilting object, the $(d+2)$-angulated cluster category, and the $d$-almost positive subcategory of the derived category (the higher analogue of the category of two-term complexes of projectives). We show that the third one, introduced by the second-named author, is the $d$-exangulated quotient of the other two, introduced by Oppermann and Thomas, by the ideals generated by morphisms factoring through morphisms from injective to projective objects, thus providing an algebraic connection between the two models of Oppermann-Thomas. This is a $d$-exangulated version in type $A$ of a result of Brüstle and Yang and its interpretation by the first-named author together with Fang, Palu, Plamondon and Pressland. It also explains a well-known coincidence between the number of 2-term silting complexes in type $A_{n}$ and of tilting modules in type $A_{n+1}$ from the $0$-Auslander perspective. We expect this to serve as a prototypical example of the $0$-Auslander correspondence in higher homological algebra.

We present a fully extrinsic, parametrization-free variant of tensor calculus on embedded, possibly evolving, submanifolds with boundary in arbitrary dimension and codimension. The proposed approach is component-free and, for general rank tensors, covers fundamental concepts such as tangential projection, extrinsic and covariant derivatives, the extrinsic Stokes' formula, and the Laplace-Beltrami operator. The distinctive features of the developed framework are its algorithmic recursivity and the transparency provided by the special row representation of tensors reminiscent of the recursive data structure of complete trees. Consequently, the suggested tensor calculus is amenable to computations and theoretical analysis, and the latter is demonstrated for general dimension and codimension through three standalone applications. First, we derive a new extrinsic conservation law, namely the principle of vanishing extrinsic momentum, for incompressible Euler flows on Riemannian manifolds. Second, we revisit the concept of Cauchy stress on embedded submanifolds with positive codimension and argue that the conservation of angular momentum implies tangentiality and symmetry of the stress tensors only when they are restricted to act on tangential orientations. Third, for evolving submanifolds, we introduce the material derivative of tensor fields of general rank in an extrinsic manner and derive an expression for the rate of change of the associated tensorial Dirichlet energy. The paper provides a practical notation and tools that are immediately usable in mathematical modeling, analysis of geometry-aware PDEs and in numerical methods on embedded submanifolds.

We consider predictive density estimation under logarithmic score for $d$-dimensional infinitely divisible location models. Taking the formal Bayes predictive density under the Lebesgue prior as a benchmark, we study the Kullback-Leibler regret of competing Bayes predictive densities. Our main contribution is an exact entropy-energy identity: the integrated regret of a Bayes predictive density $\hat{p}^π$ under prior $π$ relative to the benchmark admits an exact representation as the Dirichlet-form energy of the square-rooted marginal distribution $\sqrt{M^π}$ for the symmetric Markov semigroup induced by the benchmark kernel. This converts regret comparisons into a potential-theoretic problem and yields a sharp recurrence/transience characterization of when the benchmark predictive density can or cannot be uniformly improved. We introduce an $\mathcal{A}$-harmonic class of improper priors -- defined through the generator $\mathcal{A}$ of the induced process -- and give explicit tail conditions -- an integral test on the induced marginal, equivalent to power-law prior decay in heavy-tailed models -- that guarantee admissibility of the resulting Bayes predictive density. We illustrate the theory with new results for several distributions.

This position statement looks back on two decades of work on shallow embeddings of non-classical logics in classical higher-order logic (HOL), a line of research that expanded into a range of logic embeddings in HOL and inspired the LogiKEy logic-pluralistic knowledge representation and reasoning methodology. This paper advances the case for logical pluralism at object-logic level within a unifying meta-logical framework such as LogiKEy, grounding the argument in computational metaphysics. More broadly, it advocates principled support for logical pluralism in modern proof assistants, and cautions against logical imperialism -- the rigid adoption of a single foundational logic for large-scale theory developments -- which impedes the interdisciplinary reuse that LogiKEy is designed to enable.

We investigate a new notion of regularity for tensor triangulated categories, called residual regularity. We show that residual regularity descends and ascends via finite separable extensions and we classify all finite groups whose derived category of permutation modules is residually regular.

We prove the finite abelian two-generator conjecture of Darijani--Miraftab--Witte Morris: every directed Cayley digraph on a finite abelian group with two distinct nonzero generators has two arc-disjoint Hamiltonian paths. The proof uses a cut-reflection theorem for Hamiltonian cut values in the family Cay(Z_k; a, a+1): if Z is the set of such values and N=k-1, then, with N-Z={N-z : z in Z}, dist(Z,N-Z)<=1. The proof uses sector-filling inequalities for primitive-ray multiplicities and an extremal graph recording pairs at minimal reflected distance. The estimate is sharp modulo parity: exact reflection occurs for odd k, while distance one occurs for even k. The second remaining cyclic family, Cay(Z_k; -a, a+1), is treated by an explicit quotient--fiber construction. We also prove the remaining three-factor case for Cartesian products of directed cycles. Together with the two-factor and at-least-four-factor theorems of Darijani--Miraftab--Witte Morris, this resolves their directed-cycle product conjecture for all numbers of factors.

Let $d\geq 2$ and $k\geq 1$ be fixed. We prove that, for every $ε>0$ and every real $β$, there exist integers $1\leq b_1,\ldots,b_k\leq N$ such that \[ \left\|\sum_{j=1}^k b_j^{1/d}-β\right\| \ll_{d,k,ε} N^{-k/d+ε}. \] The proof combines Schmidt's Subspace Theorem with an explicit inhomogeneous transference argument. This improves Iyer's (2025) higher-root exponent $(k-d+1)/d^2$, and also the analogous $d$-ary full-basis exponent away from the cases where $k+1$ is a power of $d$, at the cost of ineffectivity. We also record a conjectural uniform exponent $k-1/d$. In the square-root case $d=2$, we give explicit integer-target constructions for $k=2,3,4$ attaining this conjectural value.

We establish that semidefinite programs (SDPs) over the unbounded positive semidefinite cone are mathematically equivalent to thermodynamic systems of independent bosonic modes: the eigenvalues of the optimization variable play the role of expected occupation numbers, the linear objective plays the role of total expected energy, and the linear equality constraints play the role of conserved non-commuting charges. Building on this perspective, we recast general SDPs as bosonic free-energy minimization problems at strictly positive temperature, regularized by the Bose-Einstein entropy; the original SDP is recovered in the zero-temperature limit. The optimal primal variable takes the form of a Bose-Einstein thermal operator parametrized by the dual variables. We prove an approximation-error bound that depends on the ground-space degeneracy and the spectral gap of the dual slack operator, improving on the linear-in-dimension worst-case duality gap of interior-point methods. We also introduce the Bose-Einstein quantum relative entropy as a Bregman divergence on the unbounded positive semidefinite cone, generated by the negative Bose-Einstein entropy. We propose it as a natural divergence for unnormalized positive operators, for which the standard Umegaki relative entropy can become negative, and we show that it satisfies a restricted monotonicity property under affine maps modeling bosonic Gaussian channels. Finally, we develop hybrid quantum-classical algorithms for the regularized SDP using only Hamiltonian simulation, Hadamard tests, and classical sampling, and bound their runtime in closed form. Unlike existing quantum SDP solvers, whose runtimes scale polynomially with an a priori upper bound on the primal trace, our framework operates directly on the unbounded cone, replacing this bound with a dependence on the spectral structure of the dual slack operator.

Problem 8.1 in Astaiza et. al. asks about the relationship between the cycle decomposition of a permutation $σ$ and that of its symmetric tensor power $σ^{\odot k}$. In this paper, we investigate this question and give formulas for computing the number of fixed points and, in the case of a permutation containing at most one cycle of length greater than one, the number of $s$-cycles.

We define metrics in space that are natural counterparts of the hyperbolic metric in plane domains, using the characterization of the hyperbolic metric due to Beardon and Pommerenke. We obtain inequalities for these metrics under quasiconformal homeomorphisms between domains in space that have at least two boundary points. We discuss the failure of the existence of such estimates for non-homeomorphic quasiregular mappings.

In Diaconis and Saloff-Coste (1996), the authors introduced the simple ``transvection" walk on $\mathrm{GL}_n(\mathbb F_2)$: at each step, choose two distinct rows and add one to the other. In Ben-Hamou (2025), the author recently proved that this walk has mixing time $O(n^2\log n)$. Inspired by applications in cryptography (see Sotiraki (2016)), Ben-Hamou and Peres (2018) conjectured that the first $k$ columns of this walk mixed in $O(nk \log(n))$ steps. Our main result is a proof of this conjecture uniformly in $n$ and $k.$ Our proof is based on a local-to-global entropy estimate, in the spirit of block factorization results such as Caputo et al (2015), Caputo et al (2021). In our setting, the kernels that correspond roughly to the block kernels of Caputo et al (2021) do not have uniformly large log-Sobolev constants, and so naively applying these techniques does not improve over Ben-Hamou (2025). We avoid these bad blocks by combining our entropy estimates with a burn-in argument similar to classical drift-and-minorization arguments of Rosenthal (1995). This method may be of broader interest, and so we illustrate it by proving an analogous result for a family of product-replacement algorithms on the Heisenberg group.

In this paper, we study the behavior of Ekedahl-Oort strata under natural embeddings between the good reductions modulo $p$ of GSpin Shimura varieties and Rapoport-Smithling-Zhang unitary Shimura varieties, a prototypical setting for the construction of special cycles in the Kudla program. In each case, we determine the EO stratum containing the image of a given EO stratum under the embedding. We also compute discrete invariants of these Shimura varieties, including their $p$-ranks and $a$-numbers; in the GSpin case, these are obtained via the Kuga-Satake embedding.

The relative entropy between two states is a key concept in quantum information theory and quantum field theory. In the setting of quantum field theory, its computation requires the handling of relative modular Hamiltonians, which are typically very difficult to compute explicitly. In this paper, we exploit locality properties of general algebraic QFTs to estimate relative modular Hamiltonians between two states, $ω$ and $\tildeω$, and hence also their relative entropy, in terms of the modular Hamiltonian of a reference state $\hatω$, which might be better understood. For suitable pairs of states we can estimate the relative modular Hamiltonian for the algebra of a region $V_2$ from above, resp. from below, in terms of the modular Hamiltonian of $\hatω$ on a larger region $V_3$, resp. a smaller region $V_1$. Pairs of states and choices of regions which are susceptible to our scheme are related to the presence of superluminal signalling in the sense of Sorkin's paradox. If $ω=\hatω$, then there exists a unitary that maps $ω$ to $\tildeω$ on $V_3$ and that does not allow superluminal signalling from the spacelike complement $V_3'$ to $V_2$, resp. from $V_1$ to $V_2'$, if our upper, resp. lower, bound applies. To investigate the strength of our estimates we consider coherent states for CCR systems, focussing particularly on free scalar fields. Our estimates apply even if the relative modular Hamiltonian cannot be computed exactly. For sufficiently regular excitations we recover an exact result by squeezing. Our method thus yields an independent proof for the relative entropy formula in cases where the relative modular Hamiltonian cannot be computed exactly. For massless fields we establish the analogous result also for double cone regions. These results indicate that our estimates do not lose too much information.

In this paper, we introduce and study shifted twisted quantum affine algebras which provide a twisted counterpart of the theory of shifted quantum affine algebras. The shifted twisted quantum affine algebra $\U_q^{μ_+,μ_-}(\hgs)$ is obtained from the Drinfeld current presentation of twisted quantum loop algebras by shifting the Cartan--Drinfeld currents $ϕ_i^\pm(z)$ according to a coweight pair $(μ_+,μ_-)$. We prove that it admits a triangular decomposition and that, up to isomorphism, they depend only on the total shift $μ=μ_+ + μ_-$. For each shift $μ$, we define a category $\mathcal O_μ$ of representations of $\U_q^μ(\hgs) = \U_q^{0,μ}(\hgs)$ and prove a rationality theorem for the Cartan currents: on every weight space, the two currents $ϕ_i^+(z)$ and $ϕ_i^-(z)$ are expansions of the same rational operator-valued function, whose degree is prescribed by $α_i(μ)$. As a consequence, we classify the simple objects of $\mathcal O_μ$ by rational $\ell$-weights of the corresponding degrees. We then construct a deformed Drinfeld coproduct and use it to define a fusion product on the direct sum $\mathcal{O}^{sh}$ of the categories $\mathcal O_μ$. This fusion product is compatible with $q$-characters. We also classify finite-dimensional simple modules in $\mathcal{O}^{sh}$ in terms of dominant rational $\ell$-weights, with a separate treatment of type $A_{2n}^{(2)}$. Finally, we construct restriction representations relating representations of twisted quantum affine Borel algebras to representations of shifted twisted quantum affine algebras, and establish a $q$-characters formula for simple finite-dimensional representations of shifted twisted quantum affine algebras in terms of the $q$-characters of the corresponding simple representations of the twisted quantum affine Borel algebra $\U_q(\bs)$.

We study order parameters in one-dimensional quantum lattice models with finite invertible or non-invertible symmetry. We investigate what properties a string operator must satisfy in order to acquire a non-vanishing expectation value in a given gapped phase. We deduce that multiplets of string order parameters organise into a Lagrangian algebra in the Drinfel'd centre of the symmetry category. In particular, we highlight the role of the multiplication rule as governing the fusion of the twisted sector local operators that constitute the string operator in the infrared limit. Our derivations exploit the tensor network approach to the classification of gapped phases and its reformulation in terms of module categories over the symmetry category. Within this framework, a gapped phase is associated with a pattern of spontaneous symmetry breaking wherein a Morita class of algebras of topological lines is preserved in the ground state subspace. The crux of the proof is to show that the expectation value of any string operator explicitly depends on the tube algebra module associated with the Lagrangian algebra, which is realised as the full centre of the corresponding module category. Finally, we demonstrate that these techniques extend to phases of symmetric mixed states.