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A two reference determinant version of the distinguishable cluster with singles and doubles (DCSD) has been developed. We have implemented the two determinant distinguishable cluster (2D-DCSD) and the corresponding traditional 2D-CCSD method in a new open-source package written in Julia called ElemCo.jl. The methods were benchmarked on singlet and triplet excited states of valence and Rydberg character, as well as for singlet–triplet gaps of diradicals. It is demonstrated that the distinguishable cluster approximation improves the accuracy of 2D-CCSD. [ABSTRACT FROM AUTHOR]

We introduce a generalization of the σ-SCF method to approximate noncollinear spin ground and excited single-reference electronic states by minimizing the Hamiltonian variance. The new method is based on the σ-SCF method, originally proposed by Ye et al. [J. Chem. Phys. 147, 214104 (2017)], and provides a prescription to determine ground and excited noncollinear spin states on an equal footing. Our implementation was carried out utilizing an initial simulated annealing stage followed by a mean-field iterative self-consistent approach to simplify the cumbersome search introduced by generalizing the spin degrees of freedom. The simulated annealing stage ensures a broad exploration of the Hilbert space spanned by the generalized spin single-reference states with random complex element-wise rotations of the generalized density matrix elements in the simulated annealing stage. The mean-field iterative self-consistent stage employs an effective Fockian derived from the variance, which is utilized to converge tightly to the solutions. This process helps us to easily find complex spin structures, avoiding manipulating the initial guess. As proof-of-concept tests, we present results for Hn (n = 3–7) planar rings and polyhedral clusters with geometrical spin frustration. We show that most of these systems have noncollinear spin excited states that can be interpreted in terms of geometric spin frustration. These states are not directly targeted by energy minimization methods, which are meant to converge to the ground state. This stresses the capability of the σ-SCF methodology to find approximate noncollinear spin structures as mean-field excited states. [ABSTRACT FROM AUTHOR]

We consider the mutation invariants of cluster algebras of rank 2. We characterize the mutation invariants of finite type. Two examples are provided for the affine type and we prove the non-existence of Laurent mutation invariants of non-affine type. As an application, a class of Diophantine equations encoded with cluster algebras are studied. [ABSTRACT FROM AUTHOR]

For a rooted cluster algebra (Q) over a valued quiver Q , a symmetric cluster variable is any cluster variable belonging to a cluster associated with a quiver σ (Q) , for some permutation σ. The subalgebra of (Q) generated by all symmetric cluster variables, is called the symmetric mutation subalgebra and is denoted by ℬ (Q). In this paper, we identify the class of cluster algebras that satisfy ℬ (Q) = (Q) , which contains almost every quiver of finite mutation type. In the process of proving the main result, we provide a classification of quivers mutations classes that relates their maximum weights to the shapes of the initial quivers. Furthermore, some properties of symmetric mutation subalgebras are given. [ABSTRACT FROM AUTHOR]

Prior clustering approaches of soccer players have employed a variety of methods based on various data categories, but none of them have focused on clustering by career paths characterized through a time series analysis of yearly performance quality. Therefore, this study aims to propose a methodology how a career path can be represented as a time series of a player's seasonal qualities and then be clustered with players that have a similar career path. The underlying data focuses on soccer players from the five largest European soccer nations (Big-5). This allows for the identification of different types of career paths of players and the investigation of significant disparities between career paths among the Big-5 nations. In line with our proposed methodological approach, we identified and interpreted 13 different clusters of player career paths. These range from the cluster with the highest player quality scores to the pattern comprising players with the weakest scores. Further, the detected clusters show significant differences regarding variables of soccer players' early career phase in adolescence (e.g., age of debut in professional soccer, years spent in a youth academy). The presented approach might represent a first step for stakeholders in soccer to get an objective insight in players' career by utilizing mainly freely available data sources. [ABSTRACT FROM AUTHOR]

Denote by N ⊂ SLk the subgroup of unipotent upper triangular matrices. In this paper, we show that the dual canonical basis of ℂ[N] (and base affine spaces) can be parametrized by semi-standard Young tableaux. Moreover, we give an explicit formula for every element in the dual canonical basis using the data of the corresponding semi-standard Young tableau. We apply our results to study cluster variables in ℂ[N]. [ABSTRACT FROM AUTHOR]

We study the T -system of type A ∞ , also known as the octahedron recurrence/equation, viewed as a 2 + 1 -dimensional discrete evolution equation. Generalizing earlier work on arctic curves for the Aztec Diamond obtained from solutions of the octahedron recurrence with 'flat' initial data, we consider initial data along parallel 'slanted' planes perpendicular to an arbitrary admissible direction (r , s , t) ∈ Z + 3 . The corresponding solutions of the T -system are interpreted as partition functions of dimer models on some suitable 'pinecone' graphs introduced by Bousquet–Mélou, Propp, and West in 2009. The T -system formulation and some exact solutions in uniform or periodic cases allow us to explore the thermodynamic limit of the corresponding dimer models and to derive exact arctic curves separating the various phases of the system. This direct approach bypasses the standard general theory of dimers using the Kasteleyn matrix approach and uses instead the theory of Analytic Combinatorics in Several Variables, by focusing on a linear system obeyed by the dimer density generating function. [ABSTRACT FROM AUTHOR]

We construct a new solution to the tetrahedron equation and the three-dimensional (3D) reflection equation by extending the quantum cluster algebra approach by Sun and Yagi concerning the former. We consider the Fock–Goncharov quivers associated with the longest elements of the Weyl groups of type |$A$| and |$C$|⁠ , and investigate the cluster transformations corresponding to changing a reduced expression into a "most distant" one. By devising a new realization of the quantum |$y$| -variables in terms of |$q$| -Weyl algebra, the solutions are extracted as the operators whose adjoint actions yield the cluster transformations of the quantum |$y$| -variables. Explicit formulas of their matrix elements are also derived for some typical representations. [ABSTRACT FROM AUTHOR]

Cluster algebras, introduced by Fomin and Zelevinsky, are connected to dual canonical bases. Geometric realization is an effective way to solve algebraic problems. In 2002, Fomin and Zelevinsky proposed geometric realizations for root systems and finite-type cluster algebras. This study examines the regulation of mutation in geometric realization. We establish a fascinating link between centrally symmetric triangulations and quivers in classical cluster algebras, showing that these triangulations and quivers are interchangeable under mutation. This connection leads to a significant finding: any cluster variable of a given cluster algebra can induce a connected subquiver within the corresponding initial quiver. Furthermore, we demonstrate that any two quivers generated by cluster variables intersect, emphasizing their connectedness. To illustrate the practical implications of this geometric realization, we present a convenient method to compute seed mutations of type Dn and define a category based on centrally symmetric triangulations. This category’s objects are ℤ-linear combinations of positive roots of the root system of type Dn, and we propose an equivalent condition for morphisms. This condition is a generalization of the case of type An by [P. Caldero, F. Chapoton and R. Schiffler, Quivers with relations arising from clusters (An case), Trans. Amer. Math. Soc. 358(3) (2006) 1347–1364]. Finally, we propose a geometric realization for cluster algebras: weighted triangulations. This approach does not require cluster algebras to be of finite type and can verify whether a matrix from a cluster algebra is equivalent to a subdiagonal matrix. [ABSTRACT FROM AUTHOR]

Advanced footwear technology featuring stack heights higher than 30 mm has been proven to improve running economy in elite and recreational runners. While it is understood that the physiological benefit is highly individual, the individual biomechanical response to different stack heights remains unclear. Thirty-one runners performed running trials with three different shoe conditions of 25 mm, 35 mm, and 45 mm stack height on an outdoor running course wearing a STRYD sensor. The STRYD running variables for each participant were normalized to the 25 mm shoe condition and used to cluster participants into three distinct groups. Each cluster showed unique running patterns, with leg spring stiffness and vertical oscillation contributing most to the variance. No significant differences were found between clusters in terms of body height, body weight, leg length, and running speed. This study indicates that runners change running patterns individually when running with footwear featuring different stack heights. Clustering these patterns can help understand subgroups of runners and potentially support running shoe recommendations. [ABSTRACT FROM AUTHOR]

The functional integral formulation of the Hubbard model when treated in its Kotliar‐Ruckenstein representation in the radial gauge involves fermionic, as well as complex and radial slave boson fields. In order to improve on the understanding of the interplay of the three types of fields, and on the nature of the latter, a comprehensive investigation of an exactly solvable two‐site cluster is performed, as it entails all pitfalls embodied in this approach. It is first shown that the exact partition function is recovered, even when incorporating in the calculation the square root factors that are at the heart of the representation, when suitably regularized. It is shown that using radial slave boson fields allows to overcome all hurdles following from the normal ordering procedure. It is then demonstrated that this applies to the Green's function as well, and to the correlation functions of physical interest, thereby answering the criticisms raised by Schönhammer [K. Schönhammer, Phys. Rev. B 199042, 2591]. In addition, the investigation generalizes the calculations to the Hubbard model extended by a non‐local Coulomb interaction. [ABSTRACT FROM AUTHOR]

We prove, under an assumption on the critical points of a real‐valued function, that the symmetric Ising perceptron exhibits the 'frozen 1‐RSB' structure conjectured by Krauth and Mézard in the physics literature; that is, typical solutions of the model lie in clusters of vanishing entropy density. Moreover, we prove this in a very strong form conjectured by Huang, Wong, and Kabashima: a typical solution of the model is isolated with high probability and the Hamming distance to all other solutions is linear in the dimension. The frozen 1‐RSB scenario is part of a recent and intriguing explanation of the performance of learning algorithms by Baldassi, Ingrosso, Lucibello, Saglietti, and Zecchina. We prove this structural result by comparing the symmetric Ising perceptron model to a planted model and proving a comparison result between the two models. Our main technical tool towards this comparison is an inductive argument for the concentration of the logarithm of number of solutions in the model. [ABSTRACT FROM AUTHOR]

We define the cluster characters for 2-Calabi-Yau Frobenius extriangulated categories with cluster tilting objects. This provides a unified framework of cluster characters in 2-Calabi-Yau triangulated categories and 2-Calabi-Yau Frobenius exact categories given by Palu and Fu–Keller, respectively. [ABSTRACT FROM AUTHOR]

We generalize the positivity conjecture on (Kauffman bracket) skein algebras to Roger–Yang skein algebras. To generalize it, we use Chebyshev polynomials of the first kind to give candidates of positive bases. Moreover, the polynomials form a lower bound in the sense of [9] and [10]. We also discuss a relation between the polynomials and the centers of Roger–Yang skein algebras when the quantum parameter is a complex root of unity. [ABSTRACT FROM AUTHOR]

Weakly Supervised Object Localization (WSOL) aims to utilize the features learned by a classifier on the image-level labels to locate target objects. However, these existing channel selection methods for WSOL still cannot effectively select the important channels and remove the unimportant ones. To address this issue, we propose a Clustering-inspired Channel Selection method based on Class Activation Maps (CCS-CAM). Compared with the traditional methods, the advantage of CCS-CAM is that it is very simple yet effective for channel selection due to the K-means clustering based on Class Activation Maps. It can effectively ensure both object localization and classification accuracy. The effectiveness of the proposed CCS-CAM method has been demonstrated using multiple public datasets, with GT-Know Loc reaching 87.9% and 63.71% on the CUB200-2011 and ImageNet-1k respectively, which is superior to the other state-of-the-art methods. • WSOL network extends image classification for precise object location. • CCS module filters channels, using K-means for effective object localization. • Tested on CUB200-2011, ImageNet-1K, FGVC-Aircraft, Stanford Cars with success. [ABSTRACT FROM AUTHOR]

We study quivers supporting twisted graded Calabi–Yau algebras. Specifically, we classify quivers on four vertices in which the Nakayama automorphism acts on the degree zero part by either a four-cycle, a three-cycle, or two two-cycles. In order to realize algebras associated to some of these quivers, we show that a graded twist of a twisted graded Calabi–Yau algebra is another algebra of the same type. [ABSTRACT FROM AUTHOR]

Let A be a finite-dimensional algebra, and M be a d -cluster tilting subcategory of mod A. From the viewpoint of higher homological algebra, a natural question to ask is when M induces a d -cluster tilting subcategory in Mod A. In this paper, we investigate this question in a more general form. We consider M as an essentially small d -abelian category, known to be equivalent to a d -cluster tilting subcategory of an abelian category A. The completion of M , denoted by Ind (M) , is defined as the universal completion of M with respect to filtered colimits. We explore Ind (M) and demonstrate its equivalence to the full subcategory L d (M) of Mod M , comprising left d -exact functors. Notably, Ind (M) as a subcategory of Mod M Eff (M) falls short of being a d -cluster tilting subcategory since it satisfies all properties of a d -cluster tilting subcategory except d -rigidity. For a d -cluster tilting subcategory M of mod A , M → consists of all filtered colimits of objects from M , is a generating-cogenerating, functorially finite subcategory of Mod A. The question of whether M → is a d -rigid subcategory remains unanswered. However, if it is indeed d -rigid, it qualifies as a d -cluster tilting subcategory. In the case d = 2 , employing cotorsion theory, we establish that M → is a 2-cluster tilting subcategory if and only if M is of finite type. Thus, the question regarding whether M → is a d -cluster tilting subcategory of Mod A appears to be equivalent to Iyama's question about the finiteness of M. Furthermore, for general d , we address the problem and present several equivalent conditions for Iyama's question. [ABSTRACT FROM AUTHOR]

Evolutionary game theory, encompassing discrete, continuous, and mixed strategies, is pivotal for understanding cooperation dynamics. Discrete strategies involve deterministic actions with a fixed probability of one, whereas continuous strategies employ intermediate probabilities to convey the extent of cooperation and emphasize expected payoffs. Mixed strategies, though akin to continuous ones, calculate immediate payoffs based on the action chosen at a given moment within intermediate probabilities. Although previous research has highlighted the distinct impacts of these strategic approaches on fostering cooperation, the reasons behind the differing levels of cooperation among these approaches have remained somewhat unclear. This study explores how these strategic approaches influence cooperation in the context of the prisoner's dilemma game, particularly in networked populations with varying clustering coefficients. Our research goes beyond existing studies by revealing that the differences in cooperation levels between these strategic approaches are not confined to finite populations; they also depend on the clustering coefficients of these populations. In populations with nonzero clustering coefficients, we observed varying degrees of stable cooperation for each strategic approach across multiple simulations, with mixed strategies showing the most variability, followed by continuous and discrete strategies. However, this variability in cooperation evolution decreased in populations with a clustering coefficient of zero, narrowing the differences in cooperation levels among the strategies. These findings suggest that in more realistic settings, the robustness of cooperation systems may be compromised, as the evolution of cooperation through mixed and continuous strategies introduces a degree of unpredictability. [ABSTRACT FROM AUTHOR]

We introduce a new family of real simple modules over the quantum affine algebras, called the affine determinantial modules, which contains the Kirillov-Reshetikhin (KR)-modules as a special subfamily, and then prove T-systems among them which generalize the T-systems among KR-modules and unipotent quantum minors in the quantum unipotent coordinate algebras simultaneously. We develop new combinatorial tools: admissible chains of i -boxes which produce commuting families of affine determinantial modules, and box moves which describe the T-system in a combinatorial way. Using these results, we prove that various module categories over the quantum affine algebras provide monoidal categorifications of cluster algebras. As special cases, Hernandez-Leclerc categories C g 0 and C g − provide monoidal categorifications of the cluster algebras for an arbitrary quantum affine algebra. [ABSTRACT FROM AUTHOR]

This paper discusses the problem of control laws for Discrete Event Dynamic Systems (DEDSs) represented by Timed Event Graphs (TEGs) under Generalized Marking Constraints (GMCs). The behaviour of TEGs is described using linear equations while GMCs are expressed by weighted inequalities in Min-Plus algebra. We formulate the problem in terms of control linear Min-Plus models. Hence, an algebraic method to calculate control laws ensuring the respect of GMCs is suggested. And once sufficient conditions are satisfied, we propose casual feedbacks to guarantee these marking specifications. Besides, the proposed control strategy is applied to a dual-armed cluster tool, a well-known industrial application. [ABSTRACT FROM AUTHOR]

We study the reflections of locally free Caldero–Chapoton functions associated to representations of Geiß–Leclerc–Schröer’s quivers with relations for symmetrizable Cartan matrices. We prove that for rank 2 cluster algebras, non-initial cluster variables are expressed as locally free Caldero–Chapoton functions of locally free indecomposable rigid representations. Our method gives rise to a new proof of the locally free Caldero–Chapoton formulas obtained by Geiß–Leclerc–Schröer in Dynkin cases. For general acyclic skew-symmetrizable cluster algebras, we prove the formula for any non-initial cluster variable obtained by almost sink and source mutations. [ABSTRACT FROM AUTHOR]

Deep clustering is an approach that uses deep learning to cluster data, since it involves training a neural network model to become familiar with a data representation that is suitable for clustering. Deep clustering has been applied to a wide range of data types, including images, texts, time series and has the advantage of being able to automatically learn features from the data, which can be more effective than using hand-crafted features. It is also able to handle high-dimensional data, such as time series with many variables, which can be challenging for traditional clustering techniques. In this paper, we introduce a novel deep neural network type to improve the performance of the auto-encoder part by ignoring the unnecessary extra-noises and labelling the input data. Our approach is helpful when just a limited amount of labelled data is available, but labelling a big amount of data would be costly or time-consuming. It also applies for the data in high-dimensional and difficult to define a good set of features for clustering. [ABSTRACT FROM AUTHOR]

We introduce a cluster algebraic generalization of Thurston's earthquake map for the cluster algebras of finite type, which we call the cluster earthquake map. It is defined by gluing exponential maps, which is modeled after the earthquakes along ideal arcs. We prove an analogue of the earthquake theorem, which states that the cluster earthquake map gives a homeomorphism between the spaces of |$\mathbb {R}^{\textrm {trop}}$| - and |$\mathbb {R}_{>0}$| -valued points of the cluster |$\mathcal {X}$| -variety. For those of type |$A_{n}$| and |$D_{n}$|⁠ , the cluster earthquake map indeed recovers the earthquake maps for marked disks and once-punctured marked disks, respectively. Moreover, we investigate certain asymptotic behaviors of the cluster earthquake map, which give rise to "continuous deformations" of the Fock–Goncharov fan. [ABSTRACT FROM AUTHOR]

The SL (2 , Z) -symmetry of Cherednik's spherical double affine Hecke algebras in Macdonald theory includes a distinguished generator which acts as a discrete time evolution of Macdonald operators, which can also be interpreted as a torus Dehn twist in type A. We prove for all twisted and untwisted affine algebras of type ABCD that the time-evolved q-difference Macdonald operators, in the t → ∞ q-Whittaker limit, form a representation of the associated discrete integrable quantum Q-systems, which are obtained, in all but one case, via the canonical quantization of suitable cluster algebras. The proof relies strongly on the duality property of Macdonald and Koornwinder polynomials, which allows, in the q-Whittaker limit, for a unified description of the quantum Q-system variables and the conserved quantities as limits of the time-evolved Macdonald operators and the Pieri operators, respectively. The latter are identified with relativistic q-difference Toda Hamiltonians. A crucial ingredient in the proof is the use of the "Fourier transformed" picture, in which we compute time-translation operators and prove that they commute with the Pieri operators or Hamiltonians. We also discuss the universal solutions of Koornwinder-Macdonald eigenvalue and Pieri equations, for which we prove a duality relation, which simplifies the proofs further. [ABSTRACT FROM AUTHOR]

We provide a relation between the geometric framework for q-Painlevé equations and cluster Poisson varieties by using toric models of rational surfaces associated with q-Painlevé equations. We introduce the notion of seeds of q-Painlevé type by the negative semi-definiteness of symmetric bilinear forms associated with seeds, and classify the mutation equivalence classes of these seeds. This classification coincides with the classification of q-Painlevé equations given by Sakai. We realize q-Painlevé systems as automorphisms on cluster Poisson varieties associated with seeds of q-Painlevé type. [ABSTRACT FROM AUTHOR]

Frieze patterns have been introduced by Coxeter [Acta Arith. 18 (1971), pp. 297–310] in the 1970's and have recently attracted renewed interest due to their close connection with Fomin-Zelevinsky's cluster algebras. Frieze patterns can be interpreted as assignments of values to the diagonals of a triangulated polygon satisfying certain conditions for crossing diagonals (Ptolemy relations). Weak friezes, as introduced by Çanakçı and Jørgensen [Adv. in Appl. Math. 131 (2021), Paper No. 102253], are generalizing this concept by allowing to glue dissected polygons so that the Ptolemy relations only have to be satisfied for crossings involving one of the gluing diagonals. To any frieze pattern one can associate a symmetric matrix using a triangular fundamental domain of the frieze pattern in the upper and lower half of the matrix and putting zeroes on the diagonal. Broline, Crowe and Isaacs [Geometriae Dedicata 3 (1974), pp. 171–176] have found a formula for the determinants of these matrices and their work has later been generalized in various directions by other authors. These frieze pattern determinants are the main focus of our paper. As our main result we show that this determinant behaves well with respect to gluing weak friezes: the determinant is the product of the determinants for the pieces glued, up to a scalar factor coming from the gluing diagonal. Then we give several applications of this result, showing that formulas from the literature, obtained by Broline-Crowe-Isaacs, Baur-Marsh [J. Combin. Theory Ser. A 119 (2012), pp. 1110–1122], Bessenrodt-Holm-Jørgensen [J. Combin. Theory Ser. A 123 (2014), pp. 30–42] and Maldonado [ Frieze matrices and infinite frieze patterns with coefficients , Preprint, arXiv: 2207.04120 , 2022] can all be obtained as consequences of our result. [ABSTRACT FROM AUTHOR]

It is known that many (upper) cluster algebras possess different kinds of good bases which contain the cluster monomials and are parametrized by the tropical points of cluster Poisson varieties. For a large class of upper cluster algebras (injective-reachable ones with full rank coefficients), we describe all of their bases with these properties. Moreover, we show the existence of the generic basis for them. In addition, we prove that Bridgeland's representation-theoretic formula is effective for their theta functions (weak genteelness). Our results apply to (almost) all known cluster algebras arising from representation theory or higher Teichmüller theory, including quantum affine algebras, unipotent cells, double Bruhat cells, skein algebras over surfaces, where we change the coefficients if necessary so that the full rank assumption holds. [ABSTRACT FROM AUTHOR]

Cluster integrable systems are a broad class of integrable systems modelled on bipartite dimer models on the torus. Many discrete integrable dynamics arise by applying sequences of local transformations, which form the cluster modular group of the cluster integrable system. This cluster modular group was recently characterized by the first author and Inchiostro. There exist some discrete integrable dynamics that make use of non-local transformations associated with geometric \(R\)-matrices. In this article we characterize the generalized cluster modular group - which includes both local and non-local transformations - in terms of extended affine symmetric groups. We also describe the action of the generalized cluster modular group on the spectral data associated with cluster integrable systems.Mathematics Subject Classifications: 82B23, 13F60, 14H70, 20B35Keywords: Bipartite dimer model, cluster algebras, geometric \(R\)-matrices, discrete integrable systems, extended affine symmetric group

Unitary cluster expansions of the electronic wavefunction have recently gained much interest because of their use in conjunction with quantum algorithms. In this contribution, we investigate some aspects of an ansatz, using generalized two-body excitation operators, which have been considered in some recent studies on quantum algorithms for quantum chemistry. Our numerical results show that, in particular, two-body operators with effective particle–hole excitation level of one in connection with the usual particle–hole double excitation operators lead to a very accurate, yet compact representation of the wavefunction. Generalized two-body operators with effective excitation rank zero have a considerably less pronounced effect. We compare with standard and unitary coupled-cluster expansions and show that the above mentioned approach matches or even surpasses the accuracy of expansions with three-body particle–hole excitations, in particular at the onset of strong correlation. A downside of the approach is that it is rather difficult to rigorously converge it to its variational minimum. [ABSTRACT FROM AUTHOR]

In a categorification of skew-symmetric cluster algebras, each cluster variable corresponds with an indecomposable module over the associated Jacobian algebra. Buan, Marsh and Reiten studied when the denominator vector of each cluster variable in an acyclic cluster algebra coincides with the dimension vector of the corresponding module. In this paper, we give analogues of their results for cluster algebras from triangulated surfaces by comparing two kinds of intersection numbers of tagged arcs. [ABSTRACT FROM AUTHOR]

We extend the notion of y -variables (coefficients) in cluster algebras to cluster scattering diagrams (CSDs). Accordingly, we extend the dilogarithm identity associated with a period in a cluster pattern to the one associated with a loop in a CSD. We show that these identities are constructed from and reduced to trivial ones by applying the pentagon identity possibly infinitely many times. [ABSTRACT FROM AUTHOR]

The mixed spin Ising nanowire with random and dilute crystal fields has been explored based on the finite cluster approximation. Much concern has been addressed in treating the effects of the randomness with two distribution types of the crystal field on the phase diagrams. It has been demonstrated that for a low random crystal field D marked by the fluctuation d, all transition lines are of second order and are not particularly sensitive to the fluctuation; whereas, for high values of D, the transition temperature depends on the fluctuation and may display tricritical points. Under the effect of the dilute crystal field, according to the surface shell coupling strength, the tricritical behavior occurs exclusively for higher values of the concentration p where a large majority of spins are influenced by the crystal field. All these behaviors have been confirmed by examination of magnetizations. [ABSTRACT FROM AUTHOR]

We develop the quantum cluster algebra approach recently introduced by Sun and Yagi to investigate the tetrahedron equation, a three-dimensional generalization of the Yang-Baxter equation. In the case of square quiver, we devise a new realization of quantum Y -variables in terms q -Weyl algebras and obtain a solution that possesses three spectral parameters. It is expressed in various forms, comprising four products of quantum dilogarithms depending on the signs in decomposing the quantum mutations into the automorphism part and the monomial part. For a specific choice of them, our formula precisely reproduces Sergeev's R matrix, which corresponds to a vertex formulation of the Zamolodchikov-Bazhanov-Baxter model when q is specialized to a root of unity. [ABSTRACT FROM AUTHOR]

The two main approaches to the study of irreducible representations of orders (via traces and Poisson orders) have so far been applied in a completely independent fashion. We define and study a natural compatibility relation between the two approaches leading to the notion of Poisson trace orders. It is proved that all regular and reduced traces are always compatible with any Poisson order structure. The modified discriminant ideals of all Poisson trace orders are proved to be Poisson ideals and the zero loci of discriminant ideals are shown to be unions of symplectic cores, under natural assumptions (maximal orders and Cayley–Hamilton algebras). A base change theorem for Poisson trace orders is proved. A broad range of Poisson trace orders are constructed based on the proved theorems: quantized universal enveloping algebras, quantum Schubert cell algebras and quantum function algebras at roots of unity, symplectic reflection algebras, 3D and 4D Sklyanin algebras, Drinfeld doubles of pre-Nichols algebras of diagonal type, and root of unity quantum cluster algebras. [ABSTRACT FROM AUTHOR]

Palu defined the index with respect to a cluster tilting object in a suitable triangulated category, in order to better understand the Caldero–Chapoton map that exhibits the connection between cluster algebras and representation theory. We push this further by proposing an index with respect to a contravariantly finite, rigid subcategory, and we show this index behaves similarly to the classical index. Let |${\mathcal{C}}$| be a skeletally small triangulated category with split idempotents, which is thus an extriangulated category |$({\mathcal{C}},{\mathbb{E}},{\mathfrak{s}})$|⁠. Suppose |${\mathcal{X}}$| is a contravariantly finite, rigid subcategory of |${\mathcal{C}}$|⁠. We define the index |$ {\operatorname{\textrm{ind}}}{_{{\mathcal{X}}}}(C)$| of an object |$C\in{\mathcal{C}}$| with respect to |${\mathcal{X}}$| as the |$ {K}{_{0}}$| -class |$[C {]}{_{{\mathcal{X}}}}$| in Grothendieck group |$ {K}{_{0}}({\mathcal{C}}, {{\mathbb{E}}}{_{{\mathcal{X}}}}, {{\mathfrak{s}}}{_{{\mathcal{X}}}})$| of the relative extriangulated category |$({\mathcal{C}}, {{\mathbb{E}}}{_{{\mathcal{X}}}}, {{\mathfrak{s}}}{_{{\mathcal{X}}}})$|⁠. By analogy to the classical case, we give an additivity formula with error term for |$ {\operatorname{\textrm{ind}}}{_{{\mathcal{X}}}}$| on triangles in |${\mathcal{C}}$|⁠. In case |${\mathcal{X}}$| is contained in another suitable subcategory |${\mathcal{T}}$| of |${\mathcal{C}}$|⁠ , there is a surjection |$Q\colon{K}{_{0}}({\mathcal{C}}, {{\mathbb{E}}}{_{{\mathcal{T}}}}, {{\mathfrak{s}}}{_{{\mathcal{T}}}}) \twoheadrightarrow{K}{_{0}}({\mathcal{C}}, {{\mathbb{E}}}{_{{\mathcal{X}}}}, {{\mathfrak{s}}}{_{{\mathcal{X}}}})$|⁠. Thus, in order to describe |$ {K}{_{0}}({\mathcal{C}}, {{\mathbb{E}}}{_{{\mathcal{X}}}}, {{\mathfrak{s}}}{_{{\mathcal{X}}}})$|⁠ , it suffices to determine |$ {K}{_{0}}({\mathcal{C}}, {{\mathbb{E}}}{_{{\mathcal{T}}}}, {{\mathfrak{s}}}{_{{\mathcal{T}}}})$| and |$\operatorname{Ker}\nolimits Q$|⁠. We do this under certain assumptions. [ABSTRACT FROM AUTHOR]

We analyze the structure of Feigin-Stoyanovsky's principal subspaces of affine Lie algebra from the jet algebra viewpoint. For type A level one principal subspaces, we show that their shifted multi-graded Hilbert series can be expressed either using the quantum dilogarithm or as certain generating functions "counting" finite-dimensional representations of A -type quivers. This notably results in novel fermionic character formulas for these principal subspaces. Moreover, our result implies that all level one principal subspaces of type A are "classically free" as vertex algebras. We also analyze infinite jet algebras associated to principal subspaces of affine vertex algebras L 1 (so 5) , L 1 (so 8) and L 1 (g 2). We derive a new character formula for the principal subspace of L 1 (so 5) , proving that it is classically free, and present evidence that the principal subspaces of L 1 (so 8) and of L 1 (g 2) are also classically free. [ABSTRACT FROM AUTHOR]

We study a simple geometric model for local transformations of bipartite graphs. The state consists of a choice of a vector at each white vertex made in such a way that the vectors neighboring each black vertex satisfy a linear relation. The evolution for different choices of the graph coincides with many notable dynamical systems including the pentagram map, Q-nets, and discrete Darboux maps. On the other hand, for plabic graphs we prove unique extendability of a configuration from the boundary to the interior, an elegant illustration of the fact that Postnikov's boundary measurement map is invertible. In all cases there is a cluster algebra operating in the background, resolving the open question for Q-nets of whether such a structure exists. [ABSTRACT FROM AUTHOR]

In this paper, we mainly answer a Lampe's question [3] about the solutions of a Diophantine equation, that is, we give a criterion to determine which solutions of the Diophantine equation are in the orbit of the initial solution (ϵ , ϵ , ϵ , ϵ , ϵ) under the actions of the group G which is defined by mutations of a cluster algebra. In order to do this, using a rational map φ , we transform the Diophantine equation to a related equation whose all positive integral solutions form the orbit of an initial solution φ (ϵ , ϵ , ϵ , ϵ , ϵ) = (3 , 4 , 4) under the actions of the group G ˜ , and the set S (3 , 4 , 4) is shown to be the orbit of (ϵ , ϵ , ϵ , ϵ , ϵ) under the actions of a subgroup of G. Then the criterion is proved as the main conclusion. [ABSTRACT FROM AUTHOR]

We develop silting theory of a Noetherian algebra |$\Lambda $| over a commutative Noetherian ring |$R$|⁠. We study mutation theory of |$2$| -term silting complexes of |$\Lambda $|⁠ , and as a consequence, we see that mutation exists. As in the case of finite-dimensional algebras, functorially finite torsion classes of |$\Lambda $| bijectively correspond to silting |$\Lambda $| -modules, if |$R$| is complete local. We show a reduction theorem of |$2$| -term silting complexes of |$\Lambda $|⁠ , and by using this theorem, we study torsion classes of the module category of |$\Lambda $|⁠. When |$R$| has Krull dimension one, we describe the set of torsion classes of |$\Lambda $| explicitly by using the set of torsion classes of finite-dimensional algebras. [ABSTRACT FROM AUTHOR]

We construct Grassmannian categories of infinite rank, providing an infinite analogue of the Grassmannian cluster categories introduced by Jensen, King, and Su. Each Grassmannian category of infinite rank is given as the category of graded maximal Cohen–Macaulay modules over a certain hypersurface singularity. We show that generically free modules of rank |$1$| in a Grassmannian category of infinite rank are in bijection with the Plücker coordinates in an appropriate Grassmannian cluster algebra of infinite rank. Moreover, this bijection is structure preserving, as it relates rigidity in the category to compatibility of Plücker coordinates. Along the way, we develop a combinatorial formula to compute the dimension of the |$\textrm {Ext}^{1}$| -spaces between any two generically free modules of rank |$1$| in the Grassmannian category of infinite rank. [ABSTRACT FROM AUTHOR]

In this paper, we study quasi-homomorphisms of quantum cluster algebras, which are quantum analogy of quasi-homomorphisms of cluster algebras introduced by Fraser. For a quantum Grassmannian cluster algebra C q [ Gr (k , n) ] , we show that there is an associated braid group and each generator σ i of the braid group preserves the quasi-commutative relations of quantum Plücker coordinates and exchange relations of the quantum Grassmannian cluster algebra. We conjecture that σ i also preserves r -term (r ≥ 4) quantum Plücker relations of C q [ Gr (k , n) ] and other relations which cannot be derived from quantum Plücker relations (if any). Up to this conjecture, we show that σ i is a quasi-automorphism of C q [ Gr (k , n) ] and the braid group acts on C q [ Gr (k , n) ]. [ABSTRACT FROM AUTHOR]

We discuss a product formula for |$F$| -polynomials in cluster algebras and provide two proofs. One proof is inductive and uses only the mutation rule for |$F$| -polynomials. The other is based on the Fock–Goncharov decomposition of mutations. We conclude by expanding this product formula as a sum and illustrate applications. This expansion provides an explicit combinatorial computation of |$F$| -polynomials in a given seed that depends only on the |$\textbf {c}$| -vectors and |$\textbf {g}$| -vectors along a finite sequence of mutations from the initial seed to the given seed. [ABSTRACT FROM AUTHOR]

It is known that many (upper) cluster algebras are not unique factorization domains. We exhibit the local factorization properties with respect to any given seed t: any non-zero element in a full rank upper cluster algebra can be uniquely written as the product of a cluster monomial in t and another element not divisible by the cluster variables in t. Our approach is based on introducing the valuation pairing on an upper cluster algebra: it counts the maximal multiplicity of a cluster variable among the factorizations of any given element. We apply the valuation pairing to obtain many results concerning factoriality, d-vectors, F-polynomials and the combinatorics of cluster Poisson variables. In particular, we obtain that full rank and primitive upper cluster algebras are factorial; an explanation of d-vectors using valuation pairing; a cluster monomial in non-initial cluster variables is determined by its F-polynomial; the F-polynomials of non-initial cluster variables are irreducible; and the cluster Poisson variables parametrize the exchange pairs of the corresponding upper cluster algebra. [ABSTRACT FROM AUTHOR]

The notion of generalized quantum cluster algebras was introduced as a natural generalization of Berenstein and Zelevinsky's quantum cluster algebras as well as Chekhov and Shapiro's generalized cluster algebras. In this paper, we focus on a generalized quantum cluster algebra of Kronecker type which possesses infinitely many cluster variables. We obtain the cluster multiplication formulas for this algebra. As an application of these formulas, a positive bar-invariant basis is explicitly constructed. Both results generalize those known for the Kronecker cluster algebra and quantum cluster algebra. [ABSTRACT FROM AUTHOR]

Dyck path categories are introduced as a combinatorial model of the category of representations of quivers of Dynkin-type n . In particular, it is proved that there is a bijection between some Dyck paths and perfect matchings of some snake graphs. The approach allows us to give formulas for cluster variables in cluster algebras of Dynkin-type n in terms of Dyck paths. [ABSTRACT FROM AUTHOR]

A new construction of the associahedron was recently given by Arkani‐Hamed, Bai, He, and Yan in connection with the physics of scattering amplitudes. We show that their construction (suitably understood) can be applied to construct generalized associahedra of any simply laced Dynkin type. Unexpectedly, we also show that this same construction produces Newton polytopes for all the F$F$‐polynomials of the corresponding cluster algebras. In addition, we show that the toric variety associated to the g$g$‐vector fan has the property that its nef cone is simplicial. [ABSTRACT FROM AUTHOR]

Associated to a convex integral polygon N is a cluster integrable system XN constructed from the dimer model. We compute the group GN of symmetries of XN , called the (2-2) cluster modular group, showing that it is a certain abelian group conjectured by Fock and Marshakov. Combinatorially, non-torsion elements of GN are ways of shuffling the underlying bipartite graph, generalizing domino-shuffling. Algebro-geometrically, GN is a subgroup of the Picard group of a certain algebraic surface associated to N. [ABSTRACT FROM AUTHOR]

We determine, for hard spheres, the Helmholtz free energy of a liquid that contains a solid cluster as a function of the size of the solid cluster by means of the formalism of the thermodynamics of curved interfaces. This is done at the constant total number of particles, volume, and temperature. We show that under certain conditions, one may have several local minima in the free energy profile, one for the homogeneous liquid and others for the spherical, cylindrical, and planar solid clusters surrounded by liquid. The variation of the interfacial free energy with the radius of the solid cluster and the distance between equimolar and tension surfaces are inputs from simulation results of nucleation studies. This is possible because stable solid clusters in the canonical ensemble become critical in the isothermal–isobaric ensemble. At each local minimum, we find no difference in chemical potential between the phases. At local maxima, we also find equal chemical potential, albeit in this case the nucleus is unstable. Moreover, the theory allows us to describe the stable solid clusters found in simulations. Therefore, we can use it for any combination of the total number of particles, volume, and global density as long as a minimum in the Helmholtz free energy occurs. We also study under which conditions the absolute minimum in the free energy corresponds to a homogeneous liquid or to a heterogeneous system having either spherical, cylindrical, or planar geometry. This work shows that the thermodynamics of curved interfaces at equilibrium can be used to describe nucleation. [ABSTRACT FROM AUTHOR]

Let |$Q$| be a finite acyclic valued quiver. We give the cluster multiplication formulas in the quantum cluster algebra of |$Q$| with arbitrary coefficients, by applying certain quotients of derived Hall subalgebras of |$Q$|⁠. These formulas can be viewed as the quantum version of the cluster multiplication theorem in the classical cluster algebra proved by Caldero–Keller for finite type, Hubery for affine type, and Xiao–Xu for acyclic quivers. [ABSTRACT FROM AUTHOR]

We apply the method of orbit harmonics to the set of break divisors and orientable divisors on graphs to obtain the central and external zonotopal algebras, respectively. We then relate a construction of Efimov in the context of cohomological Hall algebras to the central zonotopal algebra of a graph |$G_{Q,\gamma }$| constructed from a symmetric quiver |$Q$| with enough loops and a dimension vector |$\gamma $|⁠. This provides a concrete combinatorial perspective on the former work, allowing us to identify the quantum Donaldson–Thomas (DT) invariants as the Hilbert series of the space of |$S_{\gamma }$| -invariants of the Postnikov–Shapiro slim subgraph space attached to |$G_{Q,\gamma }$|⁠. The connection with orbit harmonics in turn allows us to give a manifestly nonnegative combinatorial interpretation to numerical DT invariants as the number of |$S_{\gamma }$| -orbits under the permutation action on the set of break divisors on |$G$|⁠. We conclude with several representation-theoretic consequences, whose combinatorial ramifications may be of independent interest. [ABSTRACT FROM AUTHOR]