Search

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]

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]

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]

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]

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]

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]

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]

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]

Generalized double affine Hecke algebras (GDAHA) are flat deformations of the group algebras of 2-dimensional crystallographic groups associated to star-shaped simply laced affine Dynkin diagrams. In this paper, we first construct a functor that sends representations of the D ˜ 4 -type GDAHA to representations of the E ˜ 6 -type one for specialised parameters. Then, under no restrictions on the parameters, we construct embeddings of both GDAHAs of type D ˜ 4 and E ˜ 6 into matrix algebras over quantum cluster X -varieties, thus linking to the theory of higher Teichmüller spaces. For E ˜ 6 , the two explicit representations we provide over distinct quantum tori are shown to be related by quiver reductions and mutations. [ABSTRACT FROM AUTHOR]

Cluster algebra structures for Grassmannians and their (open) positroid strata are controlled by a Postnikov diagram D or, equivalently, a dimer model on the disc, as encoded by either a bipartite graph or the dual quiver (with faces). The associated dimer algebra A , determined directly by the quiver with a certain potential, can also be realised as the endomorphism algebra of a cluster-tilting object in an associated Frobenius cluster category. In this paper, we introduce a class of A -modules corresponding to perfect matchings of the dimer model of D and show that, when D is connected, the indecomposable projective A -modules are in this class. Surprisingly, this allows us to deduce that the cluster category associated to D embeds into the cluster category for the appropriate Grassmannian. We show that the indecomposable projectives correspond to certain matchings which have appeared previously in work of Muller–Speyer. This allows us to identify the cluster-tilting object associated to D , by showing that it is determined by one of the standard labelling rules constructing a cluster of Plücker coordinates from D. By computing a projective resolution of every perfect matching module, we show that Marsh–Scott's formula for twisted Plücker coordinates, expressed as a dimer partition function, is a special case of the general cluster character formula, and thus observe that the Marsh–Scott twist can be categorified by a particular syzygy operation in the Grassmannian cluster category. [ABSTRACT FROM AUTHOR]