Non-trivial Identities Research Articles

Necessary Conditions for the Residual Nilpotency of Certain Group Theory Constructions

Consider a graph G of groups such that each vertex group locally satisfies a nontrivial identity and each edge subgroup is properly included into the corresponding vertex groups and its index in at least one of them exceeds 2. We prove that if the fundamental group F of G is locally residually nilpotent then there exists a prime number p such that each edge subgroup is p′-isolated in the corresponding vertex group. We show also that if F is the free product of an arbitrary family of groups with one amalgamated subgroup or a multiple HNN-extension then the same result holds without restrictions on the indices of edge subgroups.

Equations in oligomorphic clones and the constraint satisfaction problem for ω-categorical structures

There exist two conjectures for constraint satisfaction problems (CSPs) of reducts of finitely bounded homogeneous structures: the first one states that tractability of the CSP of such a structure is, when the structure is a model-complete core, equivalent to its polymorphism clone satisfying a certain nontrivial linear identity modulo outer embeddings. The second conjecture, challenging the approach via model-complete cores by reflections, states that tractability is equivalent to the linear identities (without outer embeddings) satisfied by its polymorphisms clone, together with the natural uniformity on it, being nontrivial. We prove that the identities satisfied in the polymorphism clone of a structure allow for conclusions about the orbit growth of its automorphism group, and apply this to show that the two conjectures are equivalent. We contrast this with a counterexample showing that [Formula: see text]-categoricity alone is insufficient to imply the equivalence of the two conditions above in a model-complete core. Taking another approach, we then show how the Ramsey property of a homogeneous structure can be utilized for obtaining a similar equivalence under different conditions. We then prove that any polymorphism of sufficiently large arity which is totally symmetric modulo outer embeddings of a finitely bounded structure can be turned into a nontrivial system of linear identities, and obtain nontrivial linear identities for all tractable cases of reducts of the rational order, the random graph, and the random poset. Finally, we provide a new and short proof, in the language of monoids, of the theorem stating that every [Formula: see text]-categorical structure is homomorphically equivalent to a model-complete core.

Factorizations of skew braces

We introduce strong left ideals of skew braces and prove that they produce non-trivial decomposition of set-theoretic solutions of the Yang-Baxter equation. We study factorization of skew left braces through strong left ideals and we prove analogs of It\^{o}'s theorem in the context of skew left braces. As a corollary, we obtain applications to the retractability problem of involutive non-degenerate solutions of the Yang-Baxter equation. Finally, we classify skew braces that contain no non-trivial proper ideals.

Non-comaximal graph of ideals of a ring

Let R be a ring. The non-comaximal graph of R, denoted by NC(R) is an undirected graph whose vertex set is the collection of all non-trivial (left) ideals of R and any two distinct vertices I and J are adjacent if and only if $$I+J\ne R$$ . The concepts of connectedness, independent set, clique and traversability of NC(R) are discussed.

On a graph of ideals of a commutative ring

In this paper, we introduce and investigate a new graph of a commutative ring R, denoted by G(R), with all nontrivial ideals of R as vertices, and two distinct vertices I and J are adjacent if and only if ann(I∩J)=ann(I)+ann(J). In this article, the basic properties and possible structures of the graph G(R) are studied and investigated as diameter, girth, clique number, cut vertex and domination number. We characterize all rings R for which G(R) is planar, complete and complete r-partite. We show that, if (R,M) is a local Artinian ring, then G(R) is complete if and only if Soc(R) is simple. Also, it is shown that if R is a ring with G(R) is r-regular, then either G(R) is complete or null graph. Moreover, we show that if R is an Artinian ring, then R is a serial ring if and only if G(R/I) is complete for each ideal I of R.

Comments on the multi-spin solution to the Yang–Baxter equation and basic hypergeometric sum/integral identity

We present a multi-spin solution to the Yang–Baxter equation (YBE). The solution corresponds to the integrable lattice spin model of statistical mechanics with positive Boltzmann weights and parametrized in terms of the basic hypergeometric functions. We obtain this solution from a nontrivial basic hypergeometric sum-integral identity which originates from the equality of supersymmetric indices for certain three-dimensional [Formula: see text] = 2 Seiberg dual theories.

A graphic approach to gauge invariance induced identity

All tree-level amplitudes in Einstein-Yang-Mills (EYM) theory and gravity (GR) can be expanded in terms of color ordered Yang-Mills (YM) ones whose coefficients are polynomial functions of Lorentz inner products and are constructed by a graphic rule. Once the gauge invariance condition of any graviton is imposed, the expansion of a tree level EYM or gravity amplitude induces a nontrivial identity between color ordered YM amplitudes. Being different from traditional Kleiss-Kuijf (KK) and Bern-Carrasco-Johansson (BCJ) relations, the gauge invariance induced identity involves polarizations in the coefficients. In this paper, we investigate the relationship between the gauge invariance induced identity and traditional BCJ relations. By proposing a refined graphic rule, we prove that all the gauge invariance induced identities for single trace tree-level EYM amplitudes can be precisely expanded in terms of traditional BCJ relations, without referring any property of polarizations. When further considering the transversality of polarizations and momentum conservation, we prove that the gauge invariance induced identity for tree-level GR (or pure YM) amplitudes can also be expanded in terms of traditional BCJ relations for YM (or bi-scalar) amplitudes. As a byproduct, a graph-based BCJ relation is proposed and proved.

Residual Separability of Subgroups in Free Products with Amalgamated Subgroup of Finite Index

Let P be the free product of groups A and B with amalgamated subgroup H, where H is a proper subgroup of finite index in A and B. We assume that the groups A and B satisfy a nontrivial identity and for each natural n the number of all subgroups of index n in A and B is finite. We prove that all cyclic subgroups in P are residually separable if and only if P is residually finite and all cyclic subgroups in H are residually separable; and all finitely generated subgroups in P are residually separable if and only if P is residually finite and all subgroups that are the intersections of H with finitely generated subgroups of P are finitely separable in H.

Toward an Analytic Perturbative Solution for the Abjm Quantum Spectral Curve

Recently we showed how non-homogeneous second-order difference equations appearing within ABJM quantum spectral curve description could be solved using Mellin space technique. In particular we provided explicit results for anomalous dimensions of twist 1 operators in sl(2) sector at arbitrary spin values up to four loop order. It was shown that the obtained results may be expressed in terms of harmonic sums decorated by fourth root of unity factors, so that maximum transcendentality principle holds. In this note we show that the same result could be also obtained by direct solution of the mentioned equations in spectral parameter u-space. The solution involves new highly nontrivial identities between hypergeometric functions, which may have various other applications. We expect this method to be more easily generalizable to higher loop orders as well as to other theories, such as N=4 SYM.

The monoids of the patience sorting algorithm

The left patience sorting ([Formula: see text][Formula: see text]PS) monoid, also known in the literature as the Bell monoid, and the right patient sorting ([Formula: see text]PS) monoid are introduced by defining certain congruences on words. Such congruences are constructed using insertion algorithms based on the concept of decreasing subsequences. Presentations for these monoids are given.Each finite-rank [Formula: see text]PS monoid is shown to have polynomial growth and to satisfy a nontrivial identity (dependent on its rank), while the infinite rank [Formula: see text]PS monoid does not satisfy any nontrivial identity. Each [Formula: see text][Formula: see text]PS monoid of finite rank has exponential growth and does not satisfy any nontrivial identity. The complexity of the insertion algorithms is discussed.[Formula: see text]PS monoids of finite rank are shown to be automatic and to have recursive complete presentations. When the rank is [Formula: see text] or [Formula: see text], they are also biautomatic. [Formula: see text][Formula: see text]PS monoids of finite rank are shown to have finite complete presentations and to be biautomatic.

Classifying G-graded algebras of exponent two

Let F be a field of characteristic zero and let $$\mathcal{V}$$ be a variety of associative F-algebras graded by a finite abelian group G. If $$\mathcal{V}$$ satisfies an ordinary non-trivial identity, then the sequence $$c_n^G(\mathcal{V})$$ of G-codimensions is exponentially bounded. In [2, 3, 8], the authors captured such exponential growth by proving that the limit $$^G(\mathcal{V}) = {\rm{lim}}_{n \to \infty} \sqrt[n]{{c_n^G(\mathcal{V})}}$$ exists and it is an integer, called the G-exponent of $$\mathcal{V}$$ . The purpose of this paper is to characterize the varieties of G-graded algebras of exponent greater than 2. As a consequence, we find a characterization for the varieties with exponent equal to 2.

BPS Wilson loops in mathcal{N} \u2265 2 superconformal Chern-Simons-matter theories

In mathcal{N} ≥ 2 superconformal Chern-Simons-matter theories we construct the infinite family of Bogomol’nyi-Prasad-Sommerfield (BPS) Wilson loops featured by constant parametric couplings to scalar and fermion matter, including both line Wilson loops in Minkowski spacetime and circle Wilson loops in Euclidean space. We find that the connection of the most general BPS Wilson loop cannot be decomposed in terms of double-node connections. Moreover, if the quiver contains triangles, it cannot be interpreted as a super-matrix inside a superalgebra. However, for particular choices of the parameters it reduces to the well-known connections of 1/6 BPS Wilson loops in Aharony-Bergman-Jafferis-Maldacena (ABJM) theory and 1/4 BPS Wilson loops in mathcal{N} = 4 orbifold ABJM theory. In the particular case of mathcal{N} = 2 orbifold ABJM theory we identify the gravity duals of a subset of operators. We investigate the cohomological equivalence of fermionic and bosonic BPS Wilson loops at quantum level by studying their expectation values, and find strong evidence that the cohomological equivalence holds quantum mechanically, at framing one. Finally, we discuss a stronger formulation of the cohomological equivalence, which implies non-trivial identities for correlation functions of composite operators in the defect CFT defined on the Wilson contour and allows to make novel predictions on the corresponding unknown integrals that call for a confirmation.

Identities in Plactic, Hypoplactic, Sylvester, Baxter, and Related Monoids

This paper considers whether non-trivial identities are satisfied by certain `plactic-like' monoids that, like the plactic monoid, are closely connected to combinatorics. New results show that the hypoplactic, sylvester, Baxter, stalactic, and taiga monoids satisfy identities, and indeed give shortest identities satisfied by these monoids. The existing state of knowledge is discussed for the plactic monoid and left and right patience sorting monoids.

Sum rules for characters from character-preservation property of matrix models

One of the main features of eigenvalue matrix models is that the averages of characters are again characters, what can be considered as a far-going generalization of the Fourier transform property of Gaussian exponential. This is true for the standard Hermitian and unitary (trigonometric) matrix models and for their various deformations, classical and quantum ones. Arising explicit formulas for the partition functions are very efficient for practical computer calculations. However, to handle them theoretically, one needs to tame remaining finite sums over representations of a given size, which turns into an interesting conceptual problem. Already the semicircle distribution in the large-N limit implies interesting combinatorial sum rules for characters. We describe also implications to W-representations, including a character decomposition of cut-and-join operators, which unexpectedly involves only single-hook diagrams and also requires non-trivial summation identities.

Twist-field representations of W-algebras, exact conformal blocks and character identities

We study the twist-field representations of W-algebras and generalize construction of the corresponding vertex operators to D- and B-series. It is shown, how the computation of characters of these representations leads to nontrivial identities involving lattice theta-functions. We also propose a way to calculate their exact conformal blocks, expressing them for D-series in terms of geometric data of the corresponding Prym variety for covering curve with involution.

On the regular graph of ideals of total ring of fractions

The regular graph of ideals of a commutative ring [Formula: see text], denoted by [Formula: see text], is a graph whose vertex-set is the set of all nontrivial ideals of [Formula: see text] and, for every two distinct vertices [Formula: see text] and [Formula: see text], there is an edge between [Formula: see text] and [Formula: see text], whenever [Formula: see text] contains a nonzero zero-divisor on [Formula: see text] or vice versa. In this paper, we will show that [Formula: see text] is isomorphic to an induced sub-graph of [Formula: see text] which nicely can describe [Formula: see text]. This approach enables us to find the independence number of [Formula: see text] when [Formula: see text] is a reduced ring. Among other things, some basic graph theoretic properties of [Formula: see text] and relevant ring theoretic properties of [Formula: see text] will be studied.

The M-intersection graph of ideals of a commutative ring

Let [Formula: see text] be a commutative ring and [Formula: see text] be an [Formula: see text]-module, and let [Formula: see text] be the set of all nontrivial ideals of [Formula: see text]. The [Formula: see text]-intersection graph of ideals of [Formula: see text], denoted by [Formula: see text], is a graph with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. For every multiplication [Formula: see text]-module [Formula: see text], the diameter and the girth of [Formula: see text] are determined. Among other results, we prove that if [Formula: see text] is a faithful [Formula: see text]-module and the clique number of [Formula: see text] is finite, then [Formula: see text] is a semilocal ring. We denote the [Formula: see text]-intersection graph of ideals of the ring [Formula: see text] by [Formula: see text], where [Formula: see text] are integers and [Formula: see text] is a [Formula: see text]-module. We determine the values of [Formula: see text] and [Formula: see text] for which [Formula: see text] is perfect. Furthermore, we derive a sufficient condition for [Formula: see text] to be weakly perfect.

On the Lattice of Overcommutative Varieties of Monoids

It is unknown so far, whether the lattice of all varieties of monoids satisfies some non-trivial identity. The objective of this note is to give the negative answer to this question. Namely, we prove that any finite lattice is a homomorphic image of some sublattice of the lattice of overcommutative varieties of monoids (i.e., varieties that contain the variety of all commutative monoids). This implies that the lattice of overcommutative varieties of monoids and therefore, the lattice of all varieties of monoids does not satisfy any non-trivial identity.

Comments on Exchange Graphs in Cluster Algebras

ABSTRACTAn important problem in the theory of cluster algebras is to compute the fundamental group of the exchange graph. A non-trivial closed loop in the exchange graph, for example, generates a non-trivial identity for the classical and quantum dilogarithm functions. An interesting conjecture, partly motivated by dilogarithm functions, is that this fundamental group is generated by closed loops of mutations involving only two of the cluster variables. We present examples and counterexamples for this naive conjecture, and then formulate a better version of the conjecture for acyclic seeds.

Crystals and trees: Quasi-Kashiwara operators, monoids of binary trees, and Robinson–Schensted-type correspondences

Kashiwara's crystal graphs have a natural monoid structure that arises by identifying words labelling vertices that appear in the same position of isomorphic components. The celebrated plactic monoid (the monoid of Young tableaux), arises in this way from the crystal graph for the q-analogue of the general linear Lie algebra gln, and the so-called Kashiwara operators interact beautifully with the combinatorics of Young tableaux and with the Robinson–Schensted–Knuth correspondence. The authors previously constructed an analogous ‘quasi-crystal’ structure for the related hypoplactic monoid (the monoid of quasi-ribbon tableaux), which has similarly neat combinatorial properties. This paper constructs an analogous ‘crystal-type’ structure for the sylvester and Baxter monoids (the monoids of binary search trees and pairs of twin binary search trees, respectively). Both monoids are shown to arise from this structure just as the plactic monoid does from the usual crystal graph. The interaction of the structure with the sylvester and Baxter versions of the Robinson–Schensted–Knuth correspondence is studied. The structure is then applied to prove results on the number of factorizations of elements of these monoids, and to prove that both monoids satisfy non-trivial identities.