Isomorphism Types Research Articles

Ordered set partitions and the $0$-Hecke algebra

Let the symmetric group 𝔖 n act on the polynomial ring ℚ[x n ]=ℚ[x 1 ,⋯,x n ] by variable permutation. The coinvariant algebra is the graded 𝔖 n -module R n :=ℚ[x n ]/I n , where I n is the ideal in ℚ[x n ] generated by invariant polynomials with vanishing constant term. Haglund, Rhoades, and Shimozono introduced a new quotient R n,k of the polynomial ring ℚ[x n ] depending on two positive integers k≤n which reduces to the classical coinvariant algebra of the symmetric group 𝔖 n when k=n. The quotient R n,k carries the structure of a graded 𝔖 n -module; Haglund et. al. determine its graded isomorphism type and relate it to the Delta Conjecture in the theory of Macdonald polynomials. We introduce and study a related quotient S n,k of 𝔽[x n ] which carries a graded action of the 0-Hecke algebra H n (0), where 𝔽 is an arbitrary field. We prove 0-Hecke analogs of the results of Haglund, Rhoades, and Shimozono. In the classical case k=n, we recover earlier results of Huang concerning the 0-Hecke action on the coinvariant algebra.

A Friedberg enumeration of equivalence structures

We solve a problem posed by Goncharov and Knight (Problem 4 in [S. Goncharov and J. Knight, Computable structure and antistructure theorems, Algebra Logika 41(6) (2002) 639–681, 757]). More specifically, we produce an effective Friedberg (i.e. injective) enumeration of computable equivalence structures, up to isomorphism. We also prove that there exists an effective Friedberg enumeration of all isomorphism types of infinite computable equivalence structures.

On certain graded representations of filiform Lie algebras

Let be a connected complex linear algebraic group of the same dimension as V such that the poset of the Zariski closures of the orbits for its action coincides with a full flag of subspaces of V. Using the classification of graded filiform Lie algebras, we determine the isomorphism types of the unipotent radical U of G in case G is not nilpotent and U is of maximal class. In particular, if , there are, up to isomorphism, only two such unipotent groups.

Operations on arc diagrams and degenerations for invariant subspaces of linear operators. Part II

ABSTRACTFor a partition β, denote by Nβ the nilpotent linear operator of Jordan type β. Given partitions β, γ, we investigate the representation space of all short exact sequenceswhere α is any partition with each part at most 2. Due to the condition on α, the isomorphism type of a sequence ℰ is given by an arc diagram Δ; denote by 𝕍Δ the subset of of all sequences isomorphic to ℰ. Thus, the variety carries a stratification given by the subsets of type 𝕍Δ. We compute the dimension of each stratum and show that the boundary of a stratum 𝕍Δ consists exactly of those where Δ′ is obtained from Δ by a non-empty sequence of arc moves of five possible types (A)–(E). The case where all three partitions are fixed has been studied in [3, 5]. There, arc moves of types (A)–(D)suffice to describe the boundary of a 𝕍Δ in . Our fifth move (E), “explosion,” is needed to break up an arc into two poles to allow for changes in the partition α.

Box moves on Littlewood–Richardson tableaux and an application to invariant subspace varieties

In his 1951 book “Infinite Abelian Groups”, Kaplansky gives a combinatorial characterization of the isomorphism types of embeddings of a cyclic subgroup in a finite abelian group. In this paper we first use partial maps on Littlewood–Richardson tableaux to generalize this result to finite direct sums of such embeddings. Our main interest is an application to invariant subspaces of nilpotent linear operators. We develop a criterion to decide if two irreducible components in the representation space are in the boundary partial order.

The experimental phase diagram study of the binary polyols system erythritol-xylitol

A comprehensive phase diagram for the binary polyols system erythritol-xylitol has been mapped with a transparent characterization approach. Here, the phase equilibrium of the system has been studied experimentally using a combination of methods: Temperature-history (T-history), X-Ray Diffraction (XRD), and Field-Emission Scanning Electron Microscopy (FESEM), and linked to Tammann plots. Existing literature has previously shown the system to be a non-isomorphous type forming a simple eutectic, by combining experimental data with theoretical modelling. The present investigation shows that the system's phase diagram is a partially isomorphous type forming a eutectic, but not a non-isomorphous type forming a simple eutectic. Here, the eutectic was found within 25–30mol% erythritol and at 77°C, which differs from the previous studies identifying the eutectic respectively at 25 or 36mol% erythritol and at 82°C. The reasons for the differences are hard to deduce since the research approach is not presented as fully transparent from the past studies. In the present study, only the temperature-composition plot of the first melting (of the two components in a physical mix, but not of a single blend) indicated the shape of a simple eutectic in a non-isomorphous system. The cycles after the first melting in contrast started from the real blend, and displayed eutectic and solid-solid phase changes in T-history. These were verified as forming solid solutions with XRD and FESEM. This eutectic melts at a temperature suitable for low-temperature solar heating, but displayed glass transition, supercooling, and thermally activated degradation, thus affecting its practical aspects as a PCM.

X-ray radiography investigation of structural conditions of Fe-15Cr-35Ni- 11 W steel irradiated by ion-plasma fluxes

It was found that structural-phase transformations induced by radiation in the highly doped heat resistant Fe-15Cr-35Ni-11 W alloy under the effects of treatment with ion-plasma beams differ from the transformations in steels of types 0 × 18H10T and 0 × 16N15M3B widely used in nuclear power engineering. Presence of these differences was established by performing X-ray radiography analysis, which demonstrated that additional reflections on the X-ray patterns of irradiated samples of Fe-15Cr-35Ni-11 W alloy appear from the side of large angles relative to the reflections for the initial solid solution. Detailed X-ray diffraction studies carried out by the authors showed that additional peaks appeared from the side of smaller angles in the X-ray diffraction patterns of iron-chromium alloys of type 0 × 18 (10–30) H additionally doped with Ti, Mo, Nb, Al to the amount of 1–3% and irradiated with ion-plasma beams.In both cases the phase thus formed is of isomorphic matrix type and is thermally metastable and, in contrast to 0 × 18H10T steel, Fe-15Cr-35Ni-11 W alloy undergoes softening. The analysis of published data on the possible causes inflicting similar structural-phase transformations in materials subjected to intensive ion-plasma treatment was performed. Concentrations of crystalline lattice stacking faults in Fe-15Cr-35Ni-11 W alloy and in 0 × 18H10T steel in the deformed state were determined by X-ray diffraction analysis. It was found that concentration of structural stacking faults in this state is 4 times higher for 0 × 18H10T steel, which indicates the lower stacking fault energy in this steel. Conclusion was made that the observed effects are associated with the mechanism of radiation-induced plastic deformation. Structural-phase changes in Fe-15Cr-35Ni-11 W alloy are associated with deformation by twinning, in contrast to 0 × 18H10T steel, where the observed transformations are due to slip deformation.

Faces of highest weight modules and the universal Weyl polyhedron

Let V be a highest weight module over a Kac–Moody algebra g, and let convV denote the convex hull of its weights. We determine the combinatorial isomorphism type of convV, i.e. we completely classify the faces and their inclusions. In the special case where g is semisimple, this brings closure to a question studied by Cellini and Marietti (2015) [7] for the adjoint representation, and by Khare (2016, 2017) [17,18] for most modules. The determination of faces of finite-dimensional modules up to the Weyl group action and some of their inclusions also appears in previous works of Satake (1960) [25], Borel and Tits (1965) [3], Vinberg (1990) [26], and Casselman (1997) [6].For any subset of the simple roots, we introduce a remarkable convex cone which we call the universal Weyl polyhedron, which controls the convex hulls of all modules parabolically induced from the corresponding Levi factor. Namely, the combinatorial isomorphism type of the cone stores the classification of faces for all such highest weight modules, as well as how faces degenerate as the highest weight gets increasingly singular. To our knowledge, this cone is new in finite and infinite type.We further answer a question of Michel Brion, by showing that the localization of convV along a face is always the convex hull of the weights of a parabolically induced module. Finally, as we determine the inclusion relations between faces representation-theoretically from the set of weights, without recourse to convexity, we answer a similar question for highest weight modules over symmetrizable quantum groups.

Interrelations between policymakers’ intentions and school agents’ interpretation of accountability policy in Israel

The study examined the interrelations between policymakers’ intentions for test-based accountability policy, and school agents’ perceptions and actions with regard to this policy. Mixed-methods were used and encompassed 24 policymakers, 80 school principals, 168 teachers and case studies of four schools. New institutional theory, including the concept of “environmental shift” (Powell & Di Maggio, 1991) and the metaphor of “coupling” (Weick, 1976), served as the conceptual framework. Findings indicate that the interrelations between intentions, perceptions and actions are mainly tightly coupled. These are discussed by invoking three types of institutional isomorphism (DiMaggio & Powell, 1983): coercive, mimetic and normative.

Loewy lengths of blocks with abelian defect groups

We consider p p -blocks with abelian defect groups and in the first part prove a relationship between its Loewy length and that for blocks of normal subgroups of index p p . Using this, we show that if B B is a 2 2 -block of a finite group with abelian defect group D ≅ C 2 a 1 × ⋯ × C 2 a r × ( C 2 ) s D \cong C_{2^{a_1}} \times \cdots \times C_{2^{a_r}} \times (C_2)^s , where a i > 1 a_i > 1 for all i i and r ≥ 0 r \geq 0 , then d > LL ⁡ ( B ) ≤ 2 a 1 + ⋯ + 2 a r + 2 s − r + 1 d > \operatorname {LL}(B) \leq 2^{a_1}+\cdots +2^{a_r}+2s-r+1 , where | D | = 2 d |D|=2^d . When s = 1 s=1 the upper bound can be improved to 2 a 1 + ⋯ + 2 a r + 2 − r 2^{a_1}+\cdots +2^{a_r}+2-r . Together these give sharp upper bounds for every isomorphism type of D D . A consequence is that when D D is an abelian 2 2 -group the Loewy length is bounded above by | D | |D| except when D D is a Klein-four group and B B is Morita equivalent to the principal block of A 5 A_5 . We conjecture similar bounds for arbitrary primes and give evidence that it holds for principal 3 3 -blocks.

Tail Positive Words and Generalized Coinvariant Algebras

Let $n,k,$ and $r$ be nonnegative integers and let $S_n$ be the symmetric group. We introduce a quotient $R_{n,k,r}$ of the polynomial ring $\mathbb{Q}[x_1, \dots, x_n]$ in $n$ variables which carries the structure of a graded $S_n$-module. When $r \ge n$ or $k = 0$ the quotient $R_{n,k,r}$ reduces to the classical coinvariant algebra $R_n$ attached to the symmetric group. Just as algebraic properties of $R_n$ are controlled by combinatorial properties of permutations in $S_n$, the algebra of $R_{n,k,r}$ is controlled by the combinatorics of objects called tail positive words. We calculate the standard monomial basis of $R_{n,k,r}$ and its graded $S_n$-isomorphism type. We also view $R_{n,k,r}$ as a module over the 0-Hecke algebra $H_n(0)$, prove that $R_{n,k,r}$ is a projective 0-Hecke module, and calculate its quasisymmetric and nonsymmetric 0-Hecke characteristics. We conjecture a relationship between our quotient $R_{n,k,r}$ and the delta operators of the theory of Macdonald polynomials.

Conditional constructions in African languages

This paper serves as an introduction to the special issue of Studies in African Linguistics devoted to conditional constructions in African languages. I first describe the motivation for this volume and the common terminological conventions used in the papers, before discussing some of the more influential attempts to categorize conditional constructions together with some of the functions of conditional constructions. I then present an overview of conditional constructions in African languages, noting the various kinds of conditional meanings that are distinguished grammatically in different languages, and types of isomorphism between conditional constructions and other categories. I conclude with a note on concessive conditionals.

Extensionality and logicality

Tarski characterized logical notions as invariant under permutations of the domain. The outcome, according to Tarski, is that our logic, which is commonly said to be a logic of extension rather than intension, is not even a logic of extension—it is a logic of cardinality (or, more accurately, of “isomorphism type”). In this paper, I make this idea precise. We look at a scale inspired by Ruth Barcan Marcus of various levels of meaning: extensions, intensions and hyperintensions. On this scale, the lower the level of meaning, the more coarse-grained and less “intensional” it is. I propose to extend this scale to accommodate a level of meaning appropriate for logic. Thus, below the level of extension, we will have a more coarse-grained level of form. I employ a semantic conception of form, adopted from Sher, where forms are features of things “in the world”. Each expression in the language embodies a form, and by the definition we give, forms will be invariant under permutations and thus Tarskian logical notions. I then define the logical terms of a language as those terms whose extension can be determined by their form. Logicality will be shown to be a lower level analogue of rigidity. Using Barcan Marcus’s principles of explicit and implicit extensionality, we are able to characterize purely logical languages as “sub-extensional”, namely, as concerned only with form, and we thus obtain a wider perspective on both logicality and extensionality.

Classifying bicrossed products of two Taft algebras

We classify all Hopf algebras which factor through two Taft algebras Tn2(q‾) and respectively Tm2(q). To start with, all possible matched pairs between the two Taft algebras are described: if q‾≠qn−1 then the matched pairs are in bijection with the group of d-th roots of unity in k, where d=(m,n) while if q‾=qn−1 then besides the matched pairs above we obtain an additional family of matched pairs indexed by k⁎. The corresponding bicrossed products (double cross product in Majid's terminology) are explicitly described by generators and relations and classified. As a consequence of our approach, we are able to compute the number of isomorphism types of these bicrossed products as well as to describe their automorphism groups.

Topological Ramsey spaces from Fraïssé classes, Ramsey-classification theorems, and initial structures in the Tukey types of p-points

A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fraisse classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Sokic is extended to equivalence relations for finite products of structures from Fraisse classes of finite relational structures satisfying the Ramsey property and the Order-Prescribed Free Amalgamation Property. This is essential to proving Ramsey-classification theorems for equivalence relations on fronts, generalizing the Pudlak–Rodl Theorem to this class of topological Ramsey spaces. To each topological Ramsey space in this framework corresponds an associated ultrafilter satisfying some weak partition property. By using the correct Fraisse classes, we construct topological Ramsey spaces which are dense in the partial orders of Baumgartner and Taylor (Trans Am Math Soc 241:283–309, 1978) generating p-points which are k-arrow but not $$k+1$$ -arrow, and in a partial order of Blass (Trans Am Math Soc 179:145–166, 1973) producing a diamond shape in the Rudin-Keisler structure of p-points. Any space in our framework in which blocks are products of n many structures produces ultrafilters with initial Tukey structure exactly the Boolean algebra $$\mathcal {P}(n)$$ . If the number of Fraisse classes on each block grows without bound, then the Tukey types of the p-points below the space’s associated ultrafilter have the structure exactly $$[\omega ]^{<\omega }$$ . In contrast, the set of isomorphism types of any product of finitely many Fraisse classes of finite relational structures satisfying the Ramsey property and the OPFAP, partially ordered by embedding, is realized as the initial Rudin-Keisler structure of some p-point generated by a space constructed from our template.

Intersection Local Times, Loop Soups and Permanental Wick Powers

Several stochastic processes related to transient Levy processes with potential densities u(x,y) = u(y x), that need not be symmetric nor bounded on the diagonal, are defined and studied. They are real valued processes on a space of measures V endowed with a metric d. Su- cient conditions are obtained for the continuity of these processes on (V,d). The processes include n-fold self-intersection local times of tran- sient Levy processes and permanental chaoses, which are 'loop soup n- fold self-intersection local times' constructed from the loop soup of the Levy process. Loop soups are also used to define permanental Wick powers, which generalizes standard Wick powers, a class of n-th order Gaussian chaoses. Dynkin type isomorphism theorems are obtained that relate the various processes. Poisson chaos processes are defined and permanental Wick powers are shown to have a Poisson chaos decomposition. Additional proper- ties of Poisson chaos processes are studied and a martingale extension is obtained for many of the processes described above.

Independent sums of [formula omitted] and [formula omitted

We construct a new idempotent Fourier multiplier on the Hardy space on the bidisc, which could not be obtained by applying known one dimensional results. The main tool is a new L1 equivalent of the Stein martingale inequality which holds for a special filtration of periodic subsets of T with some restrictions on the functions involved. We also identify the isomorphic type of the range of the associated operator as the independent sum of dyadic Hn1, which is known to be a complemented and invariant subspace of dyadic H1.

Generalised dihedral subalgebras from the Monster

The conjugacy classes of the Monster which occur in the McKay observation correspond to the isomorphism types of certain 2 2 -generated subalgebras of the Griess algebra. Sakuma, Ivanov and others showed that these subalgebras match the classification of vertex algebras generated by two Ising conformal vectors, or of Majorana algebras generated by two axes. In both cases, the eigenvalues α , β \alpha ,\beta parametrising the theory are fixed to 1 4 \frac {1}{4} , 1 32 \frac {1}{32} . We generalise these parameters and the algebras which depend on them, in particular finding the largest possible (nonassociative) axial algebras which satisfy the same key features, by working extensively with the underlying rings. The resulting algebras admit an associating symmetric bilinear form and satisfy the same 6 6 -transposition property as the Monster; 1 4 \frac {1}{4} , 1 32 \frac {1}{32} turns out to be distinguished.

The isomorphism problem for torsion free nilpotent groups of Hirsch length at most 5

AbstractWe consider the isomorphism problem for the finitely generated torsion free nilpotent groups of Hirsch length at most five. We show how this problem translates to solving an explicitly given set of polynomial equations. Based on this, we introduce a canonical form for each isomorphism type of finitely generated torsion free nilpotent group of Hirsch length at most 5 and, using a variation of our methods, we give an explicit description of its automorphisms.

Successive Approximation of p-Class Towers

Let F be a number field and p be a prime. In the successive approximation theorem, we prove that, for each integer n ≥ 1, finitely many candidates for the Galois group of the nth stage of the p-class tower over F are determined by abelian type invariants of p-class groups C1pE of unramified extensions E/F with degree [E : F] = pn-1. Illustrated by the most extensive numerical results available currently, the transfer kernels (TE, F) of the p-class extensions TE, F : C1pF → C1pE from F to unramified cyclic degree-p extensions E/F are shown to be capable of narrowing down the number of contestants significantly. By determining the isomorphism type of the maximal subgroups S G of all 3-groups G with coclass cc(G) = 1, and establishing a general theorem on the connection between the p-class towers of a number field F and of an unramified abelian p-extension E/F, we are able to provide a theoretical proof of the realization of certain 3-groups S with maximal class by 3-tower groups of dihedral fields E with degree 6, which could not be realized up to now.