Dominating Cycles and Forbidden Pairs Containing $$P_{5}$$

A cycle C in a graph G is dominating if every edge of G is incident with at least one vertex of C. For a set \(\mathcal {H}\) of connected graphs, a graph G is said to be \(\mathcal {H}\)-free if G does not contain any member of \(\mathcal {H}\) as an induced subgraph. When \(|\mathcal {H}| = 2, \mathcal {H}\) is called a forbidden pair. In this paper, we investigate the characterization of the class of the forbidden pairs guaranteeing the existence of a dominating cycle and show the following two results: (i) Every 2-connected \(\{P_{5}, K_{4}^{-}\}\)-free graph contains a longest cycle which is a dominating cycle. (ii) Every 2-connected \(\{P_{5}, W^{*}\}\)-free graph contains a longest cycle which is a dominating cycle. Here \(P_{5}\) is the path of order \(5, K_{4}^{-}\) is the graph obtained from the complete graph of order 4 by removing one edge, and \(W^{*}\) is the graph obtained from two triangles and an edge by identifying one vertex in each.     

Largest initial segments pointwise fixed by automorphisms of models of set theory

Given a model \(\mathcal {M}\) of set theory, and a nontrivial automorphism j of \(\mathcal {M}\), let \(\mathcal {I}_{\mathrm {fix}}(j)\) be the submodel of \(\mathcal {M}\) whose universe consists of elements m of \(\mathcal {M}\) such that \(j(x)=x\) for every x in the transitive closure of m (where the transitive closure of m is computed within \(\mathcal {M}\)). Here we study the class \(\mathcal {C}\) of structures of the form \(\mathcal {I}_{\mathrm {fix}}(j)\), where the ambient model \(\mathcal {M}\) satisfies a frugal yet robust fragment of \(\mathrm {ZFC}\) known as \(\mathrm {MOST}\), and \(j(m)=m\) whenever m is a finite ordinal in the sense of \(\mathcal {M}.\) Our main achievement is the calculation of the theory of \(\mathcal {C}\) as precisely \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\). The following theorems encapsulate our principal results: Theorem A. Every structure in \(\mathcal {C}\) satisfies \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm { Collection}\). Theorem B. Each of the following three conditions is sufficient for a countable structure \(\mathcal {N}\) to be in \(\mathcal {C}\):(a) \(\mathcal {N}\) is a transitive model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\).(b) \(\mathcal {N}\) is a recursively saturated model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\).(c) \(\mathcal {N}\) is a model of \(\mathrm {ZFC}\). Theorem C. Suppose \(\mathcal {M}\) is a countable recursively saturated model of \(\mathrm {ZFC}\) and I is a proper initial segment of \(\mathrm {Ord}^{\mathcal {M}}\) that is closed under exponentiation and contains \(\omega ^\mathcal {M}\) . There is a group embedding \(j\longmapsto \check{j}\) from \(\mathrm {Aut}(\mathbb {Q})\) into \(\mathrm {Aut}(\mathcal {M})\) such that I is the longest initial segment of \(\mathrm {Ord}^{\mathcal {M}}\) that is pointwise fixed by \(\check{j}\) for every nontrivial \(j\in \mathrm {Aut}(\mathbb {Q}).\) In Theorem C, \(\mathrm {Aut}(X)\) is the group of automorphisms of the structure X, and \(\mathbb {Q}\) is the ordered set of rationals.     

On Standard Derived Equivalences of Orbit Categories

Let k be a commutative ring, \(\mathcal {A}\) and \(\mathcal {B}\) – two k-linear categories with an action of a group G. We introduce the notion of a standard G-equivalence from \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}\) to \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\), where \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\) is the homotopy category of finitely generated projective \(\mathcal {A}\)-complexes. We construct a map from the set of standard G-equivalences to the set of standard equivalences from \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}\) to \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\) and a map from the set of standard G-equivalences from \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}\) to \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\) to the set of standard equivalences from \(\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {B}/G)\) to \(\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {A}/G)\), where \(\mathcal {A}/G\) denotes the orbit category. We investigate the properties of these maps and apply our results to the case where \(\mathcal {A}=\mathcal {B}=R\) is a Frobenius k-algebra and G is the cyclic group generated by its Nakayama automorphism ν. We apply this technique to obtain the generating set of the derived Picard group of a Frobenius Nakayama algebra over an algebraically closed field.     

Inflation of Finite Lattices Along All-or-nothing Sets

We introduce a new generalization of Alan Day’s doubling construction. For ordered sets \(\mathcal {L}\) and \(\mathcal {K}\) and a subset \(E \subseteq \ \leq _{\mathcal {L}}\) we define the ordered set \(\mathcal {L} \star _{E} \mathcal {K}\) arising from inflation of \(\mathcal {L}\) along E by \(\mathcal {K}\). Under the restriction that \(\mathcal {L}\) and \(\mathcal {K}\) are finite lattices, we find those subsets \(E \subseteq \ \leq _{\mathcal {L}}\) such that the ordered set \(\mathcal {L} \star _{E} \mathcal {K}\) is a lattice. Finite lattices that can be constructed in this way are classified in terms of their congruence lattices.A finite lattice is binary cut-through codable if and only if there exists a 0?1 spanning chain \(\left \{\theta _{i}\colon 0 \leq i \leq n \right \}\) in \(Con(\mathcal {L})\) such that the cardinality of the largest block of ?? _i /?? _i?1 is 2 for every i with 1≤i≤n. These are exactly the lattices that can be constructed by inflation from the 1-element lattice using only the 2-element lattice. We investigate the structure of binary cut-through codable lattices and describe an infinite class of lattices that generate binary cut-through codable varieties.     

Irreducible Morphisms and Locally Finite Dimensional Representations

Let \(\mathcal {A}\) be a Hom-finite additive Krull-Schmidt k-category where k is an algebraically closed field. Let \(\text {mod}\mathcal {A}\) denote the category of locally finite dimensional \(\mathcal {A}\)-modules, that is, the category of covariant functors \(\mathcal {A} \to \text {mod}k\). We prove that an irreducible monomorphism in \(\text {mod}\mathcal {A}\) has a finitely generated cokernel, and that an irreducible epimorphism in \(\text {mod}\mathcal {A}\) has a finitely co-generated kernel. Using this, we get that an almost split sequence in \(\text {mod}\mathcal {A}\) has to start with a finitely co-presented module and end with a finitely presented one. Finally, we apply our results to the study of rep(Q), the category of locally finite dimensional representations of a strongly locally finite quiver. We describe all possible shapes of the Auslander-Reiten quiver of rep(Q).     

Group theoretic characterizations of Buekenhout–Metz unitals in $\mathop{\mathrm{PG}}(2,q^{2})$

Let G be the group of projectivities stabilizing a unital \(\mathcal{U}\) in \(\mathop{\mathrm{PG}}(2,q^{2})\) and let A,B be two distinct points of \(\mathcal{U}\). In this paper we prove that, if G has an elation group of order q with center A and a group of projectivities stabilizing both A and B of order a divisor of q?1 greater than \(2(\sqrt{q}-1)\), then \(\mathcal{U}\) is an ovoidal Buekenhout–Metz unital. From this result two group theoretic characterizations of orthogonal Buekenhout–Metz unitals are given.     

Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture

Let H be a real algebraic group acting equivariantly with finitely many orbits on a real algebraic manifold X and a real algebraic bundle \({\mathcal {E}}\) on X. Let \(\mathfrak {h}\) be the Lie algebra of H. Let \(\mathcal {S}(X,{\mathcal {E}})\) be the space of Schwartz sections of \({\mathcal {E}}\). We prove that \(\mathfrak {h}\mathcal {S}(X,{\mathcal {E}})\) is a closed subspace of \(\mathcal {S}(X,{\mathcal {E}})\) of finite codimension. We give an application of this result in the case when H is a real spherical subgroup of a real reductive group G. We deduce an equivalence of two old conjectures due to Casselman: the automatic continuity and the comparison conjecture for zero homology. Namely, let \(\pi \) be a Casselman–Wallach representation of G and V be the corresponding Harish–Chandra module. Then the natural morphism of coinvariants \(V_{\mathfrak {h}}\rightarrow \pi _{\mathfrak {h}}\) is an isomorphism if and only if any linear \(\mathfrak {h}\)-invariant functional on V is continuous in the topology induced from \(\pi \). The latter statement is known to hold in two important special cases: if H includes a symmetric subgroup, and if H includes the nilradical of a minimal parabolic subgroup of G.     

Gradings on Modules Over Lie Algebras of E Types

For any grading by an abelian group G on the exceptional simple Lie algebra \(\mathcal {L}\) of type E ₆ or E ₇ over an algebraically closed field of characteristic zero, we compute the graded Brauer invariants of simple finite-dimensional modules, thus completing the computation of these invariants for simple finite-dimensional Lie algebras. This yields the classification of finite-dimensional G-graded simple \(\mathcal {L}\)-modules, as well as necessary and sufficient conditions for a finite-dimensional \(\mathcal {L}\)-module to admit a G-grading compatible with the given G-grading on \(\mathcal {L}\).     

Cartan coherent configurations

The Cartan scheme \(\mathcal{X}\) of a finite group G with a (B, N)-pair is defined to be the coherent configuration associated with the action of G on the right cosets of the Cartan subgroup \(B\cap N\) by right multiplication. It is proved that if G is a simple group of Lie type, then asymptotically the coherent configuration \(\mathcal{X}\) is 2-separable, i.e., the array of 2-dimensional intersection numbers determines \(\mathcal{X}\) up to isomorphism. It is also proved that in this case, the base number of \(\mathcal{X}\) equals 2. This enables us to construct a polynomial-time algorithm for recognizing Cartan schemes when the rank of G and the order of the underlying field are sufficiently large. One of the key points in the proof is a new sufficient condition for an arbitrary homogeneous coherent configuration to be 2-separable.     

The lattice of congruence lattices of algebras on a finite set

The congruence lattices of all algebras defined on a fixed finite set A ordered by inclusion form a finite atomistic lattice \(\mathcal {E}\). We describe the atoms and coatoms. Each meet-irreducible element of \(\mathcal {E}\) being determined by a single unary mapping on A, we characterize completely those which are determined by a permutation or by an acyclic mapping on the set A. Using these characterisations we deduce several properties of the lattice \(\mathcal {E}\); in particular, we prove that \(\mathcal {E}\) is tolerance-simple whenever \(|A|\ge 4\).     

Relative Projectivity and Transferability for Partial Lattices

A partial lattice P is ideal-projective, with respect to a class \(\mathcal {C}\) of lattices, if for every \(K\in \mathcal {C}\) and every homomorphism φ of partial lattices from P to the ideal lattice of K, there are arbitrarily large choice functions f:P→K for φ that are also homomorphisms of partial lattices. This extends the traditional concept of (sharp) transferability of a lattice with respect to \(\mathcal {C}\). We prove the following: (1) A finite lattice P, belonging to a variety \(\mathcal {V}\), is sharply transferable with respect to \(\mathcal {V}\) iff it is projective with respect to \(\mathcal {V}\) and weakly distributive lattice homomorphisms, iff it is ideal-projective with respect to \(\mathcal {V}\), (2) Every finite distributive lattice is sharply transferable with respect to the class \(\mathcal {R}_{\text {mod}}\) of all relatively complemented modular lattices, (3) The gluing D ₄ of two squares, the top of one being identified with the bottom of the other one, is sharply transferable with respect to a variety \(\mathcal {V}\) iff \(\mathcal {V}\) is contained in the variety \(\mathcal {M}_{\omega }\) generated by all lattices of length 2, (4) D ₄ is projective, but not ideal-projective, with respect to \(\mathcal {R}_{\text {mod}}\) , (5) D ₄ is transferable, but not sharply transferable, with respect to the variety \(\mathcal {M}\) of all modular lattices. This solves a 1978 problem of G. Grätzer, (6) We construct a modular lattice whose canonical embedding into its ideal lattice is not pure. This solves a 1974 problem of E. Nelson.     

Notes on the Kazhdan–Lusztig Theorem on Equivalence of the Drinfeld Category and the Category of $\boldsymbol{U}_{\!\boldsymbol q}{\boldsymbol {\mathfrak g}}$-Modules

We discuss the proof of Kazhdan and Lusztig of the equivalence of the Drinfeld category \({\mathcal D}({\mathfrak g},\hbar)\) of \({\mathfrak g}\)-modules and the category of finite dimensional \(U_q{\mathfrak g}\)-modules, \(q=e^{\pi i\hbar}\), for \(\hbar\in{\mathbb C}\setminus{\mathbb Q}^*\). Aiming at operator algebraists the result is formulated as the existence for each \(\hbar\in i{\mathbb R}\) of a normalized unitary 2-cochain \({\mathcal F}\) on the dual \(\hat G\) of a compact simple Lie group G such that the convolution algebra of G with the coproduct twisted by \({\mathcal F}\) is *-isomorphic to the convolution algebra of the q-deformation G _q of G, while the coboundary of \({\mathcal F}^{-1}\) coincides with Drinfeld’s KZ-associator defined via monodromy of the Knizhnik–Zamolodchikov equations.     

Geometry of Coadjoint Orbits and Multiplicity-one Branching Laws for Symmetric Pairs

Consider the restriction of an irreducible unitary representation π of a Lie group G to its subgroup H. Kirillov’s revolutionary idea on the orbit method suggests that the multiplicity of an irreducible H-module ν occurring in the restriction π|_H could be read from the coadjoint action of H on \(\mathcal {O}^{G} \cap \text {pr}^{-1}({\mathcal {O}}^{H})\), provided π and ν are ‘geometric quantizations’ of a G-coadjoint orbit \(\mathcal {O}^{G}\) and an H-coadjoint orbit \(\mathcal {O}^{H}\), respectively, where \(\text {pr} \colon \sqrt {-1}\mathfrak {g}^{\ast } \to \sqrt {-1}\mathfrak {h}^{\ast }\) is the projection dual to the inclusion \(\mathfrak {h} \subset \mathfrak {g}\) of Lie algebras. Such results were previously established by Kirillov, Corwin and Greenleaf for nilpotent Lie groups. In this article, we highlight specific elliptic orbits \(\mathcal {O}^{G}\) of a semisimple Lie group G corresponding to highest weight modules of scalar type. We prove that the Corwin–Greenleaf number \(\sharp (\mathcal {O}^{G} \cap \text {pr}^{-1}({\mathcal {O}}^{H}))/H\) is either zero or one for any H-coadjoint orbit \(\mathcal {O}^{H}\), whenever (G,H) is a symmetric pair of holomorphic type. Furthermore, we determine the coadjoint orbits \(\mathcal {O}^{H}\) with nonzero Corwin–Greenleaf number. Our results coincide with the prediction of the orbit philosophy, and can be seen as ‘classical limits’ of the multiplicity-free branching laws of holomorphic discrete series representations (Kobayashi [Progr. Math. 2007]).     

Stringy K-theory and the Chern character

We construct two new G-equivariant rings: \(\mathcal{K}(X,G)\), called the stringy K-theory of the G-variety X, and \(\mathcal{H}(X,G)\), called the stringy cohomology of the G-variety X, for any smooth, projective variety X with an action of a finite group G. For a smooth Deligne–Mumford stack \(\mathcal{X}\), we also construct a new ring \(\mathsf{K}_{\mathrm{orb}}(\mathcal{X})\) called the full orbifold K-theory of \(\mathcal{X}\). We show that for a global quotient \(\mathcal{X} = [X/G]\), the ring of G-invariants \(K_{\mathrm{orb}}(\mathcal{X})\) of \(\mathcal{K}(X,G)\) is a subalgebra of \(\mathsf{K}_{\mathrm{orb}}([X/G])\) and is linearly isomorphic to the “orbifold K-theory” of Adem-Ruan [AR] (and hence Atiyah-Segal), but carries a different “quantum” product which respects the natural group grading.We prove that there is a ring isomorphism \(\mathcal{C}\mathbf{h}:\mathcal{K}(X,G)\to\mathcal{H}(X,G)\), which we call the stringy Chern character. We also show that there is a ring homomorphism \(\mathfrak{C}\mathfrak{h}_\mathrm{orb}:\mathsf{K}_{\mathrm{orb}}(\mathcal{X}) \rightarrow H^\bullet_{\mathrm{orb}}(\mathcal{X})\), which we call the orbifold Chern character, which induces an isomorphism \(Ch_{\mathrm{orb}}:K_{\mathrm{orb}}(\mathcal{X})\rightarrow H^\bullet_{\mathrm{orb}}(\mathcal{X})\) when restricted to the sub-algebra \(K_{\mathrm{orb}}(\mathcal{X})\). Here \(H_{\mathrm{orb}}^\bullet(\mathcal{X})\) is the Chen–Ruan orbifold cohomology. We further show that \(\mathcal{C}\mathbf{h}\) and \(\mathfrak{C}\mathfrak{h}_\mathrm{orb}\) preserve many properties of these algebras and satisfy the Grothendieck–Riemann–Roch theorem with respect to étale maps. All of these results hold both in the algebro-geometric category and in the topological category for equivariant almost complex manifolds.We further prove that \(\mathcal{H}(X,G)\) is isomorphic to Fantechi and Göttsche’s construction [FG, JKK]. Since our constructions do not use complex curves, stable maps, admissible covers, or moduli spaces, our results greatly simplify the definitions of the Fantechi–Göttsche ring, Chen–Ruan orbifold cohomology, and the Abramovich–Graber–Vistoli orbifold Chow ring.We conclude by showing that a K-theoretic version of Ruan’s Hyper-Kähler Resolution Conjecture holds for the symmetric product of a complex projective surface with trivial first Chern class.     

<Emphasis Type="Italic">l</Emphasis>-Groups <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">X</Emphasis>) in continuous logic

In the context of continuous logic, this paper axiomatizes both the class \(\mathcal {C}\) of lattice-ordered groups isomorphic to C(X) for X compact and the subclass \(\mathcal {C}^+\) of structures existentially closed in \(\mathcal {C}\); shows that the theory of \(\mathcal {C}^+\) is \(\aleph _0\)-categorical and admits elimination of quantifiers; establishes a Nullstellensatz for \(\mathcal {C}\) and \(\mathcal {C}^+\); shows that \(C(X)\in \mathcal {C}\) has a prime-model extension in \(\mathcal {C}^+\) just in case X is Boolean; and proves that in a sense relevant to continuous logic, positive formulas admit in \(\mathcal {C}^+\) elimination of quantifiers to positive formulas.     

Vizing Bound for the Chromatic Number on Some Graph Classes

We are interested in hereditary classes of graphs \({\mathcal {G}}\) such that every graph \(G \in {\mathcal {G}}\) satisfies \(\varvec{\chi }(G) \le \omega (G) + 1\), where \(\chi (G)\) (\(\omega (G)\)) denote the chromatic (clique) number of G. This upper bound is called the Vizing bound for the chromatic number. Apart from perfect graphs few classes are known to satisfy the Vizing bound in the literature. We show that if G is (\(P_6, S_{1, 2, 2}\), diamond)-free, then \(\chi (G) \le \omega (G)+1\), and we give examples to show that the bound is sharp.     

Hall Polynomials for Nakayama Algebras

Let \(\mathcal{A}\) be a representation finite algebra over finite field k. In this note we first show that the existence of Hall polynomials for \(\mathcal{A}\) equivalent to the existence of the Hall polynomial \(\varphi^{M}_{N L}\) for each \(M, L \in mod\mathcal{A}\) and \(N\in ind\mathcal{A}\). Then we show that for a basic connected Nakayama algebra \(\mathcal{A}\), \(\mathcal{H}(\mathcal{A})=\mathcal{L}(\mathcal{A})\) and Hall polynomials exist for this algebra. We also provide another proof of the existence of Hall polynomials for the representation directed split algebras.     

Malcev products of unipotent monoids and varieties of bands

Let \(\mathcal{U}\) be the class of all unipotent monoids and \(\mathcal{B}\) the variety of all bands. We characterize the Malcev product \(\mathcal{U} \circ \mathcal{V}\) where \(\mathcal{V}\) is a subvariety of \(\mathcal{B}\) low in its lattice of subvarieties, \(\mathcal{B}\) itself and the subquasivariety \(\mathcal{S} \circ \mathcal{RB}\), where \(\mathcal{S}\) stands for semilattices and \(\mathcal{RB}\) for rectangular bands, in several ways including by a set of axioms. For members of some of them we describe the structure as well. This succeeds by using the relation \(\widetilde{\mathcal{H}}= \widetilde{\mathcal{L}} \cap \widetilde{\mathcal{R}}\), where \(a\;\,\widetilde{\mathcal{L}}\;\,b\) if and only if a and b have the same idempotent right identities, and \(\widetilde{\mathcal{R}}\) is its dual.We also consider \((\mathcal{U} \circ \mathcal{RB}) \circ \mathcal{S}\) which provides the motivation for this study since \((\mathcal{G} \circ \mathcal{RB}) \circ \mathcal{S}\) coincides with completely regular semigroups, where \(\mathcal{G}\) is the variety of all groups. All this amounts to a generalization of the latter: \(\mathcal{U}\) instead of \(\mathcal{G}\).     

Numerical semigroups whose fractions are of maximal embedding dimension

Each saturated (resp., Arf) numerical semigroup S has the property that each of its fractions \(\frac{S}{k}\) is saturated (resp., Arf), but the property of being of maximal embedding dimension (MED) is not stable under formation of fractions. If S is a numerical semigroup, then S is MED (resp., Arf; resp., saturated) if and only if, for each 2≤k∈?, \(S = \frac{T}{k}\) for infinitely many MED (resp., Arf; resp., saturated) numerical semigroups T. Let \(\mathcal{A}\) (resp., \(\mathcal{F}\)) be the class of Arf numerical semigroups (resp., of numerical semigroups each of whose fractions is of maximal embedding dimension). Then there exists an infinite strictly ascending chain \(\mathcal{A} =\mathcal{C}_{1} \subset\mathcal{C}_{2} \subset\mathcal{C}_{3}\subset \,\cdots\, \subset\mathcal{F}\), where, like \(\mathcal{A}\) and \(\mathcal{F}\), each \(\mathcal{C}_{n}\) is stable under the formation of fractions.     

Flag-transitive point-quasiprimitive 2-$$(v,\, k,\, 2)$$ designs

Let \(\mathcal {D}\) be a non-trivial 2-\((v,\,k,\,2)\) design. Assume that G is a flag-transitive and point-quasiprimitive automorphism group of \(\mathcal {D},\) then G is of holomorph affine or almost simple type.