Recent zbMATH articles in MSC 20https://zbmath.org/atom/cc/202024-09-27T17:47:02.548271ZWerkzeugBook review of: A. A. Ivanov, The Mathieu groupshttps://zbmath.org/1541.000282024-09-27T17:47:02.548271Z"van der Waall, Rob"https://zbmath.org/authors/?q=ai:van-der-waall.robert-willemReview of [Zbl 1402.20003].On the symmetric 2-\((v, k, \lambda)\) designs with a flag-transitive point-imprimitive automorphism grouphttps://zbmath.org/1541.050152024-09-27T17:47:02.548271Z"Montinaro, Alessandro"https://zbmath.org/authors/?q=ai:montinaro.alessandroSummary: The symmetric 2-\((v, k, \lambda)\) designs with \(k > \lambda (\lambda - 3) / 2\) admitting a flag-transitive point-imprimitive automorphism group are completely classified: they are the known 2-designs with parameters \((16, 6, 2), (45, 12, 3), (15, 8, 4)\) or \((96, 20, 4)\).Alternating groups and point-primitive linear spaces with number of points being squarefreehttps://zbmath.org/1541.050222024-09-27T17:47:02.548271Z"Guan, Haiyan"https://zbmath.org/authors/?q=ai:guan.haiyan"Zhou, Shenglin"https://zbmath.org/authors/?q=ai:zhou.shenglinSummary: This paper is a further contribution to the classification of point-primitive finite regular linear spaces. Let \(\mathcal{S}\) be a nontrivial finite regular linear space whose number of points \(v\) is squarefree. We prove that if \(G \leq \Aut (\mathcal{S})\) is point-primitive with an alternating socle, then \(\mathcal{S}\) is the projective space \(\mathrm{PG}(3, 2)\).
{{\copyright} 2023 Wiley Periodicals LLC.}Flag-transitive, point-imprimitive symmetric \(2\)-\((v, k, \lambda)\) designs with \(k > \lambda (\lambda - 3) / 2\)https://zbmath.org/1541.050232024-09-27T17:47:02.548271Z"Montinaro, Alessandro"https://zbmath.org/authors/?q=ai:montinaro.alessandroSummary: Let \(\mathcal{D} = (\mathcal{P}, \mathcal{B})\) be a symmetric 2-\((v, k, \lambda)\) design admitting a flag-transitive, point-imprimitive automorphism group \(G\) that leaves invariant a non-trivial partition \(\Sigma\) of \(\mathcal{P}\). \textit{C. E. Praeger} and \textit{S. Zhou}, J. Comb. Theory, Ser. A 113, No. 7, 1381--1395 (2006; Zbl 1106.05012)] have shown that, there is a constant \(k_0\) such that, for each \(B \in \mathcal{B}\) and \(\Delta \in \Sigma\), the size of \(|B \cap \Delta |\) is either 0 or \(k_0\). In the present paper we show that, if \(k > \lambda (\lambda - 3) / 2\) and \(k_0 \geqslant 3\), \(\mathcal{D}\) is isomorphic to one of the known flag-transitive, point-imprimitive symmetric 2-designs with parameters \((45, 12, 3)\) or \((96, 20, 4)\).The non-braid graph of dihedral group \(D_n\)https://zbmath.org/1541.050812024-09-27T17:47:02.548271Z"Muhammad, Hubbi"https://zbmath.org/authors/?q=ai:muhammad.hubbi"Maharani, Rambu Maya Imung"https://zbmath.org/authors/?q=ai:maharani.rambu-maya-imung"Nurhayati, Sri"https://zbmath.org/authors/?q=ai:nurhayati.sri"Wadu, Mira"https://zbmath.org/authors/?q=ai:wadu.mira"Susanti, Yeni"https://zbmath.org/authors/?q=ai:susanti.yeniSummary: We introduce the non-braid graph of a group \(G\), denoted by \(\zeta(G)\), as a graph with vertex set \(G\setminus B(G)\), where \(B(G)\) is the braider of \(G\), defined as the set \(\{x \in G \mid (\forall y \in G) xyx = yxy\}\), and two distinct vertices \(x\) and \(y\) are joined by an edge if and only if \(xyx \neq yxy\). In this paper particularly we give the independent number, the vertex chromatic number, the clique number, and the minimum vertex cover of non-braid graph of dihedral group \(D_n\).Graphs connected to isotopes of inverse property quasigroups: a few applicationshttps://zbmath.org/1541.050822024-09-27T17:47:02.548271Z"Nadeem, Muhammad"https://zbmath.org/authors/?q=ai:nadeem.muhammad-faisal|nadeem.muhammad"Ali, Sharafat"https://zbmath.org/authors/?q=ai:ali.sharafat"Alam, Md. Ashraful"https://zbmath.org/authors/?q=ai:alam.md-ashrafulSummary: Many real-world applications can be modelled as graphs or networks, including social networks and biological networks. The theory of algebraic combinatorics provides tools to analyze the functioning of these networks, and it also contributes to the understanding of complex systems and their dynamics. Algebraic methods help uncover hidden patterns and properties that may not be immediately apparent in a visual representation of a graph. In this paper, we introduce left and right inverse graphs associated with finite loop structures and two mappings, P-edge labeling and V-edge labeling, of Latin squares. Moreover, this work includes some structural and graphical results of the commutator subloop, nucleus, and loop isotopes of inverse property quasigroups.Projective non-commuting graph of a grouphttps://zbmath.org/1541.050832024-09-27T17:47:02.548271Z"Pezzott, Julio C. M."https://zbmath.org/authors/?q=ai:pezzott.julio-c-mSummary: Let \(G\) be a finite non-abelian group and let \(T\) be a transversal of the center of \(G\) in \(G\).
The non-commuting graph of \(G\) on a transversal of the center is the graph whose vertices are the non-central elements of \(T\) and two vertices \(x\) and \(y\) are joined by an edge whenever \(xy \neq yx\). In this paper, we classify the groups whose non-commuting graph on a transversal of the center is projective.On the orbital regular graph of finite solvable groupshttps://zbmath.org/1541.050842024-09-27T17:47:02.548271Z"Sharma, Karnika"https://zbmath.org/authors/?q=ai:sharma.karnika"Bhat, Vijay Kumar"https://zbmath.org/authors/?q=ai:bhat.vijay-kumar"Singh, Pradeep"https://zbmath.org/authors/?q=ai:singh.pradeep.1Summary: Let \(G\) be a finite solvable group and \(\Delta\) be the subset of \(\Upsilon \times \Upsilon\), where \(\Upsilon\) is the set of all pairs of size two commuting elements in \(G\). If \(G\) operates on a transitive \(G\)-space by the action \((\upsilon_1, \upsilon_2)^g = (\upsilon_1^g, \upsilon_2^g)\); \(\upsilon_1,\upsilon_2\in\Upsilon\) and \(g \in G\), then orbits of \(G\) are called orbitals. The subset \(\Delta_o = \{(\upsilon, \upsilon); \upsilon\in\Upsilon, (\upsilon, \upsilon) \in\Upsilon\times\Upsilon\}\) represents \(G\)'s diagonal orbital. The orbital regular graph is a graph on which \(G\) acts regularly on the vertices and the edge set. In this paper, we obtain the orbital regular graphs for some finite solvable groups using a regular action. Furthermore, the number of edges for each of a group's orbitals is obtained.A census of small Schurian association schemeshttps://zbmath.org/1541.051832024-09-27T17:47:02.548271Z"Lansdown, Jesse"https://zbmath.org/authors/?q=ai:lansdown.jesseAuthor's abstract: Using the classification of transitive groups of degree \(n\), for \(2\leq n\leq 48\), we classify the Schurian association schemes of order \(n\), and as a consequence, the transitive groups of degree \(n\) that are 2-closed. In addition, we compute the character table of each association scheme and provide a census of important properties. Finally, we compute the 2-closure of each transitive group of degree \(n\), for \(2\leq n\leq 48\). The results of this classification are made available as a supplementary database.
Reviewer: Jack Koolen (Hefei)The group of boundary fixing homeomorphisms of the disc is not left-orderablehttps://zbmath.org/1541.060052024-09-27T17:47:02.548271Z"Hyde, James"https://zbmath.org/authors/?q=ai:hyde.james-tSummary: A left-order on a group \(G\) is a total order \(<\) on \(G\) such that for any \(f, g\) and \(h\) in \(G\) we have \(f<g\Leftrightarrow hf<hg\). We construct a finitely generated subgroup \(H\) of \(\mathrm{Homeo}(I^2;\delta I^2)\), the group of those homeomorphisms of the disc that fix the boundary pointwise, and show \(H\) does not admit a left-order. Since any left-order on \(\mathrm{Homeo}(I^2;\delta I^2)\) would restrict to a left-order on \(H\), this shows that \(\mathrm{Homeo}(I^2;\delta I^2)\) does not admit a left-order. Since \(\mathrm{Homeo}(I;\delta I)\) admits a left-order, it follows that neither \(H\) nor \(\mathrm{Homeo}(I^2;\delta I^2)\) embed in \(\mathrm{Homeo}(I;\delta I)\).Orders on free metabelian groupshttps://zbmath.org/1541.060062024-09-27T17:47:02.548271Z"Wang, Wenhao"https://zbmath.org/authors/?q=ai:wang.wenhaoA bi-order on a group \(G\) is a total, bi-multiplication invariant order. A subset \(S\) in an ordered group \((G, \leq)\) is convex if for all \(f \leq g\) in \(S\), every element \(h \in G\) satisfying \(f \leq h \leq g\) belongs to \(S\).
In the paper under review, the author studies the convex hull of the derived subgroup of a free metabelian group with respect to a bi-order, where the convex hull \(\overline{H}\) of a subgroup \(H\) is the smallest convex subgroup containing \(H\). Let \(M_{n}\) be the free metabelian group of rank \(n\), the first two main result are the following:
Theorem A: \(M_{2}'\) is convex with respect to any bi-invariant order on \(M_{2}\).
Theorem B: Let \(\leq\) be a bi-invariant order on \(M_{n}\) with \(n \geq 3\). Then the rank of \(M_{n}/\overline{M_{n}'}\) is greater or equal to 2, where \(\overline{M_{n}'}\) is the convex hull (with respect to \(\leq\)) of the derived subgroup of \(M_{n}\).
Other results concern the space \(\mathcal{O}(G)\) of all bi-orders in \(G\). In particular, the author proves Theorem C: The space \(\mathcal{O}(M_{n})\) is homeomorphic to the Cantor set for \(n \geq 2\). In addition, he shows that no bi-order for free metabelian groups can be recognised by a regular language (Theorem D).
Reviewer: Enrico Jabara (Venezia)Local permutation polynomials and the action of e-Klenian groupshttps://zbmath.org/1541.111032024-09-27T17:47:02.548271Z"Gutierrez, Jaime"https://zbmath.org/authors/?q=ai:gutierrez.jaime"Jiménez Urroz, Jorge"https://zbmath.org/authors/?q=ai:jimenez-urroz.jorgeLet \(q\) be a power of prime \(p\), \(\mathbb{F}_q\) be the finite field with \(q\) elements and \(\mathbb{F}_q^n\) denote the cartesian product of \(n\) copies of \(\mathbb{F}_q\), for any integer \(n\geq 1\). Let \(\overline{x}=(x_1,x_2,\ldots, x_n)\) and \(\overline{x}_i=(x_1,\ldots, x_{i-1},x_{i+1},\ldots, x_n)\).
We say that a polynomial \(f\in \mathbb{F}_q[\overline{x}]\) is a \textit{permutation polynomial} if the equation \(f(\overline{x})=a\) has \(q^{n-1}\) solutions in \(\mathbb{F}_q^n\) for each \(a\in \mathbb{F}_q\).
A polynomial \(f\in \mathbb{F}_q[\overline{x}]\) is called a \textit{local permutation polynomial} (or LPP) if for each \(i\), \(1\leq i\leq n\), the polynomial \(f(a_1,\ldots, a_{i-1},x_i,a_{i+1},\ldots, a_n)\) is a permutation polynomial in \(\mathbb{F}[x_i]\), for all choices of \(\overline{a}_i\in \mathbb{F}_q^{n-1}\). Clearly any LPP is a permutation polynomial. The opposite is not true in general.
One of the main contributions in the first part of this paper is a general construction of a family of local permutation polynomials based on a class of symmetric subgroups without fixed points, the so called e-Klenian groups.
In the second part of the paper, the authors are interested in Latin Squares, namely \(t\times t\) matrices with entries from a set \(T\) of size \(t\) such that each element of \(T\) occurs exactly once in every row and every column of the matrix. The authors use the fact that bivariate local permutation polynomials define Latin Squares, to discuss several constructions of Mutually Orthogonal Latin Squares (MOLS) and, in particular, they provide a new construction of MOLS on size a prime power.
Reviewer: Neranga Fernando (Worcester)Pseudo-quotients of algebraic actions and their application to character varietieshttps://zbmath.org/1541.140662024-09-27T17:47:02.548271Z"González-Prieto, Ángel"https://zbmath.org/authors/?q=ai:gonzalez-prieto.angelLet \(G\) be an affine algebraic group acting rationally on an algebraic variety \(X\), with structure sheaf \(\mathcal{O}\), over an algebraically closed field \(\Bbbk\). We call \(X\) a \(G\)-variety.
In this interesting paper, a \textit{pseudo-quotient} is defined to be a surjective \(G\)-invariant regular morphism \(\pi : X \to Y\) such that \(\overline{\pi(W_1)} \cap \overline{\pi(W_2)} = \emptyset\) for any disjoint \(G\)-invariant closed sets \(W_1, W_2 \subset X\).
As part of standard Geometric Invariant Theory (GIT), a \textit{good} quotient is a \(G\)-invariant morphism \(\pi:X\to Y\) such that:
\begin{enumerate}
\item[(1)] for all open \(U\subset Y\), \(\pi\) induces an isomorphism of rings \(\mathcal{O}(U)\cong \mathcal{O}(\pi^{-1}(U))^G\),
\item[(2)] \(\pi(W)\) is closed for all \(G\)-invariant closed \(W\subset X\), and
\item[(3)] \(\pi(W_1) \cap \pi(W_2) = \emptyset\) for any disjoint \(G\)-invariant closed sets \(W_1, W_2 \subset X\).
\end{enumerate}
Conditions (1)--(3) imply \(\pi:X\to Y\) is a categorical quotient. We note that condition (3) implies \(\pi\) is dominant and then condition (2) implies it is surjective.
The author of the paper under review shows that pseudo-quotients satisfy condition (2) and so they only may differ from good quotients at condition (1).
Post-composing any pseudo-quotient with a regular bijection (not necessarily an isomorphism), gives another pseudo-quotient. This observation shows there are many examples of pseudo-quotients that are not good quotients (and so where condition (1) fails), and that unlike categorical quotients pseudo-quotients are not unique.
Let \(\mathrm{K}_0(\mathbf{Var}_\Bbbk)\) be the Grothendieck ring of algebraic varieties over \(\Bbbk\). It is generated by isomorphism classes of varieties \(X\), called \textit{virtual classes} and denoted by \([X]\), subject to the formal relations: \([X]:=[X-Y]+[Y]\) where \(Y\subset X\) is closed, and where addition is disjoint union and multiplication is Cartesian product. Often \([\mathbb{A}^1_\Bbbk]\) is called the \textit{Lefschetz motive}.
The non-uniqueness of pseudo-quotients is addressed by the following theorem: if \(\Bbbk\) has characteristic 0, and \(Y_1\) and \(Y_2\) are both pseudo-quotients of the \(G\)-variety \(X\), then \([Y_1]=[Y_2]\). Additionally, pseudo-quotients behave well with respect to natural stratifications of \(G\)-varieties. Together, these properties motivate why they are useful in motivic calculations of moduli spaces.
Next, let \(\Gamma\) be a finitely generated group and \(G\) a complex reductive affine algebraic group. Then \(\mathrm{Hom}(\Gamma, G)\) is an affine variety which \(G\) acts on rationally by conjugation. The affine GIT quotient (a good quotient) of \(\mathrm{Hom}(\Gamma, G)\) by the conjugation action of \(G\) is called the \textit{\(G\)-character variety of \(\Gamma\)}. The character variety is often denoted as \(\mathfrak{X}(\Gamma, G)\) where the symbol \(\mathfrak{X}\) represents ``chi'' standing for ``character'' and the order of the symbols \(\Gamma\) and \(G\) correspond to the order of those symbols in \(\mathrm{Hom}(\Gamma, G)\) since the character variety is the GIT quotient of the latter. We warn the reader that the author of the paper under review uses similar, but different, notation.
Let \(\Sigma_{n,g}\) be a connected, compact, orientable surface of genus \(g\geq 1\) with \(n\geq 0\) boundary components. The main computation of this paper, which utilizes the developed theory of pseudo-quotients by the author, gives explicit formulae for the virtual classes \([\mathfrak{X}(\pi_1(\Sigma_{n,g}),\mathrm{SL}(2,\mathbb{C}))]\). These formulae are all polynomial in the Lefschetz motive. The author also obtains formulae for virtual classes of some \textit{relative} \(\mathrm{SL}(2,\mathbb{C})\)-character varieties of \(\pi_1(\Sigma_{n,g})\); the latter being subvarieties of \(\mathfrak{X}(\pi_1(\Sigma_{n,g}),\mathrm{SL}(2,\mathbb{C}))\) cut out by fixing \(n\) conjugation classes at the \(n\) boundaries.
We end this review by noting that in the case when \(n\geq 1\) the author's formulae for \([\mathfrak{X}(\pi_1(\Sigma_{n,g}),\mathrm{SL}(2,\mathbb{C}))]\) agree with the \(E\)-polynomials obtained by the reviewer et al. in [\textit{S. Cavazos} and \textit{S. Lawton}, Int. J. Math. 25, No. 6, 27 p. (2014; Zbl 1325.14065)] and [\textit{S. Lawton} and \textit{V. Muñoz}, Pac. J. Math. 282, No. 1, 173--202 (2016; Zbl 1335.14003)]. This is no accident. Indeed, the stratification by orbit-type used by the reviewer is exactly that used by the author. But to obtain the virtual classes the author needs to utilize the theory of pseudo-quotients in addition to this stratification. Along the same lines, the author's formulae when \(n=0\) agrees with the \(E\)-polynomials computed in [\textit{J. Martínez} and \textit{V. Muñoz}, Int. Math. Res. Not. 2016, No. 3, 926--961 (2016; Zbl 1353.14013)].
Reviewer: Sean Lawton (Fairfax)Degenerations of spherical subalgebras and spherical rootshttps://zbmath.org/1541.140722024-09-27T17:47:02.548271Z"Avdeev, Roman"https://zbmath.org/authors/?q=ai:avdeev.roman-sThe setting is the one of spherical varieties \(G/H\) over the complex numbers, where \(G\) is a connected reductive algebraic group. The main goal is to compute the set of \textit{spherical roots} of a given spherical subgroup \(H\) satisfying the following condition: the unipotent radical of \(H\) is contained in the unipotent radical of a parabolic subgroup \(P\); moreover, a Levi subgroup \(K \subset H\) satisfies \(L^\prime \subset K \subset L\), where \(L\) is a Levi subgroup of \(P\) and \(L^\prime\) its derived subgroup.
The class of spherical subgroups considered in this paper generalizes that of \textit{strongly solvable} subgroups (those contained in a Borel subgroup); for the latter class, the Luna-Vust invariants had already been computed.
A key point is that such a subgroup \(H\) is uniquely determined (up to conjugation by \(C\), the connected center of \(L\)) by the pair \((K, \Psi)\), where \(\Psi\) is the finite set of the so-called \textit{active \(C\)-roots} (Proposition 4.12).
The strategy (Section 3.9), which the authors call the \textit{base algorithm}, goes roughly as follows. First, they reduce to the case where the group \(G\) is semisimple and \(H\) coincides with its normalizer in \(G\) (Section 5.3).
Then, they consider the \textit{Demazure embedding}, namely they realize \(G/H\) as the orbit \(G \cdot \mathfrak{h}\) (of the Lie algebra of \(H\)) in the Grassmannian of \(\mathrm{dim}\mathfrak{h}\)-dimensional subspaces in \(\mathrm{Lie} G\) (Section 3.8).
If \(\mathfrak{h}\) degenerates into two subalgebras \(\mathfrak{h}_1\), \(\mathfrak{h}_2\) (by taking the limit under the action of two one-parameter subgroups), such that the corresponding orbits are distinct and both of codimension one, then the problem of finding the set of spherical roots of \(G\) reduces to the same problem for \(G/N_1\) and \(G/N_2\), where \(N_i\) is the stabilizer of the subalgebra \(\mathfrak{h}_i\).
If \(H\) has at least two active \(C\)-roots, then it is possible to construct two one-parameter subgroups satisfying the desired properties; moreover, \(N_i\) are still spherical. By following such an algorithm, one is reduced to the \textit{primitive} cases, namely those for which \(\vert \Psi \vert = 1\) (Section 5.6).
The base algorithm illustrated above being rather slow, an optimization is given, for spherical subgroups \(H\) satisfying \(K=L\) (Section 6).
Reviewer: Matilde Maccan (Rennes)Weighted homological regularitieshttps://zbmath.org/1541.160072024-09-27T17:47:02.548271Z"Kirkman, E."https://zbmath.org/authors/?q=ai:kirkman.ellen-e"Won, R."https://zbmath.org/authors/?q=ai:won.robert"Zhang, J. J."https://zbmath.org/authors/?q=ai:zhang.james-jLet \(A\) be a noetherian connected graded \(k\)-algebra with a balanced dualizing complex. For a weight \(\xi = (\xi_0,\xi_1) \in \mathbb{R}^2\) and a nonzero object \(X \in \mathsf{D}^\mathrm{b}_\mathrm{fg}(A\text{-Gr})\) the authors define the following invariants :
\begin{itemize}
\item The \(\xi\)-Castelnuovo-Mumford regularity, \(\mathrm{CMreg}_\xi(X)\)
\item The \(\xi\)-Ext-regularity, \(\mathrm{Extreg}_\xi(X)\)
\item The \(\xi\)-Tor-regularity, \(\mathrm{Torreg}_\xi(X)\)
\end{itemize}
They also define the \(\xi\)-Artin-Shelter-regularity of \(A\) as
\[
\mathrm{ASreg}_\xi(A) := \mathrm{Torreg}_\xi(k) + \mathrm{CMreg}_\xi(A).
\]
With weight \(\xi = (1,1)\) one obtains the (ordinary) Castelnuovo-Mumford, Tor, Ext, and Artin-Shelter regularities that have already been studied in the existing literature (the relevant references can be found in the paper). Main results in the paper give various (in)equalities between the weighted regularities introduced above. For example, if \(\xi_0>0\), then one has:
\[
\mathrm{Torreg}_\xi(X) \,=\, \mathrm{Extreg}_\xi(X) \,\leq\, \mathrm{CMreg}_\xi(X) + \mathrm{Torreg}_\xi(k)
\]
and
\[
\mathrm{CMreg}_\xi(X) \,\leq\, \mathrm{Extreg}_\xi(X) + \mathrm{CMreg}_\xi(A).
\]
With \(X=A\) it follows that \(\mathrm{ASreg}_\xi(A) \geq 0\) holds. Further, if \(X\) has finite projective dimension (and still \(\xi_0>0\)), then there is an equality,
\[
\mathrm{CMreg}_\xi(X) \,=\, \mathrm{Torreg}_\xi(X) + \mathrm{CMreg}_\xi(A),
\]
provided that \(0 \leq \xi_1 \leq \xi_0\) or that \(\xi_1 \ll 0\).
The authors also show that the \(\xi\)-Castelnuovo-Mumford regularity can be used to detect Artin-Shelter regularity of the algebra in various ways. For example, \(A\) is Artin-Shelter regular if and only if \(A\) is Cohen-Macaulay and there exists a \(\xi = (\xi_0,\xi_1)\) with \(\xi_0>0\) such that \(\mathrm{ASreg}_\xi(A) = 0\).
The results mentioned above generalize various known results from the literature (the relevant references can be found in the paper).
Reviewer: Henrik Holm (København)Deformation cohomology for cyclic groups acting on polynomial ringshttps://zbmath.org/1541.160112024-09-27T17:47:02.548271Z"Lawson, Colin M."https://zbmath.org/authors/?q=ai:lawson.colin-m"Shepler, Anne V."https://zbmath.org/authors/?q=ai:shepler.anne-vHochschild cohomology governs deformations of algebras: Every deformation arises from a Hochschild \(2\)-cocycle, which is called an infinitesimal deformation. When the algebra is graded, the Hochschild cohomology inherits the grading, and graded deformations all arise from infinitesimal deformations of degree \(-1\). The main result of the present paper is the classification of infinitesimal deformations of degree \(-1\) of the skew group algebra \(S(V)\rtimes G\) when the characteristic of the underlying field is not \(2\). Here \(G\) is a finite cyclic group acting on a finite dimensional vector space \(V\), and \(S(V)\) is the symmetric algebra of \(V\).
In order to achieve the goal, the authors construct a twisted product resolution for \(S(V)\rtimes G\) using a periodic resolution for the cyclic group \(G\) and the Koszul resolution for \(S(V)\). With the help of this resolution, the authors decompose the space of infinitesimal deformations of degree \(-1\) into contributions from each group element. Then they discuss the cocycle conditions in terms of the codimension of fixed point spaces. It turns out that only group elements with fixed point spaces of codimension at most \(2\) (identity, reflections, and bireflections) contribute to this space of infinitesimal deformations.
The present paper recovers the known description of the space of infinitesimal deformations of degree \(-1\) in the nonmodular setting when the underlying field is algebraically closed. It also demonstrates that the main result can be used to lift an infinitesimal deformation to a deformation.
Reviewer: Yining Zhang (Singapura)The RFD and Kac quotients of the Hopf\(^\ast\)-algebras of universal orthogonal quantum groupshttps://zbmath.org/1541.160272024-09-27T17:47:02.548271Z"Das, Biswarup"https://zbmath.org/authors/?q=ai:das.biswarup"Franz, Uwe"https://zbmath.org/authors/?q=ai:franz.uwe"Skalski, Adam"https://zbmath.org/authors/?q=ai:skalski.adam-gSummary: We determine the Kac quotient and the RFD (residually finite dimensional) quotient for the Hopf\(^\ast\)-algebras associated to universal orthogonal quantum groups.(In)decomposability of finite solutions of the Yang-Baxter equationhttps://zbmath.org/1541.160312024-09-27T17:47:02.548271Z"Kanrar, Arpan"https://zbmath.org/authors/?q=ai:kanrar.arpanThe author extends a result on the decomposability of a finite non-degenerate involutive solution of the Yang-Baxter equation and studies some examples in which the conditions for (in)decomposability are not necessary.
Reviewer: J. N. Alonso Alvarez (Vigo)A note on square-free commuting probabilities of finite ringshttps://zbmath.org/1541.160322024-09-27T17:47:02.548271Z"Mendelsohn, Andrew"https://zbmath.org/authors/?q=ai:mendelsohn.andrewThis short but interesting paper deals with the square-free commuting probabilities of finite rings. It is proved in Theorems 3.1 and 3.2 that the commuting probability of a finite ring cannot be a fraction with square-free denominator, thus resolving an unpublished conjecture due to \textit{S. Buckley} and \textit{D. MacHale} (see cf. [``Groups with \(\mathrm{Pr}(G)= 1/3\)'', Preprint] and [``Contrasting the commuting probabilities of groups and rings'', Preprint], respectively).
Reviewer: Peter Danchev (Sofia)Cyclotomic \(q\)-Schur superalgebrashttps://zbmath.org/1541.160372024-09-27T17:47:02.548271Z"Zhao, Deke"https://zbmath.org/authors/?q=ai:zhao.dekeSummary: The paper aims to introduce the cyclotomic \(q\)-Schur superalgebra via the permutation supermodules of the cyclotomic Hecke algebra and investigate its structure. In particular, we show that the cyclotomic \(q\)-Schur superalgebra is a cellular superalgebra and establish the double centralizer property between the cyclotomic Hecke algebra and the cyclotomic \(q\)-Schur superalgebra.Some nil-ai-semiring varietieshttps://zbmath.org/1541.160402024-09-27T17:47:02.548271Z"Wu, Y. N."https://zbmath.org/authors/?q=ai:wu.yingnian|wu.yina|wu.yana.1|wu.yan-ning|wu.yunna|wu.yunnan|wu.yuening|wu.yanni|wu.yuning|wu.yunan|wu.yinan|wu.yanan|wu.yining|wu.yan-na"Zhao, X. Z."https://zbmath.org/authors/?q=ai:zhao.xinzhu|zhao.xuanze|zhao.xuezeng|zhao.xizeng|zhao.xiaozhi|zhao.xianzhang|zhao.xianzhong|zhao.xuezhuang|zhao.xinzhong|zhao.xuzhe|zhao.xiangzhong|zhao.xuezhi|zhao.xianzeng|zhao.xiaozhou|zhao.xiaozhe|zhao.xinzeSummary: We study some nil-ai-semiring varieties. We establish a model for the free object in the variety \textbf{FC} generated by all commutative flat semirings. Also, we provide two sufficient conditions under which a finite ai-semiring is nonfinitely based. As a consequence, we show that the power semiring \(P_{\dot{S}_c(W)}\) of the finite nil-semigroup \({\dot{S}}_c(W)\) is nonfinitely based, where \(W\) is a finite set of words in the free commutative semigroup \(X_c^+\) over an alphabet \(X\), whenever the maximum of lengths of words in \(W\) is \(k\ge 3\) and \(W\) does not contain the \(k\)th power of a letter. This partially answers a problem raised by
\textit{M. Jackson} et al. [J. Algebra 611, 211--245 (2022; Zbl 07594495)].\(U^0\)-free quantum group representationshttps://zbmath.org/1541.170072024-09-27T17:47:02.548271Z"Chen, Hongjia"https://zbmath.org/authors/?q=ai:chen.hongjia"Gao, Yun"https://zbmath.org/authors/?q=ai:gao.yun"Liu, Xingpeng"https://zbmath.org/authors/?q=ai:liu.xingpeng"Wang, Longhui"https://zbmath.org/authors/?q=ai:wang.longhuiSummary: Let \(\mathfrak{g}\) be a finite dimensional complex simple Lie algebra, and let \(U = U_q(\mathfrak{g})\) be its quantized enveloping algebra with a triangular decomposition \(U = U^- U^0 U^+\). We classify all \(U\)-module structures on \(U^0\) with the regular action of \(U^0\) on itself, which are shown to have direct connections with the \(q\)-Weyl algebra and the bounded weight representations. We obtain that the necessary and sufficient condition for the existence of such modules is that \(U\) has to be of type \(A_n(n \geq 1)\), \(B_n(n \geq 2)\) or \(C_n(n \geq 3)\). Moreover, we study their module structures and associated weight modules for each type.Quotients of the Booleanization of an inverse semigrouphttps://zbmath.org/1541.180052024-09-27T17:47:02.548271Z"Kudryavtseva, Ganna"https://zbmath.org/authors/?q=ai:kudryavtseva.gannaA \textit{representation} of a semilattice \(E\) in a Boolean algebra \(B\) is a map \(\varphi\colon E\to B\) such that \(\varphi(0)= 0\) and \(\varphi(x\wedge y)=\varphi(x)\wedge \varphi(y)\) for all \(x, y\in E\). A representation is called \textit{proper} if for any \(y\in B\) there are \(x_1,\ldots, x_n\in E\) such that \(\varphi(x_1)\vee\ldots\vee\varphi(x_n)\geq y\). Let \(X\subseteq E\times \mathcal{P}_{fin}(E)\), where \(\mathcal{P}_{fin}(E)\) denotes the set of all finite subsets of \(E\). A representation \(\varphi\colon E\to B\) is called \textit{X-to-join}, if it is proper and \(\varphi(e)=\bigvee^n_{i=1}\varphi(e_i)\) for all \((e,\{e_1,\ldots, e_n\})\in X\). The set \(B_X(E)\) of compact-opens of the space of all \(X\)-to-join characters of \(E\) (i.e. nonzero representations of \(E\) in \(\{0, 1\}\)) forms a Boolean algebra which is called the \textit{\(X\)-to-join Booleanization} of \(E\). It is proved that if \(\varphi\colon E\to B\) is an \(X\)-to-join representation, then there is a unique morphism of Boolean algebras \(\psi\colon B_X(E)\to B\) such that \(\varphi=\psi \iota_{B_X(E)}\), where \(\iota_{B_X(E)}\colon E\to B_X(E)\) is an \(X\)-to-join representation of \(E\) in \(B_X(E)\). Universal \(X\)-to-join Booleanization of an inverse semigroup \(S\) as a weakly meet-preserving quotient of the universal Booleanization \(B(S)\) is constructed. It turnes out that all such quotients of \(B(S)\) arise via \(X\)-to-join representations. An inverse semigroup \(S\) is called \textit{Boolean} if \(E(S)\) admits the structure of a Boolean algebra, and satisfies one additional condition. For inverse semigroups, prime and core representations present examples of \(X\)-to-join representations. The results are applied to \(C^*\)-algebras of right LCM monoids (left cancellative monoids where the intersection of any two principal right ideals is either empty or a principal right ideal).
For the entire collection see [Zbl 1470.20003].
Reviewer: Peeter Normak (Tallinn)Quotient toposes of discrete dynamical systemshttps://zbmath.org/1541.180102024-09-27T17:47:02.548271Z"Hora, Ryuya"https://zbmath.org/authors/?q=ai:hora.ryuya"Kamio, Yuhi"https://zbmath.org/authors/?q=ai:kamio.yuhiThe purpose of the present paper is to describe the \textit{quotients} of the topos of discrete dynamical systems. A pair \((X, f)\) where \(X\) is a set and \(f: X\to X\) is a function is called a discrete dynamical system and a morphism from \((X, f)\) to another discrete dynamical system \((Y, g)\) is a function \(h: X \to Y\) such that \(hf = gh.\) In categorical terms, we get a category which is equivalent to the presheaf topos \(\textbf{Set}^{\mathbb{N}^{\textrm{op}}},\) where \(\mathbb{N}=\{0, 1, 2, \cdots \}\) is the additive monoid of natural numbers considered as a category with just one object.
A quotient of a topos \(\mathcal{E}\) is a full subcategory \(\mathcal{Q}\) of \(\mathcal{E}\), whose inclusion functor \(\iota : \mathcal{Q}\hookrightarrow \mathcal{E}\) preserves finite limits (i.e., is left exact) and has a right adjoint \(\pi : \mathcal{E}\to \mathcal{Q}\). A prequotient of a cocomplete topos \(\mathcal{E}\) is a full subcategory of \(\mathcal{E}\), which is closed under taking isomorphic objects, finite limits, and small colimits. Then, \(Q_{\mathcal{E}}\) and \(PQ_{\mathcal{E}}\) are denoted the posets of quotients and prequotients of the topos \(\mathcal{E}\) with the inclusion order, respectively.
In this paper the authors prove that there is a natural bijective correspondence between quotient toposes of \(\textbf{Set}^{\mathbb{N}^{\textrm{op}}}\) and (semilattice) ideals of \(\mathbb{N}\times \mathbb{N}^{\textrm {div}}\), where the poset \(\mathbb{N}\) endowed with the natural partial order and \(\mathbb{N}^{\textrm {div}}\) denotes the poset of all natural numbers equipped with the divisibility partial order. Furthermore, it defines the poset isomorphism
\[
PQ_{\textbf{Set}^{\mathbb{N}^{\textrm{op}}}}\cong Q_{\textbf{Set}^{\mathbb{N}^{\textrm{op}}}} \cong \textrm{Ideal} (\mathbb{N}\times \mathbb{N}^{\textrm {div}}).
\]
In the last section of the paper they show that if two quotients of \(\textbf{Set}^{\mathbb{N}^{\textrm{op}}}\) contain exactly the same essential quotients, then they are equal to each other. This result, suggest that the study of lax epimorphisms, which correspond to essential quotients, could play a crucial role in solving the first open problem proposed by FW. Lawvere related to quotient of Grothendieck toposes. The paper concludes with three Appendices regarding to embeddings and dense order-preserving functions, some classes of geometric morphisms, and monoid epimorphisms from \(\mathbb{N}\).
Reviewer: Ali Madanshekaf (Semnan)Presentations for tensor categorieshttps://zbmath.org/1541.180202024-09-27T17:47:02.548271Z"East, James"https://zbmath.org/authors/?q=ai:east.jamesPresentations are a tool that allow to break complex structures into simple building blocks (generators) and local moves (relations) that determine equivalence of representations by generators.
One of the main goals of the current work is to introduce a systematic and completely rigorous framework for dealing with presentations for a large class of tensor categories. The authors show how presentations for such categories can be built from presentations for endomorphism monoids and certain one-sided units and then how to rewrite these into tensor presentations.
They then illustrate this general framework on several examples, recovering examples from the literature while introducing new ones. When applying it to diagram categories they get presentations for the partition category, the Brauer category, and the Temperley-Lieb category. They also treat a number of categories related to Artin braid groups: the partial vine category, the partial braid category, and the (full) vine category.
Reviewer: Nicolas Blanco (Birmingham)Free and co-free constructions for Hopf categorieshttps://zbmath.org/1541.180222024-09-27T17:47:02.548271Z"Großkopf, Paul"https://zbmath.org/authors/?q=ai:grosskopf.paul"Vercruysse, Joost"https://zbmath.org/authors/?q=ai:vercruysse.joostLet \(\mathcal{V}\) be a monoidal base category. The paper under review presents local representability results for Hopf categories enriched in \(\mathcal{V}\), and gives conditions for the existence of free and cofree constructions.
A \textit{semi-Hopf \(\mathcal{V}\)-category} in the sense of \textit{E. Batista} et al. [Algebr. Represent. Theory 19, No. 5, 1173--1216 (2016; Zbl 1373.16057)] is a symmetric monoidal \(\mathrm{Coalg}(\mathcal{V})\)-category, where \(\mathrm{Coalg}(\mathcal{V})\) is the category of coalgebra (comonoid) objects in \(\mathcal{V}\). Such a category is \textit{Hopf} if there exists a suitable antipode. Hopf categories can be seen to generalise groupoids on one hand -- in that a Hopf \(\mathsf{Set}\)-category is nothing but a groupoid -- and Hopf algebras and algebroids over commutative base rings on the other; see Section~7 of [loc. cit.]. Further, as shown in [\textit{G. Böhm}, Theory Appl. Categ. 32, 1229--1257 (2017; Zbl 1407.18005)], Hopf categories can be seen as Hopf monads in the sense of [\textit{A. Bruguières} et al., Adv. Math. 227, No. 2, 745--800 (2011; Zbl 1233.18002)] over certain categories of spans.
One of the paper's key results is the generalisation of the classical fact that the category of Hopf algebras is a \textit{reflective} and \textit{coreflective} subcategory of the category of bialgebras -- the inclusion functor \(\mathsf{Hopf} \hookrightarrow \mathsf{Bialg}\) has both a left and a right adjoint. For a suitably nice base category \(\mathcal{V}\), the authors show that this property lifts to the ''many object case'', in the form of the inclusion \(\mathcal{V}\text{-}\mathsf{Hopf} \hookrightarrow \mathcal{V}\text{-}\mathsf{sHopf}\) of Hopf \(\mathcal{V}\)-categories into semi-Hopf \(\mathcal{V}\)-categories having a left and a right adjoint. These adjoints are furthermore explicitly constructed and studied.
Section~1 recalls the definitions and important properties of locally presentable categories, \(\mathcal{V}\)-categories, and \(\mathcal{V}\)-graphs.
Section~2 introduces the important concept of very flat monoidal products. Recall that a bimodule \(M\) over a ring \(R\) is called (left) \textit{flat} if the functor \({-} \otimes_R M\) is exact. Equivalently, this is the case when \({-} \otimes_R M\) preserves monomorphisms. The authors call an object \(A\) in a monoidal category \(\mathcal{V}\) \textit{left very flat} if \({-} \otimes A\) preserves all families of jointly monic morphisms, and the tensor product on \(\mathcal{V}\) is \textit{left very flat} if all of \(\mathcal{V}\)'s objects are. In the symmetric case, any directional prefix can be dropped, as left and right very flat coincide. In the locally presentable closed (non-symmetric) monoidal very flat setting, Theorem~2.6 proves the existence of the cofree coalgebra \(T^c(V)\) of an object \(V \in \mathcal{V}\) as a certain subobject of the product \(\prod_{k \in \mathbb{N}} V^{\otimes k}\).
Section~3 introduces semi-Hopf categories, and establishes (co)monadicity and representability results of these categories over a closed symmetric monoidal and locally presentable base. Further, Hopf categories are studied. Proposition~3.8 states that, over a symmetric monoidal locally presentable category with very flat monoidal product, the category of Hopf \(\mathcal{V}\)-categories is a complete and cocomplete full subcategory of the category of semi Hopf \(\mathcal{V}\)-categories. If, in addition, the base category \(\mathcal{V}\) is closed monoidal, Theorem~3.9 establishes that \(\mathcal{V}\text{-}\mathsf{Hopf}\) is a reflective and coreflective subcategory of \(\mathcal{V}\text{-}\mathsf{sHopf}\). The authors call these adjoints the \textit{free} and \textit{cofree} Hopf category of a semi-Hopf \(\mathcal{V}\)-category, respectively.
Section~4 describes the aforementioned left and right adjoint of the inclusion functor in greater detail. The hom-objects for the free and cofree Hopf category over a closed symmetric monoidal locally presentable base with very flat monoidal product are explicitly constructed.
Section~5 presents a visual conclusion in the form of a large diagram, detailing all categories studied in the paper, as well as their relationship via adjunctions.
Reviewer: Tony Zorman (Dresden)Koszulity of dual braid monoid algebras via cluster complexeshttps://zbmath.org/1541.180262024-09-27T17:47:02.548271Z"Josuat-Vergès, Matthieu"https://zbmath.org/authors/?q=ai:josuat-verges.matthieu"Nadeau, Philippe"https://zbmath.org/authors/?q=ai:nadeau.philippeGiven a finite Coxeter group \(W\), the associated dual braid monoid \(D(W)\) was introduced by Bessis in his work on complex reflection arrangements, as a certain generating set of positive elements inside the braid group \(B(W)\). The authors of this paper show that there is an algebraic relation between the dual braid monoid and the positive part of the cluster complex via the notion of Koszul duality. They first prove the koszulity of the dual braid monoid algebra by building explicitly the minimal free resolution of the ground field. This is done by using certain chain complexes defined in terms of the positive part of the cluster complex. Furthermore, they derive various properties of the quadratic dual algebra. They show that it is naturally graded by the non-crossing partition lattice. Then they obtain an explicit basis which is naturally indexed by positive faces of the cluster complex. Moreover, They find the structure constants via a geometric rule in terms of the cluster fan. Eventually, they realize this dual algebra as a quotient of a Nichols algebra which in turn provides a connection with the results of Zhang who used the same algebra to compute the homology of Milnor fibers of reflection arrangements.
Reviewer: Hossein Faridian (Clemson)Simple functors over the Green biset functor of section Burnside ringshttps://zbmath.org/1541.190012024-09-27T17:47:02.548271Z"Coşkun, Olcay"https://zbmath.org/authors/?q=ai:coskun.olcay"Muslumov, Ruslan"https://zbmath.org/authors/?q=ai:muslumov.ruslanGreen biset functors are introduced by Serge Bouc as a general framework to work with bisetfunctors having an additional ring structure. Well-known examples include the functors of fibered Burnside rings, trivial source rings and character rings. Serge Bouc also introduced two generalizations of Burnside rings, the so-called slice and section Burnside rings which are also Green biset functors. In this paper, the authors classify simple modules over the Green biset functor of section Burnside rings.
Reviewer: Chen Sheng (Harbin)Group theory -- what's beyondhttps://zbmath.org/1541.200012024-09-27T17:47:02.548271Z"Sury, B."https://zbmath.org/authors/?q=ai:sury.balasubramanianFor the entire collection see [Zbl 1242.00057].Group representations and algorithmic complexityhttps://zbmath.org/1541.200022024-09-27T17:47:02.548271Z"Subrahmanyam, K. V."https://zbmath.org/authors/?q=ai:subrahmanyam.k-venkataFor the entire collection see [Zbl 1242.00057].Alternating groups with inherited \(G\)-permutation subgrouphttps://zbmath.org/1541.200032024-09-27T17:47:02.548271Z"Vasilyev, A. F."https://zbmath.org/authors/?q=ai:vasilev.alexander-fedorovich"Tyutyanov, V. N."https://zbmath.org/authors/?q=ai:tyutyanov.valentin-nikolaevichSummary: It has been established that any finite simple non-abelian alternating group doesn't have any inherited \(G\)-permutation subgroup.Compositional formations with systems of complemented subformationshttps://zbmath.org/1541.200042024-09-27T17:47:02.548271Z"Bliznets, I. V."https://zbmath.org/authors/?q=ai:bliznets.i-vSummary: The paper studies compositional formations with systems of complemented subformations. In particular, the following theorem is proved: \par Let \(\mathfrak{I}\) be a compositional formation, \(\mathfrak{I}\ne(1)\). Then the following conditions are equivalent:\par (1) each atom lattice \(L_c(\mathfrak{I})\) is complemented in the lattice \(L(\mathfrak{I})\); \par 2) each \(G\) group from \(\mathfrak{I}\) has an expansion as follows, \[G=A\times A_1\times\cdots\times A_t,\] where \(A\) is a nilpotent subgroup in \(G\), while \(A_1,\dots,A_t\) being simple non-abelian groups.About the maximal subgroups close to \(\mathfrak{I}\)-abnormal oneshttps://zbmath.org/1541.200052024-09-27T17:47:02.548271Z"Borodich, E. N."https://zbmath.org/authors/?q=ai:borodich.e-n"Borodich, R. V."https://zbmath.org/authors/?q=ai:borodich.ruslan-viktorovich"Selkin, M. V."https://zbmath.org/authors/?q=ai:selkin.m-vSummary: The intersections of maximal subgroups in groups of operators are investigated in the article.On the intersection of A-admissible subgroups not belonging to a given class of groupshttps://zbmath.org/1541.200062024-09-27T17:47:02.548271Z"Borodich, R. V."https://zbmath.org/authors/?q=ai:borodich.ruslan-viktorovich"Selkin, M. V."https://zbmath.org/authors/?q=ai:selkin.m-vSummary: The intersections of the given systems of maximal subgroups of finite groups are investigated.Classification of elements of height 3 lattice \(\tau_{ab}\)-closed \(\omega\)-saturated formations of finite groupshttps://zbmath.org/1541.200072024-09-27T17:47:02.548271Z"Dergacheva, I. M."https://zbmath.org/authors/?q=ai:dergacheva.i-mSummary: All groups considered are finite. Only finite groups are considered in the paper. The purpose of the paper is to describe elements of the height 3 of the lattice of all \(\tau\)-closed \(\omega\)-saturated formations in the case when \(\tau(G)\) is a set of all abnormal groups \(G\) for any group \(G\).Superradical formationshttps://zbmath.org/1541.200082024-09-27T17:47:02.548271Z"Kamornikov, S. F."https://zbmath.org/authors/?q=ai:kamornikov.sergey-fedorovichSummary: The problem of description of superradical formations is proposed. Some general directions of its solution are discussed. Attention is focused on the connection between the theory of hereditary saturated superradical formations with the theory of critical groups. A classification of such groups is given. Hereditary saturated superradical formations with special classes of critical groups are described. A great contribution of L. A. Shemetkov to the theory of superradical formations is marked. A number of open issues and challenges that stimulate the further development of the theory superradical formations are formulated.On \(p\)-solvability of some normal subgroups of finite groupshttps://zbmath.org/1541.200092024-09-27T17:47:02.548271Z"Monakhov, V. S."https://zbmath.org/authors/?q=ai:monakhov.victor-stepanovich"Hodanovich, D. A."https://zbmath.org/authors/?q=ai:khodanovich.d-aSummary: Let \(K\) be a normal subgroup of a group \(G\) and \(p\) be a prime. Suppose that \(K\) is \(S_4\)-free for \(p=2\). We prove that \(K\) is \(p\)-solvable, if for every maximal subgroup \(M\) of \(G\), not containing \(K\), the intersection \(K\cap M\) is \(p\)-nilpotent.On the products of normal hypersolvable subgroups of finite groupshttps://zbmath.org/1541.200102024-09-27T17:47:02.548271Z"Myslovets, E. N."https://zbmath.org/authors/?q=ai:myslovets.evgeniy-nikolaevichSummary: The article studies the products of normal \(Jc\)-hypersolvable and \(Jca\)-hypersolvable subgroups of finite groups.On \(\mathfrak{F}_S\)-embedded subgroup of finite grouphttps://zbmath.org/1541.200112024-09-27T17:47:02.548271Z"Ryzhik, V. N."https://zbmath.org/authors/?q=ai:ryzhik.v-n"Adarchenko, N. M."https://zbmath.org/authors/?q=ai:adarchenko.nikita-mSummary: Let \(H\) be a subgroup of a finite group \(G\). Let \(H^{sG}\) be the intersection of all \(S\)-permutable subgroups of \(G\) which contain \(H\) and let \(H_{sG}\) be a subgroup of \(H\) which is generated by all such subgroups of \(H\) which are \(S\)-permutable in \(G\). Let \(F\) be any class of groups. Then we say that a subgroup \(H\) is \(F_S\)-embedded in \(G\) provided \(G\) has a subgroup \(T\) such that \(T\in F\) and \(H^{sG}=H_{sG}T\). In the given paper we study the influence of the \(F_S\)-embedded subgroups on the structure of finite groups. In particular, we prove that a finite group \(G\) is supersoluble if and only if every its nilpotent subgroup is \(U_S\)-embedded in \(G\).Finite groups with generalized subnormal formational subgroupshttps://zbmath.org/1541.200122024-09-27T17:47:02.548271Z"Semenchuk, V. N."https://zbmath.org/authors/?q=ai:semenchuk.vladimir-nSummary: Final groups on the set properties of some system of subgroups, in a particular -- final groups, at which Silovsky subgroups generally субнормальны are considered. In a class of final solvable groups the formations closed concerning work generally of subnormal formational subgroups of mutually simple indexes are described.On intersection of \(A\)-admissible maximal subgroups not containing \(p\)-nilpotent radicalhttps://zbmath.org/1541.200132024-09-27T17:47:02.548271Z"Borodich, R. V."https://zbmath.org/authors/?q=ai:borodich.ruslan-viktorovich"Borodich, E. N."https://zbmath.org/authors/?q=ai:borodich.e-nSummary: The intersections of the given systems of maximal subgroups of finite groups are studied in this article.Finite groups with restrictions on maximal subgroups of Sylow subgroupshttps://zbmath.org/1541.200142024-09-27T17:47:02.548271Z"Kosenok, N. S."https://zbmath.org/authors/?q=ai:kosenok.n-sSummary: Let \(F\) be a class of groups. A subgroup \(H\) of a group \(G\) is called \(F\)-\(s\)-supplemented in \(G\) if there exists a subgroup \(K\) of \(G\) such that \(G=HK\) and \(K/K\cap H_G\) belongs to \(F\). We obtain some results about the \(F\)-\(s\)-supplemented subgroups, in particular, a new criteria for \(p\)-nilpotency is obtained.On finite groups with restriction on orders of non-bicyclic Sylow subgroups of some factorshttps://zbmath.org/1541.200152024-09-27T17:47:02.548271Z"Trofimuk, A. A."https://zbmath.org/authors/?q=ai:trofimuk.aleksandr-aleksandrovichSummary: Recall that a group is bicyclic if it is the product of two cyclic subgroups. The estimations of the derived length and the nilpotent length of solvable group \(G\) with restriction on orders of non-bicyclic Sylow subgroups of the factors chain \(\Phi(G)=G_0\subset G_1\subset\cdots\subset G_{m-1}\subset G_m=F(G)\), \(G_i\vartriangleleft G\) are obtained.On saturated formations of finite groups determined by the properties of Sylow subgroup attachmentshttps://zbmath.org/1541.200162024-09-27T17:47:02.548271Z"Vegera, A. S."https://zbmath.org/authors/?q=ai:vegera.a-sSummary: The properties of the class of finite groups with \(K\)-\(\mathfrak{F}\)-subnormal Sylow subgroups are studied in the article. It is shown that the class of such groups is a saturated formation.Finite groups with modular subgroups of order 4https://zbmath.org/1541.200172024-09-27T17:47:02.548271Z"Vasil'ev, A. F."https://zbmath.org/authors/?q=ai:vasilev.alexander-fedorovichSummary: All groups considered are finite. A subgroup \(M\) of a group \(G\) is a modular subgroup in \(G\), if the following conditions are true: \par (1) \(\langle X,M\cap Z\rangle=\langle X,M\rangle\cap Z\) for all \(X\le G,Z\le G\) with \(X\le Z\), and \par (2) \(\langle M,Y\cap Z\rangle=\langle M,Y\rangle\cap Z\) for all \(Y\le G,Z\le G\) with \(M\le Z\).\par Groups with modular subgroups of order 4 of Sylow subgroups are studied.On the Shemetkov-Schmid subgroup and related subgroups of finite groupshttps://zbmath.org/1541.200182024-09-27T17:47:02.548271Z"Vasil'ev, A. F."https://zbmath.org/authors/?q=ai:vasilev.alexander-fedorovich"Murashka, V. I."https://zbmath.org/authors/?q=ai:murashka.viachaslau-iSummary: In this paper the properties of the Shemetkov-Schmidt subgroup as well as generalized Fitting subgroups related with it have been determined. We call a subgroup \(HR\)-subnormal in a group \(G\), if \(H\) is subnormal in \(\langle H,R\rangle\). Finite groups with given systems of \(R\)-subnormal subgroups have been studied for \(R\in\{F(G),F^*(G)\}\). New characterizations of nilpotent and supersolvable groups have been obtained.On some relationships between the upper and lower central series in finite groupshttps://zbmath.org/1541.200192024-09-27T17:47:02.548271Z"Kurdachenko, L. A."https://zbmath.org/authors/?q=ai:kurdachenko.leonid-a"Semko, N. N."https://zbmath.org/authors/?q=ai:semko.nikolaj-n"Pypka, A. A."https://zbmath.org/authors/?q=ai:pypka.alexsandr-a|pypka.aleksand-aSummary: \textit{N. Yu. Makarenko} [Sib. Mat. Zh. 41, No. 6, 1376--1380 (2000; Zbl 0979.20022); translation in Sib. Math. J. 41, No. 6, 1137--1140 (2000)] obtained the bound on special rank of \((k+1)\)-th term \(\gamma_{k-1}(G)\) of the lower central series of a group \(G\) as a function of special rank of the factor-group of group \(G\) modulo the \(k\)-th term \(\zeta_k(G)\) of the upper central series. In this paper we obtained the bound on section \(p\)-rank of \(\gamma_{k-1}(G)\) as a function of section \(p\)-rank of the factor-group \(G/\zeta_k(G)\). It follows as corollary Makarenko's theorem.Finite almost simple 5-primary groups and their Gruenberg-Kegel graphshttps://zbmath.org/1541.200202024-09-27T17:47:02.548271Z"Kondrat'ev, A. S."https://zbmath.org/authors/?q=ai:kondratev.anatolij-sSummary: The finite 5-primary groups with disconnected prime graph are considered. The finite almost simple 5-primary groups and their Gruenberg-Kegel graphs are determined.Groups with some minimal conditions for non-normal subgroupshttps://zbmath.org/1541.200212024-09-27T17:47:02.548271Z"Chernikov, N. S."https://zbmath.org/authors/?q=ai:chernikov.nikolai-s|chernikov.n-sSummary: The groups, satisfying the minimal conditions for non-normal, abelian non-normal, non-abelian non-normal subgroups, are considered in the present article.On transitivity of normality and related topicshttps://zbmath.org/1541.200222024-09-27T17:47:02.548271Z"Kurdachenko, L. A."https://zbmath.org/authors/?q=ai:kurdachenko.leonid-a"Subbotin, I. Ya."https://zbmath.org/authors/?q=ai:subbotin.igor-yaSummary: A group \(G\) is said to be a \(T\)-group if the property ``to be a normal subgroup'' is transitive in \(G\). If every subgroup of \(G\) is a \(T\)-group, then \(G\) is called a \(\overline{T}\)-group. The authors consider some new developments in the infinite group theory related to the transitivity of normality.The picture on the presentation of direct product group of two cyclic groupshttps://zbmath.org/1541.200232024-09-27T17:47:02.548271Z"Yanita, Yanita"https://zbmath.org/authors/?q=ai:yanita.yanita"Rudianto, Budi"https://zbmath.org/authors/?q=ai:rudianto.budiSummary: A picture in a group presentation is a geometric configuration with an arrangement of discs and arcs within a boundary disc. The drawing of this picture does not have to follow a particular rule, only using the generator as discs and the relation as arcs. It will form a picture label pattern if drawn with a particular rule. This paper discusses the label pattern of a picture in the presentation of direct product groups. Direct product presentation is used with two cyclic groups, \(\mathbb{Z}_p\) and \(\mathbb{Z}_q\) where \(p, q\in\mathbb{Z}^+\) and \(p, q \geq 2\). The method for forming a picture label pattern is to arrange the first generator in the initial arrangement, compile a second generator, and add a number of commutators. Furthermore, the pattern is used to calculate the length of the label on the picture. It is obtained that the picture's label is \(a^{q - 1}b^nab^{q - n}\) and the length of the label is \(p+2n-q\), where \(n\) is the number of commutator discs.On \(\{2,3\}\)-groups in which there are no elements of order 6https://zbmath.org/1541.200242024-09-27T17:47:02.548271Z"Lytkina, D. V."https://zbmath.org/authors/?q=ai:lytkina.daria-viktorovna|lytkina.daria-v"Mazurov, V. D."https://zbmath.org/authors/?q=ai:mazurov.vladimir-d|mazurov.victor-danilovichSummary: In this paper, we study the \(\{2,3\}\)-groups which have no elements of order 6. The following theorem is proved. \par Theorem. Let \(G\) be an infinite non-primary \(\{2,3\}\)-group which have no elements of order 6. Suppose that every subgroup of \(G\) generated by elements of order 3 is finite. Then \(G\) has on of the following properties: \begin{itemize} \item[(1)] \(G=O_3(G)\cdot T\), where \(O_3(G)\ne 1\) is an abelian group, \(T\) is either a locally cyclic group or a locally quaternion 2-group which acts freely on \(O_3(G)\). \item[(2)] \(G=O_2(G)\cdot R\), where \(O_2(G)\ne 1\) is a nilpotent 2-group and the nilpotent length of \(O_2(G)\) is at most 2, \(R\) is a 3-group with unique subgroup of order 3 and \(R\) acts freely on \(O_2(G)\). \item[(3)] \(G=0_2(G)\cdot(R\cdot\langle t\rangle)\), where \(O_2(G)\ne 1\) is a nilpotent 2-group and the nilpotent length of \(O_2(G)\) is at most 2, \(R\) is a locally cyclic 3-group which acts freely on \(O_2(G)\), \(t\) is an element of order 2 such that \(t\) transforms every element of \(R\) to inverse in conjunction in \(R\). \item[(4)] \(O_3(G)=1\), the Sylow 3-subgroup \(R\) of \(G\) is not a locally cyclic group, \(N_G(R)\) acts transitively on elements of \(R\) of order 3 in conjunction in \(R\), and every Sylow 3-subgroup of \(G\) is conjugate with \(R\).\end{itemize}Toric reflection groupshttps://zbmath.org/1541.200252024-09-27T17:47:02.548271Z"Gobet, Thomas"https://zbmath.org/authors/?q=ai:gobet.thomasSummary: Several finite complex reflection groups have a braid group that is isomorphic to a torus knot group. The reflection group is obtained from the torus knot group by declaring meridians to have order \(k\) for some \(k\geq 2\), and meridians are mapped to reflections. We study all possible quotients of torus knot groups obtained by requiring meridians to have finite order. Using the theory of \(J\)-groups of
\textit{P. N. Achar} and \textit{A.-M. Aubert} [Commun. Algebra 36, No. 6, 2092--2132 (2008; Zbl 1270.20038)],
we show that these groups behave like (in general, infinite) complex reflection groups of rank two. The large family of `toric reflection groups' that we obtain includes, among others, all finite complex reflection groups of rank two with a single conjugacy class of reflecting hyperplanes, as well as Coxeter's truncations of the 3-strand braid group. We classify these toric reflection groups and explain why the corresponding torus knot group can be naturally considered as its braid group. In particular, this yields a new infinite family of reflection-like groups admitting braid groups that are Garside groups. Moreover, we show that a toric reflection group has cyclic center by showing that the quotient by the center is isomorphic to the alternating subgroup of a Coxeter group of rank three. To this end we use the fact that the center of the alternating subgroup of an irreducible, infinite Coxeter group of rank at least three is trivial. Several ingredients of the proofs are purely Coxeter-theoretic, and might be of independent interest.On polyadic operations on Cartesian powershttps://zbmath.org/1541.200262024-09-27T17:47:02.548271Z"Galmak, A. M."https://zbmath.org/authors/?q=ai:galmak.aleksandr-mikhailovich"Rusakov, A. D."https://zbmath.org/authors/?q=ai:rusakov.a-dSummary: The polyadic operations on the cartesian powers of \(n\)-ary groupoids are defined and studied in this paper.Shannon-McMillan-Breiman theorem along almost geodesics in negatively curved groupshttps://zbmath.org/1541.220012024-09-27T17:47:02.548271Z"Nevo, Amos"https://zbmath.org/authors/?q=ai:nevo.amos"Pogorzelski, Felix"https://zbmath.org/authors/?q=ai:pogorzelski.felixSummary: Consider a non-elementary Gromov-hyperbolic group \(\Gamma\) with a suitable invariant hyperbolic metric, and an ergodic probability measure preserving (p.m.p.) action on \((X ,\mu)\). We construct special increasing sequences of finite subsets \(F_n(y) \subset \Gamma\), with \((Y, \nu)\) a suitable probability space, with the following properties.
\begin{itemize}
\item Given any countable partition \(\mathcal{P}\) of \(X\) of finite Shannon entropy, the refined partitions \(\bigvee_{\gamma \in F_{n}(y)} \gamma \mathcal{P}\) have normalized information functions which converge to a constant limit, for \(\mu\)-almost every \(x \in X\) and \(\nu\)-almost every \(y \in Y\).
\item The sets \(\mathcal{F}_{n}(y)\) constitute almost-geodesic segments, and \(\bigcup_{n \in \mathbb{N}} F_{n}(y)\) is a one-sided almost geodesic with limit point \(F^{+}(y) \in \partial \Gamma\), starting at a fixed bounded distance from the identity, for almost every \(y \in Y\).
\item The distribution of the limit point \(F^{+}(y)\) belongs to the Patterson-Sullivan measure class on \(\partial \Gamma\) associated with the invariant hyperbolic metric.
\end{itemize}
The main result of the present paper amounts therefore to a Shannon-McMillan-Breiman theorem along almost-geodesic segments in any p.m.p. action of \(\Gamma\) as above. For several important classes of examples we analyze, the construction of \(F_{n}(y)\) is purely geometric and explicit. Furthermore, consider the infimum of the limits of the normalized information functions, taken over all \(\Gamma\)-generating partitions of \(X\). Using an important inequality due to
[\textit{B. Seward}, ``Weak containment and Rokhlin entropy'', Preprint, \url{arXiv:1602.06680}],
we deduce that it is equal to the Rokhlin entropy \(\mathfrak{h}^{\mathrm{Rok}}\) of the \(\Gamma\)-action on \((X, \mu)\) defined in
[\textit{B. Seward}, Invent. Math. 215, No. 1, 265--310 (2019; Zbl 1417.37043)],
provided that the action is free. Remarkably, this property holds for every choice of invariant hyperbolic metric, every choice of suitable auxiliary space \((Y, \nu)\) and every choice of special family \(F_{n}(y)\) as above. In particular, for every \(\varepsilon > 0\), there is a generating partition \(\mathcal{P}_\varepsilon\), such that for almost every \(y \in Y\), the partition refined using the sets \(F_{n}(y)\) has most of its atoms of roughly constant measure, comparable to \(\exp (-n \mathfrak{h}^{\mathrm{Rok}} \pm \varepsilon)\). This describes an approximation to the Rokhlin entropy in geometric and dynamical terms, for actions of word-hyperbolic groups.The real spectrum compactification of character varieties: characterizations and applicationshttps://zbmath.org/1541.320022024-09-27T17:47:02.548271Z"Burger, Marc"https://zbmath.org/authors/?q=ai:burger.marc"Iozzi, Alessandra"https://zbmath.org/authors/?q=ai:iozzi.alessandra"Parreau, Anne"https://zbmath.org/authors/?q=ai:parreau.anne"Pozzetti, Maria Beatrice"https://zbmath.org/authors/?q=ai:pozzetti.maria-beatriceSummary: We announce results on a compactification of general character varieties that has good topological properties and give various interpretations of its ideal points. We relate this to the Weyl chamber length compactification and apply our results to the theory of maximal and Hitchin representations.Self-similar groups and holomorphic dynamics: renormalization, integrability, and spectrumhttps://zbmath.org/1541.370482024-09-27T17:47:02.548271Z"Dang, N.-B."https://zbmath.org/authors/?q=ai:dang.nguyen-bac"Grigorchuk, R."https://zbmath.org/authors/?q=ai:grigorchuk.rostislav-i"Lyubich, M."https://zbmath.org/authors/?q=ai:lyubich.mikhailSummary: In this paper, we explore the spectral measures of the Laplacian on Schreier graphs for several self-similar groups (the Grigorchuk, Lamplighter, and Hanoi groups) from the dynamical and algebro-geometric viewpoints. For these graphs, classical Schur renormalization transformations act on appropriate spectral parameters as rational maps in two variables. We show that the spectra in question can be interpreted as asymptotic distributions of slices by a line of iterated pullbacks of certain algebraic curves under the corresponding rational maps (leading us to a notion of a \textit{spectral current}). We follow up with a dynamical criterion for discreteness of the spectrum. In case of atomic spectrum, the precise rate of convergence of finite-scale approximands to the limiting spectral measure is given. For the three groups under consideration, the corresponding rational maps happen to be fibered over polynomials in one variable. We reveal the algebro-geometric nature of this integrability phenomenon.On reduction maps and arithmetic dynamics of Mordell-Weil type groupshttps://zbmath.org/1541.371062024-09-27T17:47:02.548271Z"Banaszak, Grzegorz"https://zbmath.org/authors/?q=ai:banaszak.grzegorz"Barańczuk, Stefan"https://zbmath.org/authors/?q=ai:baranczuk.stefanLocal-global principles are proved for the Mordell-Weil group of an abelian variety over an algebraic number field \(F\), the \(S\)-unit group of the ring of integers in \(F\) and higher \(K\)-groups of \(F\). The results are given in detail for Mordell-Weil groups, the formulations for the other two types of objects are relegated to an appendix. For technical reasons, the authors assume that the abelian variety \(A\) has the endomorphim ring \(\mathrm{End}_{\overline{F}}(A) = {\mathbb Z}\).
For the first local-global principle, the authors consider orbits of a point \(P\in A(F)\) under an endomorphim \(\phi\) and orbits of a point \(Q\in A(F)\) under an endomorphism \(\psi\). Theorem 1.1 states that a nonzero intersection of the two orbits modulo almost all prime ideals is equivalent to either the two orbits having a nonzero intersection over \(F\) or a special situation. This special situation consists of both endomorphims having the same prime divisors and a simple combination of \(P\) and \(Q\) being a torsion point whose order has only prime divisors which also divide the orders of the endomorphisms \(\phi\) and \(\psi\) (in the statement of the theorem only \(\phi\) is mentioned, which appears to be a misprint).
The second local-global principle is about an analogue to the set of prime divisors of Lehmer-Pierce sequences for abelian varieties.
Definition. For \(X \subset A(F)\) with \(\mathrm{End}_{\overline{F}}(A) = {\mathbb Z}\), let \(\text{supp}(X)\) be the set of all prime ideals \(v\) such that \(P\bmod v \equiv 0\) for some \(P\in X\).
The theorem then reads as follows:
Theorem 1.2. Let \(P_1, P_2, \ldots, P_r, Q_1, Q_2, \ldots, Q_s \in A(F)\) be points of infinite order. Assume that for each positive integer \(n\),
\[
\text{supp}(\{ nP_1, nP_2, \ldots, nP_r\}) \subset \text{supp}(\{ nQ_1, nQ_2, \ldots, nQ_s\}).\]
Then for every \(i \in \{1,2, \ldots, r\}\) there exists a \(j \in \{1,2, \ldots, s\}\) such that
\[
f_j Q_j = g_i P_i
\]
for some nonzero integers \(f_j, g_i\). If the torsion part of the subgroup of \(A(F)\) generated by the \(r+s\) points \(P_i, Q_j\) is trivial or the \(s\) points \(Q_j\) are pairwise linearly independent, then we can take \(f_j = 1\) for every \(i\in \{ 1,\ldots, r \}\).
The authors reprove Theorem 2.1 on the order of reduction images by algebraic rather than geometric means, which allows its generalization to all three types of objects which are considered. The tools used are Kummer theory for abelian varieties and the Chebotarev density theorem.
The formulation of the second local-global principle for algebraic number rings yields Corollary A.8 in the appendix, which answers an open question stated in [\textit{M. Skałba}, Bull. Aust. Math. Soc. 97, No. 1, 11--14 (2018; Zbl 1432.11015)] on the standard Lehmer-Pierce sequences for integers.
Reviewer: Andreas Bender (Pavia)On the Dales-Żelazko conjecture for Beurling algebras on discrete groupshttps://zbmath.org/1541.430042024-09-27T17:47:02.548271Z"White, Jared T."https://zbmath.org/authors/?q=ai:white.jared-tSummary: Let \(G\) be a group that is either virtually soluble or virtually free, and let \(\omega\) be a weight on \(G\). We prove that if \(G\) is infinite, then there is some maximal left ideal of finite codimension in the Beurling algebra \(\ell^1(G, \omega)\), which fails to be (algebraically) finitely generated. This implies that a conjecture of Dales and Żelazko holds for these Banach algebras [\textit{H.~G. Dales} and \textit{W.~Żelazko}, Stud. Math. 212, No.~2, 173--193 (2012; Zbl 1269.46028)]. We then go on to give examples of weighted groups for which this property fails in a strong way. For instance, we describe a Beurling algebra on an infinite group in which every closed left ideal of finite codimension is finitely generated and which has many such ideals in the sense of being residually finite dimensional. These examples seem to be hard cases for proving Dales and Żelazko's conjecture.\(C^{\star}\)-algebras of higher-rank graphs from groups acting on buildings, and explicit computation of their \(K\)-theoryhttps://zbmath.org/1541.460342024-09-27T17:47:02.548271Z"Mutter, Sam A."https://zbmath.org/authors/?q=ai:mutter.sam-a"Radu, Aura-Cristiana"https://zbmath.org/authors/?q=ai:radu.aura-cristiana"Vdovina, Alina"https://zbmath.org/authors/?q=ai:vdovina.alina-aSummary: We unite elements of category theory, \(K\)-theory, and geometric group theory, by defining a class of groups called \(k\)-cube groups, which act freely and transitively on the product of \(k\) trees, for arbitrary \(k\). The quotient of this action on the product of trees defines a \(k\)-dimensional cube complex, which induces a higher-rank graph. We make deductions about the K-theory of the corresponding rank-\(k\) graph \(C^{\star}\)-algebras, and give examples of \(k\)-cube groups and their K-theory. These are among the first explicit computations of \(K\)-theory for an infinite family of rank-\(k\) graphs for \(k\geq 3\), which is not a direct consequence of the Künneth theorem for tensor products.Weakly first-countability in strongly topological gyrogroupshttps://zbmath.org/1541.540132024-09-27T17:47:02.548271Z"Zhang, Jing"https://zbmath.org/authors/?q=ai:zhang.jing.5"Lin, Kaixiong"https://zbmath.org/authors/?q=ai:lin.kaixiong"Xi, Wenfei"https://zbmath.org/authors/?q=ai:xi.wenfeiThe authors prove that if \((G, \tau, \oplus)\) is a strongly topological gyrogroup and \(H\) is a closed neutral strong subgyrogroup of \(G\), then the following are true: (1) \(G/H\) is \(\kappa\)-Fréchet-Urysohn if and only if \(G/H\) is strongly \(\kappa\)-Fréchet-Urysohn; and (2) \(\Delta(G/H)\) = \(\psi(G/H)\). Furthermore, if \((G, \tau, \oplus)\) is a sequential strongly topological gyrogroup having a point-countable \(k\)-network, then \(G\) is metrizable or it is a topological sum of cosmic subspaces. These results improve the related results in topological groups.
Reviewer: Watchareepan Atiponrat (Chiang Mai)Assouad-Nagata dimension and gap for ordered metric spaceshttps://zbmath.org/1541.540162024-09-27T17:47:02.548271Z"Erschler, Anna"https://zbmath.org/authors/?q=ai:erschler.anna"Mitrofanov, Ivan"https://zbmath.org/authors/?q=ai:mitrofanov.ivan-viktorovichAssouad-Nagata dimension is a notion that provides a control of both local and global properties of a metric space. This paper studies the relationship between the Assouad-Nagata dimension of a metric space \(X\) and orderings of \(X\) that give good solutions for the traveling salesman problem. In this approach, one orders all points of a metric space and then, given a \(k\)-point subset, visit its points in the corresponding order.
Given an ordered metric space \((M, d, T)\), for each finite subset \(X\) of \(M\), consider the ratio of the length of the corresponding path on \(X\) over the minimal length of a path that visits all points of \(X\). For each \(k\geq 1\), the order ratio function \(\mathrm {OR}_{M, T}(k)\) is the supremum of the ratios for all subsets \(X\) of \(M\) with \(2\leq \sharp X \leq k+1\). The break point \(\mathrm {Br}(M, T)\) is the smallest integer \(s\) such that \(\mathrm {OR}_{M, T}(s) < s\).
The main result states: If \(M\) is a metric space of finite Assouad-Nagata dimension \(m\) with \(m\)-dimensional control function at most \(Kr\), then there exists an order \(T\) such that for all \(k\geq 2\), the order ratio function \(\mathrm {OR}_{M, T}(k) \leq C \ln k\), where \(C\) depends only on \(m\) and \(K\); and the order break point \(\mathrm {Br}(M, T) \leq 2m + 2\).
The authors give two sufficient conditions for a metric space to have infinite break point:
1) Let \(\Gamma_i\) be a sequence of finite graphs of degree \(d_i\geq 3\) on \(n_i\) vertices, and let \(T_i\) be an order on \(\Gamma_i\). If the normalized spectral gap satisfies some condition, the order breakpoint of the sequence \((\Gamma_i, T_i)\) is infinite.
2) If a metric space \(M\) weakly contains a sequence of arbitrary large cubes, then for any order \(T\) on \(M\) the order breakpoint of \((M,T)\) is infinite.
The authors then raise a question: For a metric space \(M\) of infinite Assouad-Nagata dimension, is the order breakpoint of \(M\) infinite?
If the answer to this question is positive, it will provide a positive answer to the following gap problem for the existence of an order: For any metric space \(M\), is it true that either for any order \(T\) on \(M\) and for all \(k \geq 1\), \(\mathrm {OR}_{M, T}(k) = k\), or there exists an order \(T\) such that for all \(k\geq 2\), \(\mathrm {OR}_{M, T}(k) \leq C\ln k\)?
The authors formulate a stronger gap problem and prove it for metric spaces with doubling property: For any metric space \(M\) with doubling property and an order \(T\), either for all \(s\), \(\mathrm {OR}_{M, T}(s) = s\), or there exists \(C\) such that for all \(k\geq 1\), \(\mathrm {OR}_{M, T}(k) \leq C\ln k\).
Reviewer: Takahisa Miyata (Kobe)Completion preserves homotopy fibre squares of connected nilpotent spaceshttps://zbmath.org/1541.550112024-09-27T17:47:02.548271Z"Ronan, Andrew"https://zbmath.org/authors/?q=ai:ronan.andrew-sThe author proves that completion preserves homotopy fibre squares of connected nilpotent spaces. Similar results can be found in the literature. In particular, \textit{E. D. Farjoun} [Topology 42, No. 5, 1083--1099 (2003; Zbl 1041.55007)] showed an analogue in the case of disconnected spaces. The presented result can also be viewed as a generalisation of the connected fibre lemma of \textit{A. K. Bousfield} and \textit{D. M. Kan} [Homotopy limits, completions and localizations. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0259.55004)].
As an application, the Hasse fracture square associated to a connected nilpotent space is deduced. Along the way, the author records some closure properties of the category of \(T\)-complete nilpotent groups, where \(T\) is a non-empty set of primes.
Reviewer: Marek Golasiński (Olsztyn)Virtual and arrow Temperley-Lieb algebras, Markov traces, and virtual link invariantshttps://zbmath.org/1541.570102024-09-27T17:47:02.548271Z"Paris, Luis"https://zbmath.org/authors/?q=ai:paris.luis"Rabenda, Loïc"https://zbmath.org/authors/?q=ai:rabenda.loicSummary: Let \(R^f=\mathbb{Z}[ A^{\pm 1}]\) be the algebra of Laurent polynomials in the variable \(A\) and let \(R^a=\mathbb{Z}[ A^{\pm 1}, z_1, z_2,\ldots]\) be the algebra of Laurent polynomials in the variable \(A\) and standard polynomials in the variables \(z_1, z_2,\ldots.\) For \(n\geq1\) we denote by \(\operatorname{VB}_n\) the virtual braid group on \(n\) strands. We define two towers of algebras \(\{ \operatorname{VTL}_n ( R^f ) \}_{n = 1}^\infty\) and \(\{ \operatorname{ATL}_n ( R^a ) \}_{n = 1}^\infty\) in terms of diagrams. For each \(n\geq1\) we determine presentations for both, \( \operatorname{VTL}_n( R^f)\) and \(\operatorname{ATL}_n( R^a)\). We determine sequences of homomorphisms \(\{ \rho_n^f : R^f [ \operatorname{VB}_n ] \to \operatorname{VTL}_n ( R^f ) \}_{n = 1}^\infty\) and \(\{ \rho_n^a : R^a [ \operatorname{VB}_n ] \to \operatorname{ATL}_n ( R^a ) \}_{n = 1}^\infty \), we determine Markov traces \(\{ T_n^{\prime f} : \operatorname{VTL}_n ( R^f ) \to R^f \}_{n = 1}^\infty\) and \(\{ T_n^{\prime a} : \operatorname{ATL}_n ( R^a ) \to R^a \}_{n = 1}^\infty \), and we show that the invariants for virtual links obtained from these Markov traces are the \(f\)-polynomial for the first trace and the arrow polynomial for the second trace. We show that, for each \(n\geq1,\) the standard Temperley-Lieb algebra \(\operatorname{TL}_n\) embeds into both, \( \operatorname{VTL}_n( R^f)\) and \(\operatorname{ATL}_n( R^a)\), and that the restrictions to \(\{ \operatorname{TL}_n \}_{n = 1}^\infty\) of the two Markov traces coincide.Open books decompositions of links of minimally elliptic singularitieshttps://zbmath.org/1541.570142024-09-27T17:47:02.548271Z"Bhupal, Mohan"https://zbmath.org/authors/?q=ai:bhupal.mohanSummary: We present an explicit Milnor open book decomposition supporting the canonical contact structure on the link of each minimally elliptic singularity whose fundamental cycle \(Z\) satisfies \(- 3 \leq Z \cdot Z \leq - 1\). For the Milnor open books whose pages have genus less than three, we give a factorization of the monodromy which does not involve any left-handed Dehn twists around interior curves. Necessary results regarding the roots of reducible, and irreducible elements in mapping class groups are proved, and some new relations in the mapping class groups are presented.
{{\copyright} 2021 Wiley-VCH GmbH}Norm-controlled cohomology of transformation groupshttps://zbmath.org/1541.570342024-09-27T17:47:02.548271Z"Kimura, Mitsuaki"https://zbmath.org/authors/?q=ai:kimura.mitsuakiLet \(M\) be a complete connected Riemannian manifold and \(\mu\) the measure induced by the metric. Denote by \(\mathrm{Homeo}_0(M,\mu)\) the identity component of the group of measure preserving homeomorphisms of \(M\) with the compact open topology. \textit{M. Brandenbursky} and \textit{M. Marcinkowski} [Math. Ann. 382, No. 3--4, 1181--1197 (2022; Zbl 1490.57040)] gave a construction of bounded cohomology classes of \(\mathrm{Homeo}_0(M,\mu)\) when \(M\) is of finite volume, and they showed that for certain transformation groups of non-positively curved manifolds the third bounded cohomology is infinite dimensional.
The author generalizes this discussion. For a group \(G\) one may define the concept of a (conjugation-invariant, pseudo) norm \(\nu\), which leads to the idea of its norm-controlled cohomology \(H^*_\nu(G)\). The exact norm-controlled cohomology \(EH_\nu^n(G)\) is the kernel of the comparison map \(H^*_\nu(G) \to H^*(G)\). For non-negative integers \(d\) there are variants \(H^n_{(d)}(G,\nu)\) so that \(H^n_{(0)}(G,\nu) = H^n_\nu(G)\). Relevant examples \({\mathcal T}_M\) for \(G\) are \(\mathrm{Homeo}_0(M,\mu)\), and its subgroups \(\mathrm{Diffeo}_0(M, \mathrm{vol})\) of volume preserving diffeomorphism with compact support, and, for a symplectic manifold \((M,\omega)\), the group \(\mathrm{Symp}_0 (M,\omega)\) of symplectomorphisms with compact support. Assuming the existence of a finite volume open subset \(U\) of \(M\) with certain properties, the author shows that \(EH^3_{(d)} ({\mathcal T}, \nu_U)\) is uncountably infinite dimensional for \(d=0\), \(1\), and \(2\).
Reviewer: Karl Heinz Dovermann (Honolulu)The second step in characterizing a three-word codehttps://zbmath.org/1541.681892024-09-27T17:47:02.548271Z"Cao, Chunhua"https://zbmath.org/authors/?q=ai:cao.chunhua"Xu, Jiao"https://zbmath.org/authors/?q=ai:xu.jiao"Liao, Lei"https://zbmath.org/authors/?q=ai:liao.lei"Yang, Di"https://zbmath.org/authors/?q=ai:yang.di.5"Jia, Guichuan"https://zbmath.org/authors/?q=ai:jia.guichuan"Du, Qian"https://zbmath.org/authors/?q=ai:du.qianSummary: In the fields of combinatorics on words and theory of codes, a two-word language \(\{x, y\}\) is a code if and only if \(xy \neq yx\). But up to now, corresponding characterizations for a three-word language, which forms a code, have not been completely found. Let \(X = \{x, y, z\}\) be a three-word language and \(|x|\), \(|y|\), \(|z|\) be their lengths. When \(|x| = |y| < |z|\), a necessary and sufficient condition for \(X\) to be a code was obtained in [\textit{C. Chunhua} et al., Acta Inf. 55, No. 5, 445--457 (2018; Zbl 1398.68351)]. If \(|x| < |y| = |z| \leq 2|x|\), a necessary and sufficient condition for \(X\) to be a code is proposed in this paper.Efficient analysis of unambiguous automata using matrix semigroup techniqueshttps://zbmath.org/1541.681982024-09-27T17:47:02.548271Z"Kiefer, Stefan"https://zbmath.org/authors/?q=ai:kiefer.stefan"Widdershoven, Cas"https://zbmath.org/authors/?q=ai:widdershoven.casSummary: We introduce a novel technique to analyse unambiguous Büchi automata quantitatively, and apply this to the model checking problem. It is based on linear-algebra arguments that originate from the analysis of matrix semigroups with constant spectral radius. This method can replace a combinatorial procedure that dominates the computational complexity of the existing procedure by \textit{C. Baier} et al. [Lect. Notes Comput. Sci. 9779, 23--42 (2016; Zbl 1411.68051)]. We analyse the complexity in detail, showing that, in terms of the set \(Q\) of states of the automaton, the new algorithm runs in time \(O(|Q|^4)\), improving on an efficient implementation of the combinatorial algorithm by a factor of \(|Q|\).
For the entire collection see [Zbl 1423.68036].Uniformisation gives the full strength of regular languageshttps://zbmath.org/1541.682012024-09-27T17:47:02.548271Z"Lhote, Nathan"https://zbmath.org/authors/?q=ai:lhote.nathan"Michielini, Vincent"https://zbmath.org/authors/?q=ai:michielini.vincent"Skrzypczak, Michał"https://zbmath.org/authors/?q=ai:skrzypczak.michalSummary: Given \(R\) a binary relation between words (which we treat as a language over a product alphabet \(\mathbb{A}\times\mathbb{B}\)), a uniformisation of it is another relation \(L\) included in \(R\) which chooses a single word over \(\mathbb{B}\), for each word over \(\mathbb{A}\) whenever there exists one. It is known that MSO, the full class of regular languages, is strong enough to define a uniformisation for each of its relations. The quest of this work is to see which other formalisms, weaker than MSO, also have this property. In this paper, we solve this problem for pseudo-varieties of semigroups: we show that no nonempty pseudo-variety weaker than MSO can provide uniformisations for its relations.
For the entire collection see [Zbl 1423.68036].Reset thresholds of transformation monoidshttps://zbmath.org/1541.682322024-09-27T17:47:02.548271Z"Rystsov, I."https://zbmath.org/authors/?q=ai:rystsov.i-k|rystsov.igor"Szykuła, M."https://zbmath.org/authors/?q=ai:szykula.marekSummary: Motivated by the Černý conjecture for automata, we introduce the concept of monoidal automata, which allows us to formulate the Černý conjecture for monoids. We obtain the upper bounds on the reset threshold of monoids with certain properties. In particular, we obtain a quadratic upper bound if the transformation monoid contains a primitive group of permutations and a singular of maximal rank with only one point of contraction.The intersection of 3-maximal submonoidshttps://zbmath.org/1541.683032024-09-27T17:47:02.548271Z"Castiglione, Giuseppa"https://zbmath.org/authors/?q=ai:castiglione.giusi"Holub, Štěpán"https://zbmath.org/authors/?q=ai:holub.stepanSummary: Very little is known about the structure of the intersection of two \(k\)-generated monoids of words, even for \(k=3\). Here we investigate the case of \(k\)-maximal monoids, that is, monoids whose basis of cardinality \(k\) cannot be non-trivially decomposed into at most \(k\) words. We characterize the intersection in the case of two \(3\)-maximal monoids.PATCH graphs: an efficient data structure for completion of finitely presented groupshttps://zbmath.org/1541.684532024-09-27T17:47:02.548271Z"Lynch, Christopher"https://zbmath.org/authors/?q=ai:lynch.christopher-a"Strogova, Polina"https://zbmath.org/authors/?q=ai:strogova.polinaSummary: Based on a new data structure called PATCH Graph, an efficient completion procedure for finitely presented groups is described. A PATCH Graph represents rules and their symmetrized forms as cycles in a Cayley graph structure. Completion is easily performed directly on the graph, and structure sharing is enforced. The structure of the graph allows us to avoid certain redundant inferences. The PATCH Graph data structure and inference rules complement other extensions of Knuth-Bendix completion for finitely presented groups.
For the entire collection see [Zbl 0853.68027].An alternative foundation of quantum theoryhttps://zbmath.org/1541.810042024-09-27T17:47:02.548271Z"Helland, Inge S."https://zbmath.org/authors/?q=ai:helland.inge-sSummary: A new approach to quantum theory is proposed in this paper. The basis is taken to be theoretical variables, variables that may be accessible or inaccessible, i.e., it may be possible or impossible for an observer to assign arbitrarily sharp numerical values to them. In an epistemic process, the accessible variables are just ideal observations connected to an observer or to some communicating observers. Group actions are defined on these variables, and group representation theory is the basis for developing the Hilbert space formalism here. Operators corresponding to accessible theoretical variables are derived, and in the discrete case, it is proved that the possible physical values are the eigenvalues of these operators. The focus of the paper is some mathematical theorems paving the ground for the proposed foundation of quantum theory. It is shown here that the groups and transformations needed in this approach can be constructed explicitly in the case where the accessible variables are finite-dimensional. This simplifies the theory considerably: To reproduce the Hilbert space formulation, it is enough to assume the existence of two complementary variables. The interpretation inferred from the proposed foundation here may be called a general epistemic interpretation of quantum theory. A special case of this interpretation is QBism; it also has a relationship to several other interpretations.On Buschman-Erdelyi and Mehler-Fock transforms related to the group \(SO_0(3,1)\)https://zbmath.org/1541.810092024-09-27T17:47:02.548271Z"Shilin, Il'ya Anatol'evich"https://zbmath.org/authors/?q=ai:shilin.ilya-anatolevichSummary: By using a functional defined on a pair of the assorted represention spaces of the connected subgroup of the proper Lorentz group, a formula for the Buschman-Erdelyi transform of the Legendre function (up to a factor) is derived. Also a formula for the Mehler-Fock transform of the Legendre function of an inverse argument is obtained. Moreover, a generalization of one known formula for the Mehler-Fock transform is derived.\(q\)-analog qudit Dicke stateshttps://zbmath.org/1541.810152024-09-27T17:47:02.548271Z"Raveh, David"https://zbmath.org/authors/?q=ai:raveh.david"Nepomechie, Rafael I."https://zbmath.org/authors/?q=ai:nepomechie.rafael-iSummary: Dicke states are completely symmetric states of multiple qubits (2-level systems), and qudit Dicke states are their \(d\)-level generalization. We define here \(q\)-deformed qudit Dicke states using the quantum algebra \(su_q(d)\). We show that these states can be compactly expressed as a weighted sum over permutations with \(q\)-factors involving the so-called inversion number, an important permutation statistic in Combinatorics. We use this result to compute the bipartite entanglement entropy of these states. We also discuss the preparation of these states on a quantum computer, and show that introducing a \(q\)-dependence does not change the circuit gate count.
{{\copyright} 2024 The Author(s). Published by IOP Publishing Ltd}Erratum to: ``Triple series evaluated in \(\pi\) and \(\ln 2\) as well as Catalan's constant \(G\)''https://zbmath.org/1541.810432024-09-27T17:47:02.548271Z"Li, Chunli"https://zbmath.org/authors/?q=ai:li.chunli"Chu, Wenchang"https://zbmath.org/authors/?q=ai:chu.wenchangErratum to the authors' paper [ibid. 63, No. 11, 2005--2023 (2023; Zbl 1536.81037)] .Row-column duality and combinatorial topological stringshttps://zbmath.org/1541.811392024-09-27T17:47:02.548271Z"Padellaro, Adrian"https://zbmath.org/authors/?q=ai:padellaro.adrian"Radhakrishnan, Rajath"https://zbmath.org/authors/?q=ai:radhakrishnan.rajath"Ramgoolam, Sanjaye"https://zbmath.org/authors/?q=ai:ramgoolam.sanjayeSummary: Integrality properties of partial sums over irreducible representations, along columns of character tables of finite groups, were recently derived using combinatorial topological string theories (CTST). These CTST were based on Dijkgraaf-Witten theories of flat \(G\)-bundles for finite groups \(G\) in two dimensions, denoted \(G\)-TQFTs. We define analogous combinatorial topological strings related to two dimensional topological field theories (TQFTs) based on fusion coefficients of finite groups. These TQFTs are denoted as \(R(G)\)-TQFTs and allow analogous integrality results to be derived for partial row sums of characters over conjugacy classes along fixed rows. This relation between the \(G\)-TQFTs and \(R(G)\)-TQFTs defines a row-column duality for character tables, which provides a physical framework for exploring the mathematical analogies between rows and columns of character tables. These constructive proofs of integrality are complemented with the proof of similar and complementary results using the more traditional Galois theoretic framework for integrality properties of character tables. The partial row and column sums are used to define generalised partitions of the integer row and column sums, which are of interest in combinatorial representation theory.
{{\copyright} 2024 IOP Publishing Ltd}