A method to find generators of a semi-simple Lie group via the topology of its flag manifolds

In this paper we continue to develop the topological method to get semigroup generators of semi-simple Lie groups. Consider a subset \(\Gamma \subset G\) that contains a semi-simple subgroup \(G_{1}\) of G. If one can show that \( \Gamma \) does not leave invariant a contractible subset on any flag manifold of G, then \(\Gamma \) generates G if \(\mathrm {Ad}\left( \Gamma \right) \) generates a Zariski dense subgroup of the algebraic group \(\mathrm {Ad}\left( G\right) \). The proof is reduced to check that some specific closed orbits of \(G_{1}\) in the flag manifolds of G are not trivial in the sense of algebraic topology. Here, we consider three different cases of semi-simple Lie groups G and subgroups \(G_{1}\subset G\).     

A characterization of the desarguesian and the Figueroa planes of order $$q^{3}$$

Let \(\Pi \) be a plane of order \(q^{3}\), \(q>2\), admitting \(G\cong PGL(3,q)\) as a collineation group. By Dempwolff (Geometriae Dedicata 18:101–112, 1985) the plane \(\Pi \) contains a G-invariant subplane \(\pi _{0}\) isomorphic to PG(2, q) on which G acts 2-transitively. In this paper it is shown that, if the homologies of \(\pi _{0}\) contained in G extend to \(\Pi \) then \(\Pi \) is either the desarguesian or the Figueroa plane.     

Homomorphism reductions on Polish groups

In an earlier paper, we introduced the following pre-order on the subgroups of a given Polish group: if G is a Polish group and \(H,L \subseteq G\) are subgroups, we say H is homomorphism reducible to L iff there is a continuous group homomorphism \(\varphi : G \rightarrow G\) such that \(H = \varphi ^{-1} (L)\). We previously showed that there is a \(K_\sigma \) subgroup L of the countable power of any locally compact Polish group G such that every \(K_\sigma \) subgroup of \(G^\omega \) is homomorphism reducible to L. In the present work, we show that this fails in the countable power of the group of increasing homeomorphisms of the unit interval.     

Finite groups with <Emphasis Type="Italic">SS</Emphasis>-supplement

Let G be a finite group. A subgroup H of G is said to be SS-quasinormal in G if there is a subgroup K such that \(G=HK\) and \(HS=SH\), for all \(S\in \) Syl(K), where Syl(K) denotes the collection of all Sylow subgroups of K. A subgroup H of G is said to be SS-supplemented in G if there is a subgroup K such that \(G=HK\) and \(H\cap K\) is SS-quasinormal in G. In this paper, we investigate the SS-supplemented subgroups and strengthen a result of Skiba which gives a positive answer to an open question of Shemetkov.     

On $$\mathfrak {F}_{hq}$$-supplemented subgroups of a finite group

A subgroup H of a finite group G is quasinormal in G if it permutes with every subgroup of G. A subgroup H of a finite group G is \(\mathfrak {F}_{hq}\)-supplemented in G if G has a quasinormal subgroup N such that HN is a Hall subgroup of G and \((H\cap N)H_{G}/ H_{G} \le Z_{\mathfrak {F}}(G/H_{G})\), where \(H_{G}\) is the core of H in G and \({Z}_{\mathfrak {F}} (G/H_{G})\) is the \(\mathfrak {F}\)-hypercenter of \({G/H}_{G}\). This paper concerns the structure of a finite group G under the assumption that some subgroups of G are \(\mathfrak {F}_{hq}\)-supplemented in G.     

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.     

On <Emphasis Type="Italic">ss</Emphasis>-quasinormal or weakly <Emphasis Type="Italic">s</Emphasis>-permutably embedded subgroups of finite groups

Suppose that G is a finite group and H is a subgroup of G. H is said to be an ss-quasinormal subgroup of G if there is a subgroup B of G such that \(G=HB\) and H permutes with every Sylow subgroup of B; H is said to be weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup \(H_{se}\) of G contained in H such that \(G=HT\) and \(H\cap T\le H_{se}\). We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying \(1<|D|<|P|\) and study the structure of G under the assumption that every subgroup H of P with \(|H|=|D|\) is either ss-quasinormal or weakly s-permutably embedded in G. Some recent results are generalized and unified.     

The Gray image of constacyclic codes over the finite chain ring $$F_{p^m}[u]/\langle u^k\rangle $$

Let \(\mathbb {F}_{p^m}\) be a finite field of cardinality \(p^m\), where p is a prime, and k, N be any positive integers. We denote \(R_k=F_{p^m}[u]/\langle u^k\rangle =F_{p^m}+uF_{p^m}+\cdots +u^{k-1}F_{p^m}\) (\(u^k=0\)) and \(\lambda =a_0+a_1u+\cdots +a_{k-1}u^{k-1}\) where \(a_0, a_1,\ldots , a_{k-1}\in F_{p^m}\) satisfying \(a_0\ne 0\) and \(a_1=1\). Let r be a positive integer satisfying \(p^{r-1}+1\le k\le p^r\). First we define a Gray map from \(R_k\) to \(F_{p^m}^{p^r}\), then prove that the Gray image of any linear \(\lambda \)-constacyclic code over \(R_k\) of length N is a distance preserving linear \(a_0^{p^r}\)-constacyclic code over \(F_{p^m}\) of length \(p^rN\). Furthermore, the generator polynomials for each linear \(\lambda \)-constacyclic code over \(R_k\) of length N and its Gray image are given respectively. Finally, some optimal constacyclic codes over \(F_{3}\) and \(F_{5}\) are constructed.     

Uniquely Tree-saturated Graphs

Let H be a graph. A graph G is uniquely H-saturated if G contains no subgraph isomorphic to H, but for every edge e in the complement of G (i.e., for each “nonedge” of G), \(G+e\) contains exactly one subgraph isomorphic to H. The double star \(D_{s,t}\) is the graph formed by beginning with \(K_2\) and adding \(s+t\) new vertices, making s of these adjacent to one endpoint of the \(K_2\) and the other t adjacent to the other endpoint; \(D_{s,s}\) is a balanced double star. Our main result is that the trees T for which there exist an infinite number of uniquely T-saturated graphs are precisely the balanced double stars. In addition we completely characterize uniquely tree-saturated graphs in the case where the tree is a star \(K_{1,t} = D_{0, t-1}\). We show that if T is a double star, then there exists a nontrivial uniquely T-saturated graph. We conjecture that the converse holds; we verify this conjecture for all trees of order at most 6. We conclude by giving some open problems.     

Bergman interpolation on finite Riemann surfaces. Part II: Poincaré-Hyperbolic Case

Let M be a 3-manifold with torus boundary components \(T_{1}\) and \(T_2\). Let \(\phi :T_{1} \rightarrow T_{2}\) be a homeomorphism, \(M_\phi \) the manifold obtained from M by gluing \(T_{1}\) to \(T_{2}\) via the map \(\phi \), and T the image of \(T_{1}\) in \(M_\phi \). We show that if \(\phi \) is “sufficiently complicated” then any incompressible or strongly irreducible surface in \(M_\phi \) can be isotoped to be disjoint from T. It follows that every Heegaard splitting of a 3-manifold admitting a “sufficiently complicated” JSJ decomposition is an amalgamation of Heegaard splittings of the components of the JSJ decomposition.     

On the density of images of the power maps in Lie groups

Let G be a connected Lie group. In this paper, we study the density of the images of individual power maps \(P_k:G\rightarrow G:g\mapsto g^k\). We give criteria for the density of \(P_k(G)\) in terms of regular elements, as well as Cartan subgroups. In fact, we prove that if \(\mathrm{Reg}(G)\) is the set of regular elements of G, then \(P_k(G)\cap \mathrm{Reg}(G)\) is closed in \(\mathrm{Reg}(G)\). On the other hand, the weak exponentiality of G turns out to be equivalent to the density of all the power maps \(P_k\). In linear Lie groups, weak exponentiality reduces to the density of \(P_2(G)\). We also prove that the density of the image of \(P_k\) for G implies the same for any connected full rank subgroup.     

Relative Difference Sets and Hadamard Matrices from Perfect Quaternionic Arrays

Let \(G=\mathbf{C}_{n_1}\times \cdots \times \mathbf{C}_{n_m}\) be an abelian group of order \(n=n_1\dots n_m\), where each \(\mathbf{C}_{n_t}\) is cyclic of order \(n_t\). We present a correspondence between the (4n, 2, 4n, 2n)-relative difference sets in \(G\times Q_8\) relative to the centre \(Z(Q_8)\) and the perfect arrays of size \(n_1\times \dots \times n_m\) over the quaternionic alphabet \(Q_8\cup qQ_8\), where \(q=(1+i+j+k)/2\). In view of this connection, for \(m=2\) we introduce new families of relative difference sets in \(G\times Q_8\), as well as new families of Williamson and Ito Hadamard matrices with G-invariant components.     

On Star–Wheel Ramsey Numbers

For two given graphs \(G_1\) and \(G_2\), the Ramsey number \(R(G_1,G_2)\) is the least integer r such that for every graph G on r vertices, either G contains a \(G_1\) or \(\overline{G}\) contains a \(G_2\). In this note, we determined the Ramsey number \(R(K_{1,n},W_m)\) for even m with \(n+2\le m\le 2n-2\), where \(W_m\) is the wheel on \(m+1\) vertices, i.e., the graph obtained from a cycle \(C_m\) by adding a vertex v adjacent to all vertices of the \(C_m\).     

Triviality of Iwasawa module associated to some abelian fields of prime conductors

Let p be an odd prime number and \(\ell \) an odd prime number dividing \(p-1\). We denote by \(F=F_{p,\ell }\) the real abelian field of conductor p and degree \(\ell \), and by \(h_F\) the class number of F. For a prime number \(r \ne p,\,\ell \), let \(F_{\infty }\) be the cyclotomic \(\mathbb {Z}_r\)-extension over F, and \(M_{\infty }/F_{\infty }\) the maximal pro-r abelian extension unramified outside r. We prove that \(M_{\infty }\) coincides with \(F_{\infty }\) and consequently \(h_F\) is not divisible by r when r is a primitive root modulo \(\ell \) and r is smaller than an explicit constant depending on p.     

On color-preserving automorphisms of Cayley graphs of odd square-free order

An automorphism \(\alpha \) of a Cayley graph \(\mathrm{Cay}(G,S)\) of a group G with connection set S is color-preserving if \(\alpha (g,gs) = (h,hs)\) or \((h,hs^{-1})\) for every edge \((g,gs)\in E(\mathrm{Cay}(G,S))\). If every color-preserving automorphism of \(\mathrm{Cay}(G,S)\) is also affine, then \(\mathrm{Cay}(G,S)\) is a Cayley color automorphism (CCA) graph. If every Cayley graph \(\mathrm{Cay}(G,S)\) is a CCA graph, then G is a CCA group. Hujdurovi? et al. have shown that every non-CCA group G contains a section isomorphic to the non-abelian group \(F_{21}\) of order 21. We first show that there is a unique non-CCA Cayley graph \(\Gamma \) of \(F_{21}\). We then show that if \(\mathrm{Cay}(G,S)\) is a non-CCA graph of a group G of odd square-free order, then \(G = H\times F_{21}\) for some CCA group H, and \(\mathrm{Cay}(G,S) = \mathrm{Cay}(H,T)\mathbin {\square }\Gamma \).     

Perfect state transfer in Laplacian quantum walk

For a graph G and a related symmetric matrix M, the continuous-time quantum walk on G relative to M is defined as the unitary matrix \(U(t) = \exp (-itM)\), where t varies over the reals. Perfect state transfer occurs between vertices u and v at time \(\tau \) if the (u, v)-entry of \(U(\tau )\) has unit magnitude. This paper studies quantum walks relative to graph Laplacians. Some main observations include the following closure properties for perfect state transfer. If an n-vertex graph has perfect state transfer at time \(\tau \) relative to the Laplacian, then so does its complement if \(n\tau \in 2\pi {\mathbb {Z}}\). As a corollary, the join of \(\overline{K}_{2}\) with any m-vertex graph has perfect state transfer relative to the Laplacian if and only if \(m \equiv 2\pmod {4}\). This was previously known for the join of \(\overline{K}_{2}\) with a clique (Bose et al. in Int J Quant Inf 7:713–723, 2009). If a graph G has perfect state transfer at time \(\tau \) relative to the normalized Laplacian, then so does the weak product \(G \times H\) if for any normalized Laplacian eigenvalues \(\lambda \) of G and \(\mu \) of H, we have \(\mu (\lambda -1)\tau \in 2\pi {\mathbb {Z}}\). As a corollary, a weak product of \(P_{3}\) with an even clique or an odd cube has perfect state transfer relative to the normalized Laplacian. It was known earlier that a weak product of a circulant with odd integer eigenvalues and an even cube or a Cartesian power of \(P_{3}\) has perfect state transfer relative to the adjacency matrix. As for negative results, no path with four vertices or more has antipodal perfect state transfer relative to the normalized Laplacian. This almost matches the state of affairs under the adjacency matrix (Godsil in Discret Math 312(1):129–147, 2011).     

On the family of elliptic curves $$\varvec{y^2=x^3-m^2x+p^2}$$

In this paper, we study the torsion subgroup and rank of elliptic curves for the subfamilies of \(E_{m,p} : y^2=x^3-m^2x+p^2\), where m is a positive integer and p is a prime. We prove that for any prime p, the torsion subgroup of \(E_{m,p}(\mathbb {Q})\) is trivial for both the cases {\(m\ge 1\), \(m\not \equiv 0\pmod 3\)} and {\(m\ge 1\), \(m \equiv 0 \pmod 3\), with \(gcd(m,p)=1\)}. We also show that given any odd prime p and for any positive integer m with \(m\not \equiv 0\pmod 3\) and \(m\equiv 2\pmod {32}\), the lower bound for the rank of \(E_{m,p}(\mathbb {Q})\) is 2. Finally, we find curves of rank 9 in this family.     

Families of conditionally approximately convex functions

Let \((G,+)\) be an Abelian topological group, which is also a \(T_{0}\)-space and a Baire space simultaneously, D be an open connected subset of G and \(\alpha : D-D \rightarrow {\mathbb R}\) be a function continuous at zero and such that \(\alpha (0)=0\). We show that if \((f_n)\) is a sequence of continuous functions \(f_n : D \rightarrow {\mathbb R}\) such that \(f_n(z) \le \frac{1}{2} f_n(x)+\frac{1}{2}f(y)+\alpha (x-y)\) for \(n\in {\mathbb N}\) and \(x,y,z\in D\) such that \(2z=x+y\) and if \((f_n)\) is pointwise convergent [bounded] then it is convergent uniformly on compact subsets of D [in the case when G is additionally a separable space, it contains a subsequence which is convergent on compact subsets of D].     

Partial permutation decoding for binary linear and $$Z_4$$-linear Hadamard codes

In this paper, s-\({\text {PD}}\)-sets of minimum size \(s+1\) for partial permutation decoding for the binary linear Hadamard code \(H_m\) of length \(2^m\), for all \(m\ge 4\) and \(2 \le s \le \lfloor {\frac{2^m}{1+m}}\rfloor -1\), are constructed. Moreover, recursive constructions to obtain s-\({\text {PD}}\)-sets of size \(l\ge s+1\) for \(H_{m+1}\) of length \(2^{m+1}\), from an s-\({\text {PD}}\)-set of the same size for \(H_m\), are also described. These results are generalized to find s-\({\text {PD}}\)-sets for the \({\mathbb {Z}}_4\)-linear Hadamard codes \(H_{\gamma , \delta }\) of length \(2^m\), \(m=\gamma +2\delta -1\), which are binary Hadamard codes (not necessarily linear) obtained as the Gray map image of quaternary linear codes of type \(2^\gamma 4^\delta \). Specifically, s-PD-sets of minimum size \(s+1\) for \(H_{\gamma , \delta }\), for all \(\delta \ge 3\) and \(2\le s \le \lfloor {\frac{2^{2\delta -2}}{\delta }}\rfloor -1\), are constructed and recursive constructions are described.     

Nonsplit Hecke algebras and perverse sheaves

Let H be a Hecke algebra arising as an endomorphism algebra of the representation of a Chevalley group G over \(F_{q}\) induced by a unipotent cuspidal representation of a Levi quotient L of a parabolic subgroup. We assume that L is not a torus. In this paper we outline a geometric interpretation of the coefficients of the canonical basis of H in terms of perverse sheaves. We illustrate this in detail in the case where the Weyl group of G is ot type \(B_{4}\) and that of L is of type \(B_{2}\).