Irreducible representations of the classical Lie superalgebra of type A(0,n) in prime characteristic

Let $\mathfrak{g }=\mathfrak{s }\mathfrak{l }(1|n+1)$ be the classical Lie superalgebra of type $A(0,n)$ over an algebraically closed field of prime characteristic $p>2$ . A sufficient condition is provided for baby Kac $\mathfrak{g }$ -modules to be simple. Moreover, simple $\mathfrak{g }$ -modules with (quasi) regular semisimple characters are classified. In particular, up to isomorphism, all the simple modules for $\mathfrak{s }\mathfrak{l }(1|2)$ are determined, and representatives and dimensions of simples are precisely given. As an application, simple modules for the general linear Lie superalgebra $\mathfrak{g }\mathfrak{l }(1|n+1)$ with certain $p$ -characters are classified. In particular, a complete classification of simple $\mathfrak{g }\mathfrak{l }(1|2)$ -modules is given.     

Lax embeddings of the Hermitian unital

In this paper, we prove that every lax generalized Veronesean embedding of the Hermitian unital ${\mathcal{U}}$ of ${\mathsf{PG}(2,\mathbb{L}), \mathbb{L}}$ a quadratic extension of the field ${\mathbb{K}}$ and ${|\mathbb{K}| \geq 3}$ , in a ${\mathsf{PG}(d,\mathbb{F})}$ , with ${\mathbb{F}}$ any field and d ≥ 7, such that disjoint blocks span disjoint subspaces, is the standard Veronesean embedding in a subgeometry ${\mathsf{PG}(7,\mathbb{K}^{\prime})}$ of ${\mathsf{PG}(7,\mathbb{F})}$ (and d = 7) or it consists of the projection from a point ${p \in \mathcal{U}}$ of ${\mathcal{U}{\setminus} \{p\}}$ from a subgeometry ${\mathsf{PG}(7,\mathbb{K}^{\prime})}$ of ${\mathsf{PG}(7,\mathbb{F})}$ into a hyperplane ${\mathsf{PG}(6,\mathbb{K}^{\prime})}$ . In order to do so, when ${|\mathbb{K}| >3 }$ we strongly use the linear representation of the affine part of ${\mathcal{U}}$ (the line at infinity being secant) as the affine part of the generalized quadrangle ${\mathsf{Q}(4,\mathbb{K})}$ (the solid at infinity being non-singular); when ${|\mathbb{K}| =3}$ , we use the connection of ${\mathcal{U}}$ with the generalized hexagon of order 2.     

On unitals in PG(2,q^2) stabilized by a homology group

The linear collineation group of a classical unital of $\mathrm{PG}(2,q^2)$ contains a group of homologies of order $q+1$ . In this paper we prove that if $\mathcal{U }$ is a unital of PG $(2,q^2)$ stabilized by a homology group of order $q+1$ and $q$ is a prime number, then $\mathcal{U }$ is classical.     

Characterization of projective linear groups by the order of the normalizer of a Sylow subgroup

Let $r$ be a prime and $G$ be a finite group, and let $R, \,S$ be Sylow $r$ -subgroups of $G$ and $\text{ PGL }(2, r)$ respectively. We prove the following results: (1) If $|G|=|\text{ PGL }(2, r)|$ and $|N_{G}(R)|=|N_{\mathrm{PGL}(2, r)} (S)|$ and $r$ is not a Mersenne prime, then $G$ is isomorphic to $\text{ PSL } (2, r) \times C_{2}, \,\text{ SL }(2, r)$ or $\text{ PGL }(2, r)$ . (2) If $|G|=|\text{ PGL }(2, r)|, \,|N_{G}(R)|=|N_{\mathrm{PGL}(2, r)}(S)|$ where $r>3$ is a Mersenne prime and $r$ is an isolated vertex of the prime graph of $G$ , then $G\cong \text{ PGL }(2, r)$ .     

On the singular set of minimizers of p(x)-energies

We treat the partial regularity of locally bounded local minimizers $u$ for the $p(x)$ -energy functional $$\begin{aligned} \mathcal{E }(v;\Omega ) = \int \left( g^{\alpha \beta }(x)h_{ij}(v) D_\alpha v^i (x) D_\beta v^j (x) \right) ^{p(x)/2} dx, \end{aligned}$$ defined for maps $v : \Omega (\subset \mathbb R ^m) \rightarrow \mathbb R ^n$ . Assuming the Lipschitz continuity of the exponent $p(x) \ge 2$ , we prove that $u \in C^{1,\alpha }(\Omega _0)$ for some $\alpha \in (0,1)$ and an open set $\Omega _0 \subset \Omega $ with $\dim _\mathcal{H }(\Omega \setminus \Omega _0) \le m-[\gamma _1]-1$ , where $\dim _\mathcal{H }$ stands for the Hausdorff dimension, $[\gamma _1]$ the integral part of $\gamma _1$ , and $\gamma _1 = \inf p(x)$ .     

Growth estimates for orbits of self-adjoint groups

Let $G$ denote a closed, connected, self-adjoint, noncompact subgroup of $GL(n,\mathbb R )$ , and let $d_{R}$ and $d_{L}$ denote respectively the right and left invariant Riemannian metrics defined by the canonical inner product on $M(n,\mathbb R ) = T_{I} GL(n,\mathbb R )$ . Let $v$ be a nonzero vector of $\mathbb R ^{n}$ such that the orbit $G(v)$ is unbounded in $\mathbb R ^{n}$ . Then the function $g \rightarrow d_{R}(g, G_{v})$ is unbounded, where $G_{v} = \{g \in G : g(v) = v \}$ , and we obtain algebraically defined upper and lower bounds $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ for the asymptotic behavior of the function $\frac{log|g(v)|}{d_{R}(g, G_{v})}$ as $d_{R}(g, G_{v}) \rightarrow \infty $ . The upper bound $\lambda ^{+}(v)$ is at most 1. The orbit $G(v)$ is closed in $\mathbb R ^{n} \Leftrightarrow \lambda ^{-}(w)$ is positive for some w $\in G(v)$ . If $G_{v}$ is compact, then $g \rightarrow |d_{R}(g,I) - d_{L}(g,I)|$ is uniformly bounded in $G$ , and the exponents $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ are sharp upper and lower asymptotic bounds for the functions $\frac{log|g(v)|}{d_{R}(g,I)}$ and $\frac{log|g(v)|}{d_{L}(g,I)}$ as $d_{R}(g,I) \rightarrow \infty $ or as $d_{L}(g,I) \rightarrow \infty $ . However, we show by example that if $G_{v}$ is noncompact, then there need not exist asymptotic upper and lower bounds for the function $\frac{log|g(v)|}{d_{L}(g, G_{v})}$ as $d_{L}(g, G_{v}) \rightarrow \infty $ . The results apply to representations of noncompact semisimple Lie groups $G$ on finite dimensional real vector spaces. We compute $\lambda ^{+}$ and $\lambda ^{-}$ for the irreducible, real representations of $SL(2,\mathbb R )$ , and we show that if the dimension of the $SL(2,\mathbb R )$ -module $V$ is odd, then $\lambda ^{+} = \lambda ^{-}$ on a nonempty open subset of $V$ . We show that the function $\lambda ^{-}$ is $K$ -invariant, where $K = O(n,\mathbb R ) \cap G$ . We do not know if $\lambda ^{-}$ is $G$ -invariant.     

On the Linear Dependence of a Finite Set of Additive Functions

In this note we prove the following: Let n?≥ 2 be a fixed integer. A system of additive functions ${A_{1},A_{2},\ldots,A_{n}:\mathbb{R} \to\mathbb{R}}$ is linearly dependent (as elements of the ${\mathbb{R}}$ vector space ${\mathbb{R}^{\mathbb{R}}}$ ), if and only if, there exists an indefinite quadratic form ${Q:\mathbb{R}^{n}\to\mathbb{R} }$ such that ${Q(A_{1}(x),A_{2}(x),\ldots,A_{n}(x))\geq 0}$ or ${Q(A_{1}(x),A_{2}(x),\ldots,A_{n}(x))\leq 0}$ holds for all ${x\in\mathbb{R}}$ .     

On Suborbital Graphs for the Extended Modular Group {\hat{\Gamma}}

In this paper, we show that the extended modular group ${\hat{\Gamma}}$ acts on ${\hat{\mathbb{Q}}}$ transitively and imprimitively. Then the number of orbits of ${\hat{\Gamma} _{0}(N)}$ on ${\hat{\mathbb{Q}}}$ is calculated and compared with the number of orbits of ${\Gamma _{0}(N)}$ on ${\hat{\mathbb{Q}}}$ . Especially, we obtain the graphs ${\hat{G}_{u, N}}$ of ${\hat{\Gamma}_{0}(N)}$ on ${\hat{\mathbb{Q}}}$ , for each ${N\in\mathbb{N}}$ and each unit ${u \in U_{N} }$ , then we determine the suborbital graph ${\hat{F}_{u,N}}$ . We also give the edge conditions in ${\hat{G}_{u, N}}$ and the necessary and sufficient conditions for a circuit to be triangle in ${\hat{F}_{u, N}.}$      

Parametrization of \text{ PSL }(2,\mathbb C )-representations of surface groups

For an oriented surface of genus $g$ with $b$ boundary components, we construct a rational map from a subset of $\mathbb{C }^{6g-6+3b}$ onto an open algebraic subset of the $\text{ PSL }(2,\mathbb C )$ -character variety as an analogue of the Fenchel-Nielsen coordinates. After taking the quotient by an action of a finite group, we obtain a parametrization of a subset of the $\text{ PSL }(2,\mathbb C )$ -character variety, and similarly for the $\text{ SL }(2,\mathbb C )$ -character variety. We can systematically calculate a set of matrix generators by rational functions of the parameters. We give transformation formulae under elementary moves of pants decompositions.     

Pfaffian representations of cubic surfaces

Let $\mathbb{K }$ be a field of characteristic zero. We describe an algorithm which requires a homogeneous polynomial $F$ of degree three in $\mathbb{K }[x_{0},x_1,x_{2},x_{3}]$ and a zero ${\mathbf{a }}$ of $F$ in $\mathbb{P }^{3}_{\mathbb{K }}$ and ensures a linear Pfaffian representation of $\text{ V}(F)$ with entries in $\mathbb{K }[x_{0},x_{1},x_{2},x_{3}]$ , under mild assumptions on $F$ and ${\mathbf{a }}$ . We use this result to give an explicit construction of (and to prove the existence of) a linear Pfaffian representation of $\text{ V}(F)$ , with entries in $\mathbb{K }^{\prime }[x_{0},x_{1},x_{2},x_{3}]$ , being $\mathbb{K }^{\prime }$ an algebraic extension of $\mathbb{K }$ of degree at most six. An explicit example of such a construction is given.     

A converse of Baer’s theorem

Schur’s classical theorem states that for a group $G$ , if $G/Z(G)$ is finite, then $G'$ is finite. Baer extended this theorem for the factor group $G/Z_n(G)$ , in which $Z_n(G)$ is the $n$ -th term of the upper central series of $G$ . Hekster proved a converse of Baer’s theorem as follows: If $G$ is a finitely generated group such that $\gamma _{n+1}(G)$ is finite, then $G/Z_n(G)$ is finite where $\gamma _{n+1}(G)$ denotes the $(n+1)$ st term of the lower central series of $G$ . In this paper, we generalize this result by obtaining the same conclusion under the weaker hypothesis that $G/Z_n(G)$ is finitely generated. Furthermore, we show that the index of the subgroup $Z_n(G)$ is bounded by a precisely determined function of the order of $\gamma _{n+1}(G)$ . Moreover, we prove that the mentioned theorem of Hekster is also valid under a weaker condition that $Z_{2n}(G)/Z_{n}(G)$ is finitely generated. Although in this case the bound for the order of $\gamma _{n+1}(G)$ is not achieved.     

On the Intersection of the Normalizers of the \boldsymbol{\cal{F}}-Residuals of Subgroups of a Finite Group

In this paper we give criteria for a finite group to belong to a formation. As applications, recent theorems of Li, Shen, Shi and Qian are generalized. Let G  be a finite group, $\cal F$ a formation and p  a prime. Let $D_{\mathcal {F}}(G)$ be the intersection of the normalizers of the $\cal F$ -residuals of all subgroups of G, and let $D_{\mathcal {F}}^{p}(G)$ be the intersection of the normalizers of $(H^{\cal F}O_{p'}(G))$ for all subgroups H of G. We then define $D_{\mathcal F}^{0}(G)=D_{\mathcal F, p}^{~0}(G)=1$ and $D_{\mathcal F}^{i+1}(G)/D_{\mathcal F}^{i}(G)=D_{\mathcal F}(G/D_{\mathcal F}^{i}(G))$ , $D_{\mathcal F, p}^{i+1}(G)/D_{\mathcal F, p}^{~i}(G)=D_{\mathcal F, p}(G/D_{\mathcal F, p}^{~i}(G))$ . Let $D_{\mathcal {F}}^{\infty}(G)$ and $D_{\mathcal {F}, p}^{~\infty}(G)$ denote the terminal member of the ascending series of $D_{\mathcal F}^{i}(G)$ and $D_{\mathcal F, p}^{~i}(G)$ respectively. In this paper we prove that under certain hypotheses, the the $\cal F$ -residual $G^{\cal F}$ is nilpotent (respectively,p-nilpotent) if and only if $G=D_{\mathcal {F}}^{\infty}(G)$ (respectively, $G=D_{\mathcal {F}, p}^{~\infty}(G)$ ). Further more, if the formation $\cal F$ is either the class of all nilpotent groups or the class of all abelian groups, then $G^{\cal F}$ is p-nilpotent if and only if and only if every cyclic subgroup of G order p and 4 (if p?=?2) is contained in $D_{\mathcal {F}, p}^{~\infty}(G)$ .     

Large free sets in powers of universal algebras

We prove that for each universal algebra ${(A, \mathcal{A})}$ of cardinality ${|A| \geq 2}$ and infinite set X of cardinality ${|X| \geq | \mathcal{A}|}$ , the X-th power ${(A^{X}, \mathcal{A}^{X})}$ of the algebra ${(A, \mathcal{A})}$ contains a free subset ${\mathcal{F} \subset A^{X}}$ of cardinality ${|\mathcal{F}| = 2^{|X|}}$ . This generalizes the classical Fichtenholtz–Kantorovitch–Hausdorff result on the existence of an independent family ${\mathcal{I} \subset \mathcal{P}(X)}$ of cardinality ${|\mathcal{I}| = |\mathcal{P}(X)|}$ in the Boolean algebra ${\mathcal{P}(X)}$ of subsets of an infinite set X.     

On the largest conjugacy class length of a finite group

Let $G$ be a finite group and $\mathrm{bcl}(G)$ the largest conjugacy class length of $G$ . In this note we slightly improve He and Shi’s lower bound for $\mathrm{bcl}(G)$ , showing that $|\mathrm{bcl}(G)|\ge p^{\frac{1}{p}}(|G:O_{p}(G)|_{p})^{\frac{p-1}{p}}$ .     

Maximal Regularity for Non-Autonomous Second Order Cauchy Problems

We consider non-autonomous wave equations $$\left\{\begin{array}{ll}\ddot{u}(t) + \mathcal{B}(t) \dot{u}(t) + \mathcal{A}(t)u(t) = f(t) \quad t{\text -}{\rm a.e.}\\ u(0) = u_{0},\, \dot{u}(0) = u_{1}.\\\end{array}\right.$$ where the operators ${\mathcal{A}(t)}$ and ${\mathcal{B}(t)}$ are associated with time-dependent sesquilinear forms ${\mathfrak{a}(t, ., .)}$ and ${\mathfrak{b}}$ defined on a Hilbert space H with the same domain V. The initial values satisfy ${u_0 \in V}$ and ${u_1 \in H}$ . We prove well-posedness and maximal regularity for the solution both in the spaces V′ and H. We apply the results to non-autonomous Robin-boundary conditions and also use maximal regularity to solve a quasilinear problem.     

Privileged Regions in Critical Strips of Non-lattice Dirichlet Polynomials

This paper shows, by means of Kronecker’s theorem, the existence of infinitely many privileged regions called $r$ -rectangles (rectangles with two semicircles of small radius $r$ ) in the critical strip of each function $L_{n}(z)\!:=\!$ $1-\sum _{k=2}^{n}k^{z}$ , $n\!\ge \!2$ , containing exactly $\left[ \dfrac{T\log n}{2\pi }\right] +1$ zeros of $L_{n}(z)$ , where $T$ is the height of the $r$ -rectangle and $\left[\cdot \right]$ represents the integer part.     

Homogeneous almost quaternion-Hermitian manifolds

An almost quaternion-Hermitian structure on a Riemannian manifold $(M^{4n},g)$ is a reduction of the structure group of $M$ to $\mathrm{Sp}(n)\mathrm{Sp}(1)\subset \text{ SO }(4n)$ . In this paper we show that a compact simply connected homogeneous almost quaternion-Hermitian manifold of non-vanishing Euler characteristic is either a Wolf space, or $\mathbb{S }^2\times \mathbb{S }^2$ , or the complex quadric $\text{ SO }(7)/\mathrm{U}(3)$ .