The Schur functor on tensor powers

Let M be a left module for the Schur algebra S(n, r), and let \({s \in \mathbb{Z}^+}\) . Then \({M^{\otimes s}}\) is a \({(S(n,\,rs), F{\mathfrak{S}_{s}})}\) -bimodule, where the symmetric group \({{\mathfrak{S}_s}}\) on s letters acts on the right by place permutations. We show that the Schur functor f _rs sends \({M^{\otimes s}}\) to the \({(F{\mathfrak{S}_{rs}},F{\mathfrak{S}_s})}\) -bimodule \({F\mathfrak{S}_{rs}\otimes_{F(\mathfrak{S}_{r}\wr{\mathfrak{S}_s})} ((f_rM)^{\otimes s}\otimes_{F} F{\mathfrak{S}_s})}\) . As a corollary, we obtain the image under the Schur functor of the Lie power L ^s (M), exterior power \({\bigwedge^s(M)}\) of M and symmetric power S ^s (M).     

Security games on matroids

Two players, the Defender and the Attacker play the following game. A matroid \(M=(S,\mathcal {I})\), a weight function \(d:S\rightarrow \mathbb {R}^+\) and a cost function \(c:S\rightarrow \mathbb {R}\) are given. The Defender chooses a base B of the matroid M and the Attacker chooses an element \(s\in S\) of the ground set. In all cases, the Attacker pays the Defender the cost of attack c(s). Besides that, if \(s\in B\) then the Defender pays the Attacker the amount d(s); if, on the other hand, \(s\notin B\) then there is no further payoff. Special cases of this two-player, zero-sum game were considered and solved in various security-related applications. In this paper we show that it is also possible to compute Nash-equilibrium mixed strategies for both players in strongly polynomial time in the above general matroid setting. We also consider a further generalization where common bases of two matroids are chosen by the Defender and apply this to define and efficiently compute a new reliability metric on digraphs.     

Enveloping Algebras of the Nilpotent Malcev Algebra of Dimension Five

Pérez-Izquierdo and Shestakov recently extended the PBW theorem to Malcev algebras. It follows from their construction that for any Malcev algebra M over a field of characteristic ≠ 2, 3 there is a representation of the universal nonassociative enveloping algebra U(M) by linear operators on the polynomial algebra P(M). For the nilpotent non-Lie Malcev algebra \(\mathbb{M}\) of dimension 5, we use this representation to determine explicit structure constants for \(U(\mathbb{M})\); from this it follows that \(U(\mathbb{M})\) is not power-associative. We obtain a finite set of generators for the alternator ideal \(I(\mathbb{M}) \subset U(\mathbb{M})\) and derive structure constants for the universal alternative enveloping algebra \(A(\mathbb{M}) = U(\mathbb{M})/I(\mathbb{M})\), a new infinite dimensional alternative algebra. We verify that the map \(\iota\colon \mathbb{M} \to A(\mathbb{M})\) is injective, and so \(\mathbb{M}\) is special.     

A Compactness Theorem of the Space of Free Boundary <Emphasis Type="Italic">f</Emphasis>-Minimal Surfaces in Three-Dimensional Smooth Metric Measure Space with Boundary

Let \((M^3,g,e^{-f}d\mu _M)\) be a compact three-dimensional smooth metric measure space with nonempty boundary. Suppose that M has nonnegative Bakry–Émery Ricci curvature and the boundary \(\partial M\) is strictly f-mean convex. We prove that there exists a properly embedded smooth f-minimal surface \(\Sigma \) in M with free boundary \(\partial \Sigma \) on \(\partial M\). If we further assume that the boundary \(\partial M\) is strictly convex, then we prove that \(M^3\) is diffeomorphic to the 3-ball \(B^3\), and a compactness theorem for the space of properly embedded f-minimal surfaces with free boundary in such \((M^3,g,e^{-f}d\mu _M)\), when the topology of these f-minimal surfaces is fixed.     

Characterization of finite $$\varvec{}$$-grous by their Schur multilier

Let G be a finite p-group of order \(p^n\) and M(G) be its Schur multiplier. It is a well known result by Green that \(|M(G)|= p^{\frac{1}{2}n(n-1)-t(G)}\) for some \(t(G) \ge 0\). In this article, we classify non-abelian p-groups G of order \(p^n\) for \(t(G)=\log _p(|G|)+1\).     

Certain families of polynomials arising in the study of hyperelliptic Lie algebras

The commutative ring \(R(P(t))=\mathbb C[t^{\pm 1},u \,|\, u^2=P(t)]\), where \(P(t)=\sum _{i=0}^na_it^i=\prod _{k=1}^n(t-\alpha _i)\) with \(\alpha _i\in \mathbb C\) pairwise distinct, is the coordinate ring of a hyperelliptic curve when \(n>4\). The Lie algebra \(\mathcal {R}(P(t))={\text {Der}}(R(P(t)))\) of derivations is called the hyperelliptic Lie algebra associated to P(t) and is a particular type of multipoint Krichever–Novikov algebra. In this paper, we describe the universal central extension of \({\text {Der}}(R(P(t)))\) in terms of certain families of polynomials which in a particular case are associated Legendre polynomials. Moreover we describe certain families of polynomials that arise in the study of the group of units for the ring R(P(t)),  where \(P(t)=t^4-2bt^2+1\). In this study, pairs of Chebyshev polynomials \((U_n,T_n)\) arise as particular cases of a pairs \((r_n,s_n)\) with \(r_n+s_n\sqrt{P(t)}\) a unit in R(P(t)). We explicitly describe these polynomial pairs as coefficients of certain generating functions and show that certain of these polynomials satisfy particular second-order linear differential equations.     

Chern characters for twisted matrix factorizations and the vanishing of the higher Herbrand difference

We develop a theory of “ad hoc” Chern characters for twisted matrix factorizations associated to a scheme X, a line bundle \(\mathcal {L}\), and a regular global section \(W \in \Gamma (X, \mathcal {L})\). As an application, we establish the vanishing, in certain cases, of \(h_c^R(M,N)\), the higher Herbrand difference, and, \(\eta _c^R(M,N)\), the higher codimensional analogue of Hochster’s theta pairing, where R is a complete intersection of codimension c with isolated singularities and M and N are finitely generated R-modules. Specifically, we prove such vanishing if \(R = Q/(f_1, \dots , f_c)\) has only isolated singularities, Q is a smooth k-algebra, k is a field of characteristic 0, the \(f_i\)’s form a regular sequence, and \(c \ge 2\). Such vanishing was previously established in the general characteristic, but graded, setting in Moore et al. (Math Z 273(3–4):907–920, 2013).     

Spectral Properties of <Emphasis Type="Italic">k</Emphasis>-Quasi-<Emphasis Type="Italic">M</Emphasis>-hyponormal Operators

In this paper, we study the k-quasi-M-hyponormal operator and mainly prove that if T is a k-quasi-M-hyponormal operator, then \(\sigma _{ja}(T)\backslash \{0\}=\sigma _{a}(T)\backslash \{0\}\), and the spectrum is continuous on the class of all k-quasi-M-hyponormal operators; let \(d_{AB}\in B(B(H))\) denote either the generalized derivation \(\delta _{AB}= L_{A}-R_{B}\) or the elementary operator \(\Delta _{AB} =L_{A}R_{B}- I\), we show that if A and \(B^{*}\) are k-quasi-M-hyponormal operators, then \(d_{AB}\) is polaroid and generalized Weyl’s theorem holds for \(f(d_{AB})\), where f is an analytic function on \(\sigma (d_{AB})\) and f is not constant on each connected component of the open set U containing \(\sigma (d_{AB})\). In additon, we discuss the hyperinvariant subspace problem for k-quasi-M-hyponormal operators.     

2-Domination number of generalized Petersen graphs

Let \(G=(V,E)\) be a graph. A subset \(S\subseteq V\) is a k-dominating set of G if each vertex in \(V-S\) is adjacent to at least k vertices in S. The k-domination number of G is the cardinality of the smallest k-dominating set of G. In this paper, we shall prove that the 2-domination number of generalized Petersen graphs \(P(5k+1, 2)\) and \(P(5k+2, 2)\), for \(k>0\), is \(4k+2\) and \(4k+3\), respectively. This proves two conjectures due to Cheng (Ph.D. thesis, National Chiao Tung University, 2013). Moreover, we determine the exact 2-domination number of generalized Petersen graphs P(2k, k) and \(P(5k+4,3)\). Furthermore, we give a good lower and upper bounds on the 2-domination number of generalized Petersen graphs \(P(5k+1, 3), P(5k+2,3)\) and \(P(5k+3, 3).\)     

On finitely generated modules over quasi-Euclidean rings

Let R be a unital commutative ring, and let M be an R-module that is generated by k elements but not less. Let \(\text {E}_n(R)\) be the subgroup of \(\text {GL}_n(R)\) generated by the elementary matrices. In this paper we study the action of \(\text {E}_n(R)\) by matrix multiplication on the set \(\text {Um}_n(M)\) of unimodular rows of M of length \(n \ge k\). Assuming R is moreover Noetherian and quasi-Euclidean, e.g., R is a direct product of finitely many Euclidean rings, we show that this action is transitive if \(n > k\). We also prove that \(\text {Um}_k(M) /\text {E}_k(R)\) is equipotent with the unit group of \(R/\mathfrak {a}_1\) where \(\mathfrak {a}_1\) is the first invariant factor of M. These results encompass the well-known classification of Nielsen non-equivalent generating tuples in finitely generated Abelian groups.     

ON THE COADJOINT ORBITS OF MAXIMAL UNIPOTENT SUBGROUPS OF REDUCTIVE GROUPS

Let G be a simple algebraic group defined over an algebraically closed field of characteristic 0 or a good prime for G. Let U be a maximal unipotent subgroup of G and \( \mathfrak{u} \) its Lie algebra. We prove the separability of orbit maps and the connectedness of centralizers for the coadjoint action of U on (certain quotients of) the dual \( \mathfrak{u} \)* of \( \mathfrak{u} \). This leads to a method to give a parametrization of the coadjoint orbits in terms of so-called minimal representatives which form a disjoint union of quasi-affine varieties. Moreover, we obtain an algorithm to explicitly calculate this parametrization which has been used for G of rank at most 8, except E₈.When G is defined and split over the field of q elements, for q the power of a good prime for G, this algorithmic parametrization is used to calculate the number k(U(q); \( \mathfrak{u} \)*(q)) of coadjoint orbits of U(q) on \( \mathfrak{u} \)*(q). Since k(U(q), \( \mathfrak{u} \)*(q)) coincides with the number k(U(q)) of conjugacy classes in U(q), these calculations can be viewed as an extension of the results obtained in [11]. In each case considered here there is a polynomial h(t) with integer coefficients such that for every such q we have k(U(q)) = h(q). We also explain implications of our results for a parametrization of the irreducible complex characters of U(q).     

$$L^p$$-Analysis of the Hodge–Dirac Operator Associated with Witten Laplacians on Complete Riemannian Manifolds

We prove R-bisectoriality and boundedness of the \(H^\infty \)-functional calculus in \(L^p\) for all \(1<p<\infty \) for the Hodge–Dirac operator associated with Witten Laplacians on complete Riemannian manifolds with non-negative Bakry–Emery Ricci curvature on k-forms.     

Harmonic <Emphasis Type="Italic">p</Emphasis>-forms on Hadamard manifolds with finite total curvature

In the present note, the geometric structures and topological properties of harmonic p-forms on a complete noncompact submanifold \(M^{n}(n\ge 4)\) immersed in Hadamard manifold \(N^{n+m}\) are discussed, where \(M^{n}\) and \(N^{n+m}\) are assumed to have flat normal bundle and pure curvature tensor, respectively. Firstly, under the assumption that \(M^{n}\) satisfies the \((\mathcal {P}_\rho )\) property (i.e., the weighted Poincaré inequality holds on \(M^{n}\)) and the \((p,n-p)\)-curvature of \(N^{n+m}\) is not less than a given negative constant, using Moser iteration, the space of all \(L^{2}\) harmonic p-forms on \(M^{n}\) is proven to have finite dimensions if \(M^{n}\) has finite total curvature. Furthermore, if the total curvature is small enough or \(M^{n}\) has at most Euclidean volume growth, two vanishing theorems are, respectively, established for harmonic p-forms. Note that the two vanishing theorems extend several previous results obtained by H. Z. Lin.     

A unifying approach to the Margolis–Meakin and Birget–Rhodes group expansion

Let G be a group. We show that the Birget–Rhodes prefix expansion \(G^{Pr}\) and the Margolis–Meakin expansion M(X; f) of G with respect to \(f:X\rightarrow G\) can be regarded as inverse subsemigroups of a common E-unitary inverse semigroup P. We construct P as an inverse subsemigroup of an E-unitary inverse monoid \(U/\zeta \) which is a homomorphic image of the free product U of the free semigroup \(X^+\) on X and G. The semigroup P satisfies a universal property with respect to homomorphisms into the permissible hull C(S) of a suitable E-unitary inverse semigroup S, with \(S/\sigma _S=G\), from which the characterizing universal properties of \(G^{Pr}\) and M(X; f) can be recaptured easily.     

Weak regularity of group algebras

A Banach algebra \(\mathcal {A}\) is called weakly regular if its multiplicative semigroup is E-inversive. We show that for a unimodular group G which admits an integrable unitary representation, \(L^1(G)\) is weakly regular. Moreover for a locally compact Abelian group, \(L^1(G)\) is weakly regular if and only if G is compact; while \(L^1(G)^{**}\) is weakly regular if and only if G is finite. All of our results hold, if we replace \(L^1(G)\) with M(G).     

Complexities of normal bases constructed from Gauss periods

Let q be a power of a prime p, and let \(r=nk+1\) be a prime such that \(r\not \mid q\), where n and k are positive integers. Under a simple condition on q, r and k, a Gauss period of type (n, k) is a normal element of \({\mathbb {F}}_{q}^{n}\) over \({\mathbb {F}}_q\); the complexity of the resulting normal basis of \({\mathbb {F}}_{q}^{n}\) over \({\mathbb {F}}_q\) is denoted by C(n, k; p). Recent works determined C(n, k; p) for \(k\le 7\) and all qualified n and q. In this paper, we show that for any given \(k>0\), C(n, k; p) is given by an explicit formula except for finitely many primes \(r=nk+1\) and the exceptional primes are easily determined. Moreover, we describe an algorithm that allows one to compute C(n, k; p) for the exceptional primes \(r=nk+1\). Our numerical results cover C(n, k; p) for \(k\le 20\) and all qualified n and q.     

Isometric multipliers of a vector valued Beurling algebra on a discrete semigroup

Let (S,ω) be a weighted abelian semigroup, let M _ω (S) be the semigroup of ω-bounded multipliers of S, and let \(\mathcal {A}\) be a strictly convex commutative Banach algebra with identity. It is shown that T is an onto isometric multiplier of \(\ell ^{1}(S,\omega , \mathcal {A})\) if and only if there exists an invertible σ ∈ M _ω (S), a unitary point \(a \in \mathcal {A}\), and a k>0 such that \(T(f)= ka{\sum }_{x \in S} f(x)\delta _{\sigma (x)}\) for each \(f={\sum }_{x \in S}f(x)\delta _{x} \in \ell ^{1}(S,\omega ,\mathcal {A})\). It is also shown that an isomorphism from \(\ell ^{1}(S_{1},\omega _{1},\mathcal {A})\) onto \(\ell ^{1}(S_{2},\omega _{2}, \mathcal {B})\) induces an isomorphism from \(M(\ell ^{1}(S_{1},\omega _{1},\mathcal {A}))\), the set of all multipliers of \(\ell ^{1}(S_{1},\omega _{1},\mathcal {A})\), onto \(M(\ell ^{1}(S_{2},\omega _{2},\mathcal {B}))\).     

DISTINGUISHED PRE-NICHOLS ALGEBRAS

We formally define and study the distinguished pre-Nichols algebra \( \tilde{B} \)(V) of a braided vector space of diagonal type V with finite-dimensional Nichols algebra B(V). The algebra \( \tilde{B} \)(V) is presented by fewer relations than B(V), so it is intermediate between the tensor algebra T(V) and B(V). Prominent examples of distinguished pre-Nichols algebras are the positive parts of quantized enveloping (super)algebras and their multiparametric versions. We prove that these algebras give rise to new examples of Noetherian pointed Hopf algebras of finite Gelfand-Kirillov dimension. We investigate the kernel (in the sense of Hopf algebras) of the projection from \( \tilde{B} \)(V) to B(V), generalizing results of De Concini and Procesi on quantum groups at roots of unity.