New weighted Rogers-Ramanujan partition theorems and their implications
Krishnaswami
Alladi;
Alexander
Berkovich
2557-2577
Abstract: This paper has a two-fold purpose. First, by considering a reformulation of a deep theorem of Göllnitz, we obtain a new weighted partition identity involving the Rogers-Ramanujan partitions, namely, partitions into parts differing by at least two. Consequences of this include Jacobi's celebrated triple product identity for theta functions, Sylvester's famous refinement of Euler's theorem, as well as certain weighted partition identities. Next, by studying partitions with prescribed bounds on successive ranks and replacing these with weighted Rogers-Ramanujan partitions, we obtain two new sets of theorems - a set of three theorems involving partitions into parts $\not \equiv 0, \pm i$ (mod 6), and a set of three theorems involving partitions into parts $\not \equiv 0, \pm i$ (mod 7), $i=1,2,3$.
Core and residual intersections of ideals
Alberto
Corso;
Claudia
Polini;
Bernd
Ulrich
2579-2594
Abstract: D. Rees and J. Sally defined the core of an $R$-ideal $I$ as the intersection of all (minimal) reductions of $I$. However, it is not easy to give an explicit characterization of it in terms of data attached to the ideal. Until recently, the only case in which a closed formula was known is the one of integrally closed ideals in a two-dimensional regular local ring, due to C. Huneke and I. Swanson. The main result of this paper explicitly describes the core of a broad class of ideals with good residual properties in an arbitrary local Cohen-Macaulay ring. We also find sharp bounds on the number of minimal reductions that one needs to intersect to get the core.
Macaulay style formulas for sparse resultants
Carlos
D'Andrea
2595-2629
Abstract: We present formulas for computing the resultant of sparse polynomials as a quotient of two determinants, the denominator being a minor of the numerator. These formulas extend the original formulation given by Macaulay for homogeneous polynomials.
Surfaces with $p_g=q=3$
Christopher
D.
Hacon;
Rita
Pardini
2631-2638
Abstract: We classify minimal complex surfaces of general type with $p_g=q=3$. More precisely, we show that such a surface is either the symmetric product of a curve of genus $3$ or a free $\mathbb{Z} _2-$quotient of the product of a curve of genus $2$ and a curve of genus $3$. Our main tools are the generic vanishing theorems of Green and Lazarsfeld and the characterization of theta divisors given by Hacon in Corollary 3.4 of Fourier transforms, generic vanishing theorems and polarizations of abelian varieties.
On Fano manifolds with Nef tangent bundles admitting 1-dimensional varieties of minimal rational tangents
Ngaiming
Mok
2639-2658
Abstract: Let $X$ be a Fano manifold of Picard number 1 with numerically effective tangent bundle. According to the principal case of a conjecture of Campana-Peternell's, $X$ should be biholomorphic to a rational homogeneous manifold $G/P$, where $G$ is a simple Lie group, and $P \subset G$ is a maximal parabolic subgroup. In our opinion there is no overriding evidence for the Campana-Peternell Conjecture for the case of Picard number 1 to be valid in its full generality. As part of a general programme that the author has undertaken with Jun-Muk Hwang to study uniruled projective manifolds via their varieties of minimal rational tangents, a new geometric approach is adopted in the current article in a special case, consisting of (a) recovering the generic variety of minimal rational tangents $\mathcal{C}_x$, and (b) recovering the structure of a rational homogeneous manifold from $\mathcal{C}_x$. The author proves that, when $b_4(X) = 1$ and the generic variety of minimal rational tangents is 1-dimensional, $X$ is biholomorphic to the projective plane $\mathbb{P} ^2$, the 3-dimensional hyperquadric $Q^3$, or the 5-dimensional Fano homogeneous contact manifold of type $G_2$, to be denoted by $K(G_2)$. The principal difficulty is part (a) of the scheme. We prove that $\mathcal{C}_x \subset \mathbb{P} T_x(X)$ is a rational curve of degrees $\leq 3$, and show that $d = 1$ resp. 2 resp. 3 corresponds precisely to the cases of $X = \mathbb{P} ^2$ resp. $Q^3$ resp. $K(G_2)$. Let $\mathcal{K}$ be the normalization of a choice of a Chow component of minimal rational curves on $X$. Nefness of the tangent bundle implies that $\mathcal{K}$ is smooth. Furthermore, it implies that at any point $x \in X$, the normalization $\mathcal{K}_x$of the corresponding Chow space of minimal rational curves marked at $x$is smooth. After proving that $\mathcal{K}_x$ is a rational curve, our principal object of study is the universal family $\mathcal{U}$ of $\mathcal{K}$, giving a double fibration $\rho: \mathcal{U} \to \mathcal{K}, \mu: \mathcal{U} \to X$, which gives $\mathbb{P} ^1$-bundles. There is a rank-2 holomorphic vector bundle $V$ on $\mathcal{K}$ whose projectivization is isomorphic to $\rho: \mathcal{U} \to \mathcal{K}$. We prove that $V$ is stable, and deduce the inequality $d \leq 4$ from the inequality $c_1^2(V) \leq 4c_2(V)$ resulting from stability and the existence theorem on Hermitian-Einstein metrics. The case of $d = 4$ is ruled out by studying the structure of the curvature tensor of the Hermitian-Einstein metric on $V$ in the special case where $c_1^2(V) = 4c_2(V)$.
A Pieri-type formula for isotropic flag manifolds
Nantel
Bergeron;
Frank
Sottile
2659-2705
Abstract: We give the formula for multiplying a Schubert class on an odd orthogonal or symplectic flag manifold by a special Schubert class pulled back from the Grassmannian of maximal isotropic subspaces. This is also the formula for multiplying a type $B$ (respectively, type $C$) Schubert polynomial by the Schur $P$-polynomial $p_m$ (respectively, the Schur $Q$-polynomial $q_m$). Geometric constructions and intermediate results allow us to ultimately deduce this formula from formulas for the classical flag manifold. These intermediate results are concerned with the Bruhat order of the infinite Coxeter group ${\mathcal B}_\infty$, identities of the structure constants for the Schubert basis of cohomology, and intersections of Schubert varieties. We show that most of these identities follow from the Pieri-type formula, and our analysis leads to a new partial order on the Coxeter group ${\mathcal B}_\infty$ and formulas for many of these structure constants.
Self-tilting complexes yield unstable modules
Alexander
Zimmermann
2707-2724
Abstract: Let $G$ be a group and $R$ a commutative ring. Let $TrPic_R(RG)$ be the group of isomorphism classes of standard self-equivalences of the derived category of bounded complexes of $RG$-modules. The subgroup $HD_R(G)$ of $TrPic_R(RG)$ consisting of self-equivalences fixing the trivial $RG$-module acts on the cohomology ring $H^*(G,R)$. The action is functorial with respect to $R$. The self-equivalences which are 'splendid' in a sense defined by J. Rickard act naturally with respect to transfer and restriction to centralizers of $p$-subgroups in case $R$ is a field of characteristic $p$. In the present paper we prove that this action of self-equivalences on $H^*(G,R)$ commutes with the action of the Steenrod algebra, and study the behaviour of the action of splendid self-equivalences with respect to Lannes' $T$-functor.
Enright's completions and injectively copresented modules
Steffen
König;
Volodymyr
Mazorchuk
2725-2743
Abstract: Let $\ensuremath{\mathfrak{A}}$be a finite-dimensional simple Lie algebra over the complex numbers. It is shown that a module is complete (or relatively complete) in the sense of Enright if and only if it is injectively copresented by certain injective modules in the BGG-category ${\mathcal O}$. Let $A$ be the finite-dimensional algebra associated to a block of ${\mathcal O}$. Then the corresponding block of the category of complete modules is equivalent to the category of $eAe$-modules for a suitable choice of the idempotent $e$. Using this equivalence, a very easy proof is given for Deodhar's theorem (also proved by Bouaziz) that completion functors satisfy braid relations. The algebra $eAe$ is left properly and standardly stratified. It satisfies a double centralizer property similar to Soergel's ``combinatorial description'' of ${\mathcal O}$. Its simple objects, their characters and their multiplicities in projective or standard objects are determined.
Representations of exceptional simple alternative superalgebras of characteristic 3
M.
C.
López-Díaz;
Ivan
P.
Shestakov
2745-2758
Abstract: We study representations of simple alternative superalgebras $B(1,2)$ and $B(2,4)$. The irreducible bimodules and bimodules with superinvolution over these superalgebras are classified, and some analogues of the Kronecker factorization theorem are proved for alternative superalgebras that contain $B(1,2)$ and $B(4,2)$.
Vertices for characters of $p$-solvable groups
Gabriel
Navarro
2759-2773
Abstract: Suppose that $G$ is a finite $p$-solvable group. We associate to every irreducible complex character $\chi \in \operatorname{Irr}(G)$ of $G$ a canonical pair $(Q,\delta)$, where $Q$ is a $p$-subgroup of $G$ and $\delta \in \operatorname{Irr}(Q)$, uniquely determined by $\chi$ up to $G$-conjugacy. This pair behaves as a Green vertex and partitions $\operatorname{Irr}(G)$ into ``families" of characters. Using the pair $(Q, \delta)$, we give a canonical choice of a certain $p$-radical subgroup $R$ of $G$ and a character $\eta \in \operatorname{Irr}(R)$ associated to $\chi$ which was predicted by some conjecture of G. R. Robinson.
Applications of Langlands' functorial lift of odd orthogonal groups
Henry
H.
Kim
2775-2796
Abstract: Together with Cogdell, Piatetski-Shapiro and Shahidi, we proved earlier the existence of a weak functorial lift of a generic cuspidal representation of $SO_{2n+1}$ to $GL_{2n}$. Recently, Ginzburg, Rallis and Soudry obtained a more precise form of the lift using their integral representation technique, namely, the lift is an isobaric sum of cuspidal representations of $GL_{n_i}$ (more precisely, cuspidal representations of $GL_{2n_i}$ such that the exterior square $L$-functions have a pole at $s=1$). One purpose of this paper is to give a simpler proof of this fact in the case that a cuspidal representation has one supercuspidal component. In a separate paper, we prove it without any condition using a result on spherical unitary dual due to Barbasch and Moy. We give several applications of the functorial lift: First, we parametrize square integrable representations with generic supercuspidal support, which have been classified by Moeglin and Tadic. Second, we give a criterion for cuspidal reducibility of supercuspidal representations of $GL_m\times SO_{2n+1}$. Third, we obtain a functorial lift from generic cuspidal representations of $SO_5$ to automorphic representations of $GL_5$, corresponding to the $L$-group homomorphism $Sp_4(\mathbb{C} )\longrightarrow GL_5(\mathbb{C} )$, given by the second fundamental weight.
Characterization of the Hilbert ball by its automorphism group
Kang-Tae
Kim;
Steven
G.
Krantz
2797-2818
Abstract: Let $\Omega$ be a bounded, convex domain in a separable Hilbert space. The authors prove a version of the theorem of Bun Wong, which asserts that if such a domain admits an automorphism orbit accumulating at a strongly pseudoconvex boundary point, then it is biholomorphic to the ball. Key ingredients in the proof are a new localization argument using holomorphic peaking functions and the use of new ``normal families'' arguments in the construction of the limit biholomorphism.
Regularity properties of solutions of a class of elliptic-parabolic nonlinear Levi type equations
G.
Citti;
A.
Montanari
2819-2848
Abstract: In this paper we prove the smoothness of solutions of a class of elliptic-parabolic nonlinear Levi type equations, represented as a sum of squares plus a vector field. By means of a freezing method the study of the operator is reduced to the analysis of a family $L_{\xi_0}$ of left invariant operators on a free nilpotent Lie group. The fundamental solution $\Gamma_{\xi_0}$ of the operator $L_{\xi_0}$ is used as a parametrix of the fundamental solution of the Levi operator, and provides an explicit representation formula for the solution of the given equation. Differentiating this formula and applying a bootstrap method, we prove that the solution is $C^\infty$.
A Markov partition that reflects the geometry of a hyperbolic toral automorphism
Anthony
Manning
2849-2863
Abstract: We show how to construct a Markov partition that reflects the geometrical action of a hyperbolic automorphism of the $n$-torus. The transition matrix is the transpose of the matrix induced by the automorphism in $u$-dimensional homology, provided this is non-negative. (Here $u$ denotes the expanding dimension.) That condition is satisfied, at least for some power of the original automorphism, under a certain non-degeneracy condition on the Galois group of the characteristic polynomial. The $(^n_u)$ rectangles are constructed by an iterated function system, and they resemble the product of the projection of a $u$-dimensional face of the unit cube onto the unstable subspace and the projection of minus the orthogonal $(n-u)$-dimensional face onto the stable subspace.
A priori degeneracy of one-dimensional rotation sets for periodic point free torus maps
Jaroslaw
Kwapisz
2865-2895
Abstract: Diffeomorphisms of the two torus that are isotopic to the identity have rotation sets that are convex compact subsets of the plane. We show that certain line segments (including all rationally sloped segments with no rational points) cannot be realized as a rotation set.
The inverse problem of the calculus of variations: The use of geometrical calculus in Douglas's analysis
W.
Sarlet;
G.
Thompson;
G.
E.
Prince
2897-2919
Abstract: The main objective of this paper is to work out a full-scale application of the integrability analysis of the inverse problem of the calculus of variations, as developed in recent papers by Sarlet and Crampin. For this purpose, the celebrated work of Douglas on systems with two degrees of freedom is taken as the reference model. It is shown that the coordinate-free, geometrical calculus used in Sarlet and Crampin's general theoretical developments provides effective tools also to do the practical calculations. The result is not only that all subcases distinguished by Douglas can be given a more intrinsic characterization, but also that in most of the cases, the calculations can be carried out in a more efficient way and often lead to sharper conclusions.
On the dimensional structure of hereditarily indecomposable continua
Roman
Pol;
Miroslawa
Renska
2921-2932
Abstract: Any hereditarily indecomposable continuum $X$ of dimension $n\geq 2$ is split into layers $B_r$ consisting of all points in $X$ that belong to some $r$-dimensional continuum but avoid any non-trivial continuum of dimension less than $r$. The subjects of this paper are the dimensional and the descriptive properties of the layers $B_r$.
Transition Probabilities for Symmetric Jump Processes
Richard
F.
Bass;
David
A.
Levin
2933-2953
Abstract: We consider symmetric Markov chains on the integer lattice in $d$ dimensions, where $\alpha \in (0,2)$ and the conductance between $x$ and $y$ is comparable to $\vert x-y\vert^{-(d+\alpha )}$. We establish upper and lower bounds for the transition probabilities that are sharp up to constants.
Commensurability of 1-cusped hyperbolic 3-manifolds
Danny
Calegari;
Nathan
M.
Dunfield
2955-2969
Abstract: We give examples of non-fibered hyperbolic knot complements in homology spheres that are not commensurable to fibered knot complements in homology spheres. In fact, we give many examples of knot complements in homology spheres where every commensurable knot complement in a homology sphere has non-monic Alexander polynomial.
A non-fixed point theorem for Hamiltonian lie group actions
Christopher
Allday;
Volker
Hauschild;
Volker
Puppe
2971-2982
Abstract: We prove that, under certain conditions, if a compact connected Lie group acts effectively on a closed manifold, then there is no fixed point. Because two of the main conditions are satisfied by any Hamiltonian action on a closed symplectic manifold, the theorem applies nicely to such actions. The method of proof, however, is cohomological; and so the result applies more generally.