b-Functions associated with quivers of type A

We study the b-functions of relative invariants of the prehomogeneous vector spaces associated with quivers of type A. By applying the decomposition formula for b-functions, we determine explicitly the b-functions of one variable for each irreducible relative invariant. Moreover, we give a graphical algorithm to determine the b-functions of several variables.     

An Endpoint Estimate for the Commutator of Singular Integrals

It is well known that the commutator Tb of the singular integral operator T with a BMO function b is bounded on L^P（R^n）, 1 〈 p 〈 ∞. In this paper, we consider the endpoint estimates for a kind of commutator of singular integrals. A BMO-type estimate for Tb is obtained under the assumption b ∈ LMO.     

On the divisibility of the discriminant of an integral polynomial by prime powers

The discriminant of an integral polynomial is one of its main characteristics. It influences the distribution of its roots, the structure of the finite extension of the rational field generated by the polynomial's roots. In the paper, we show that, for any given prime power p ^b , there exists an irreducible polynomial with discriminant being a multiple of p ^b .     

Irreducible Representations of the Quantum Weyl Algebra at Roots of Unity Given by Matrices

To describe the representation theory of the quantum Weyl algebra at an lth primitive root γ of unity, Boyette, Leyk, Plunkett, Sipe, and Talley found all nonsingular irreducible matrix solutions to the equation yx ? γxy = 1, assuming yx ≠ xy. In this note, we complete their result by finding and classifying, up to equivalence, all irreducible matrix solutions (X, Y), where X is singular.     

A note on preconditioned GMRES for solving singular linear systems

For solving a singular linear system Ax=b by GMRES, it is shown in the literature that if A is range-symmetric, then GMRES converges safely to a solution. In this paper we consider preconditioned GMRES for solving a singular linear system, we construct preconditioners by so-called proper splittings, which can ensure that the coefficient matrix of the preconditioned system is range-symmetric.     

The ring of physical states in the <Emphasis Type="Italic">M</Emphasis>(2, 3) minimal Liouville gravity

We consider the M(2, 3) minimal Liouville gravity, whose state space in the gravity sector is realized as irreducible modules of the Virasoro algebra. We present a recursive construction for BRST cohomology classes based on using an explicit form of singular vectors in irreducible modules of the Virasoro algebra. We find a certain algebra acting on the BRST cohomology space and use this algebra to find the operator algebra of physical states.     

The moduli component of the space of semistable rank-2 sheaves on ℙ3 with singularities of mixed dimension

A new irreducible component of the Gieseker–Maruyama moduli scheme M(3) of semistable coherent sheaves of rank 2 with Chern classes c₁ = 0, c₂ = 3, and c₃ = 0 on P³ such that its general point corresponds to a sheaf whose singular locus contains components of dimensions 0 and 1 is described. These sheaves are obtained by elementary transformations of stable reflexive sheaves of rank 2 with Chern classes c₁ = 0, c₂ = 2, and c₃ = 2 along the projective line. The constructed family of sheaves is the first example of an irreducible component of a Gieseker–Maruyama scheme whose general point corresponds to a sheaf with singularities of mixed dimension.

     

On Martos' and Chaehes-Coopeb's approach vis-a-vis

In this paper a linear fractional programming problem is studied in presence of “singular-points”. It is proved that “singular points”, if present, exist at an extreme point of S: = {x ? R ⁿ | Ax = b, x ≧0}

It is also shown that a “singular point” is adjacent to an optimal point of S and a characterization of a non-basic vector is obtained, whose entry into the optimal basis in Martos' approach yields the “singular point”.     

Singularities of Toric Varieties Associated with Finite Distributive Lattices

With each finite lattice L we associate a projectively embedded scheme V(L); as Hibi has shown, the lattice D is distributive if and only if V(D) is irreducible, in which case it is a toric variety. We first apply Birkhoff's structure theorem for finite distributive lattices to show that the orbit decomposition of V(D) gives a lattice isomorphic to the lattice of contractions of the bounded poset of join-irreducibles of D. Then we describe the singular locus of V(D) by applying some general theory of toric varieties to the fan dual to the order polytope of P: V(D) is nonsingular along an orbit closure if and only if each fibre of the corresponding contraction is a tree. Finally, we examine the local rings and associated graded rings of orbit closures in V(D). This leads to a second (self-contained) proof that the singular locus is as described, and a similar combinatorial criterion for the normal link of an orbit closure to be irreducible.     

Irreducible Triangulations of Surfaces with Boundary

A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus g ≥ 0 with b ≥ 0 boundary components is O(g + b). So far, the result was known only for surfaces without boundary (b = 0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boundary.     

Partial differential equation approach to F 4

Singular vectors of a representation of a finite-dimensional simple Lie algebra are weight vectors in the underlying module that are nullified by positive root vectors. In this article, we use partial differential equations to explicitly find all the singular vectors of the polynomial representation of the simple Lie algebra of type F ₄ over its 26-dimensional basic irreducible module, which also supplements a proof of the completeness of Brion’s abstractly described generators. Moreover, we show that the number of irreducible submodules contained in the space of homogeneous harmonic polynomials with degree k ⩾ 2 is greater than or equal to 〚k/3〛 + 〚(k − 2)/3〛 + 2.     

Semigroups of linear endomorphisms closed under conjugation

Let a,b be singular endomorphisms of a. finite dimensional vector space V and denote by S _a the semigroup generated by all the elements g ^-1ag, where g?Aut(V).The aim of this paper is to prove that b?S _a if and only if rank(b) ≥ rank(a).     

Witten deformation and the equivariant index

Let M be a compact Riemannian manifold endowed with an isometric action of a compact, connected Lie group. The method of the Witten deformation is used to compute the virtual representation-valued equivariant index of a transversally elliptic, first order differential operator on M. The multiplicities of irreducible representations in the index are expressed in terms of local quantities associated to the isolated singular points of an equivariant bundle map that is locally Clifford multiplication by a Killing vector field near these points.      

The largest character degree,conjugacy class size and subgroups of finite groups

Let G be a finite group, let b(G) denote the largest irreducible character degree of the group G and let bcl(G) denote the largest conjugacy class size of the group G. We study the relations between the sizes of the nilpotent and solvable subgroups of G and b(G). We also study the relations between the sizes of the nilpotent and solvable subgroups of G and bcl(G).     

The Real Part of the Character Table of a Finite Group

In the ordinary character table of a finite group G, the values of the real valued irreducible characters on the real conjugacy classes form a sub-table which is square by Brauer's permutation lemma. We call this table the real part of the character table of G. Unlike the ordinary character table, viewed as a square matrix the real part of the character table is often singular. We present some results linking nonsingularity of this table to other properties of G.     

Closed geodesics in compact nilmanifolds

We study the density of closed geodesics property on 2-step nilmanifolds Γ\N, where N is a simply connected 2-step nilpotent Lie group with a left invariant Riemannian metric and Lie algebra ?, and Γ is a lattice in N. We show the density of closedgeodesics property holds for quotients of singular, simply connected, 2-step nilpotent Lie groups N which are constructed using irreducible representations of the compact Lie group SU(2). Received: 8 November 2000 / Revised version: 9 April 2001     

Rational curves of degree at most 9 on a general quintic threefold

ABSTRACT.

We prove the following form of the Clemens conjecture in low degree. Let d ≤ 9, and let F be a general quintic threefold in P ⁴. Then (1) the Hilbert scheme of rational, smooth and irreducible curves of degree d on F is finite, nonempty, and reduced; moreover, each curve is embedded in F with normal bundle (?1) ⊕ (?1), and in P ⁴ with maximal rank. (2) On F, there are no rational, singular, reduced and irreducible curves of degree d, except for the 17,601,000 six-nodal plane quintics (found by Vainsencher). (3) On F, there are no connected, reduced and reducible curves of degree d with rational components.     

Dendriform Analogues of Lie and Jordan Triple Systems

We use computer algebra to determine all the multilinear polynomial identities of degree ≤7 satisfied by the trilinear operations (a·b)·c and a·(b·c) in the free dendriform dialgebra, where a·b is the pre-Lie or the pre-Jordan product. For the pre-Lie triple products, we obtain one identity in degree 3, and three independent identities in degree 5, and we show that every identity in degree 7 follows from the identities of lower degree. For the pre-Jordan triple products, there are no identities in degree 3, five independent identities in degree 5, and ten independent irreducible identities in degree 7. Our methods involve linear algebra on large matrices over finite fields, and the representation theory of the symmetric group.