Search

In this paper we prove that any finite nilpotent alternative ring with planar zero-divisor graph is associative.

Let [Formula: see text] be a finite simple group of Lie type. In this paper, we study characters of [Formula: see text] that vanish at the non-semisimple elements and whose degree is equal to the order of a maximal unipotent subgroup of [Formula: see text]. Such characters can be viewed as a natural generalization of the Steinberg character. For groups [Formula: see text] of small rank we also determine the characters of this degree vanishing only at the non-identity unipotent elements.

Unitriangular factorization is a presentation of a linear group as a product of unipotent radicals of a Borel subgroup and its opposite. Whether this decomposition is known for Chevalley groups over rings of stable rank 1 and some Dedekind rings of arithmetic type, the case of twisted groups has been studied only over finite fields. In the present paper we give a much simpler proof for twisted groups over finite fields and the field of complex numbers.

We give an $AB$-factorization of the supercharacter table of the group of $n\times n$ unipotent upper triangular matrices over $\FF_q$, where $A$ is a lower-triangular matrix with entries in $\ZZ[q]$ and $B$ is a unipotent upper-triangular matrix with entries in $\ZZ[q^{-1}]$. To this end we introduce a $q$ deformation of a new power-sum basis of the Hopf algebra of symmetric functions in noncommutative variables. The factorization is obtain from the transition matrices between the supercharacter basis, the $q$-power-sum basis and the superclass basis. This is similar to the decomposition of the character table of the symmetric group $S_n$ given by the transition matrices between Schur functions, monomials and power-sums. We deduce some combinatorial results associated to this decomposition. In particular we compute the determinant of the supercharacter table., (example added) 10 pages, Final version To Appear in int. J. of Algebra and Computation (Accepted 2012)

In this paper, we describe the structure of proper left GC-lpp semigroups. Our main result not only generalizes a result of Takizawa on E-unitary [Formula: see text]-unipotent semigroups but also those of Fountain and Gomes on proper left ample semigroups. Ultimately, we extend McAlister's P-theorem, a fundamental structure theorem of inverse semigroups.

We prove several results about groups of finite Morley rank without unipotent p-torsion: p-torsion always occurs inside tori, Sylow p-subgroups are conjugate, and p is not the minimal prime divisor of our approximation to the "Weyl group". These results are quickly finding extensive applications within the classification project.

We consider pairs (V, H) of subgroups of a connected unipotent complex Lie group G for which the induced V × H-action on G by multiplication from the left and from the right is free. We prove that this action is proper if the Lie algebra 𝔤 of G is 3-step nilpotent. If 𝔤 is 2-step nilpotent, then there is a global slice of the action that is isomorphic to ℂn. Furthermore, a global slice isomorphic to ℂn exists if dim V = 1 = dim H or dim V = 1 and 𝔤 is 3-step nilpotent. We give an explicit example of a 3-step nilpotent Lie group and a pair of 2-dimensional subgroups such that the induced action is proper but the corresponding geometric quotient is not affine.

In this paper, we investigate essentially n-ary term operations of nilpotent extensions of algebras. We detect the connection between term operations of an original algebra and its nilpotent extensions. This structural point of view easily leads to the conclusion that the number of distinct essentially n-ary term operations of a proper algebraic nilpotent extension 𝔄 of an algebra ℑ is given by the formula \[ p_{n}({\frak A}) = \left\{ \begin{array}{ll} {p_{n}({\frak I}) + 1 \quad \mbox{for}\ n = 1,\\[3pt] p_{n}({\frak I}) \qquad \kern8pt\mbox{otherwise.} \end{array}\right. \] We show that in general the converse theorem is not true. However, we suppose that if a variety ${\cal V}$ is uniquely determined by its pn-sequences, the converse theorem is also satisfied. In the second part of the paper, we characterize generics of nilpotent shifts of varieties and describe cardinalities of minimal generics. We give a number of examples and pose some problems.

In the nineties the author, A. Stepanov and E. Plotkin developed a geometric approach towards calculations in exceptional groups at the level of K 1, decomposition of unipotents. However, it relied on the presence of large classical embeddings, such as A 5 ≤ E 6 or A 7 ≤ E 7. Recently the author, M. Gavrilovich and S. Nikolenko devised a sharper geometric method which only uses embeddings A 2 ≤ E 6, E 7, F 4. Here we show that one can make a further step and simultaneously stabilize two columns of a root unipotent by an element of type A 3 ≤ E 6, E 7. This opens the way to applications of this method at the level of K 2.

A semigroup S is called an eventually regular semigroup if for every a ∈ S, there exists a positive integer n such that an is regular. In this paper, the R-unipotent, inverse semigroup and group congruences on an eventually regular semigroup S are described by means of certain congruence pairs (ξ, K), where ξ is a normal congruence on the subsemigroup 〈E(S)〉 generated by E(S), and K is a normal subsemigroup of S.

Given a fibration f : S → ℙ1, and the associated foliation [Formula: see text], the problem of bounding the genus of the general fiber of f in terms of the sheaf [Formula: see text] is studied. Using unipotent reduction of f, several bounds are obtained, under positivity assumptions on [Formula: see text]. In Sec.4, the Poincaré problem is solved, for non-degenerate [Formula: see text], assuming that all the eigenvalues of the linear part of [Formula: see text] near singularities are greater than 3.

We let G be a group, and we let k be a natural number. We assume that G is nilpotent of class at most k, and that every (k + 1)-ary congruence preserving function on G is a polynomial function. We show that then every congruence preserving function on G (of any finite arity) is a polynomial function.

Using known results on 4-Engel groups one can see that a 4-Engel group is locally nilpotent if and only if all its 3-generator subgroups are nilpotent. As a step towards settling the question whether all 4-Engel groups are locally nilpotent we show that all 2-generator 4-Engel groups are nilpotent.

There is a unipotent subgroup of Sl(n, C) whose action on the flag manifold of Sl(n, C) completes the flows of the complex Kostant–Toda lattice (a Hamiltonian system in Lax form) through initial conditions where all the eigenvalues coincide. The action preserves the Bruhat cells, which are in one-to-one correspondence with the elements of the permutation group Σn. A generic orbit in a given cell is homeomorphic to Cm, where m is determined by the "gap sequence" of the permutation, which lists the number inversions of each degree.

This paper takes a detailed look at a subject that occurs in various contexts in mathematics, the fixed-point sets of torus actions on flag manifolds, and considers it from the (perhaps nontraditional) perspective of moment maps and length functions on Weyl groups. The approach comes from earlier work of the author where it is shown that certain singular flows in the Hamiltonian system known as the Toda lattice generate the action of a group A on a flag manifold, where A is a direct product of a non-maximal torus and unipotent group. As a first step in understanding the orbits of A in connection with the Toda lattice, this paper seeks to understand the fixed points of the non-maximal tori in this setting.

The main purpose of this paper is to study theta correspondence from representation theoretic point of view. There are two problems we have in mind. One is the construction of unipotent representations of semisimple Lie group. The other is the parametrization of unitary dual of semisimple Lie group. In the first paper of this series, we define semistable range in the domain of theta correspondence. Roughly speaking, semistable range is a range where one can define certain averaging operator analytically. In this paper, we prove that if the averaging operator is not vanishing, then it produces the theta correspondence. This paper pave the way to study theta correspondence using analytic machinery.

Much of the structure theory of inverse semigroups is based on constructing arbitrary inverse semigroups from groups and semilattices. It is known that E-unitary (or proper) inverse semigroups may be described as P-semigroups (McAlister), or inverse subsemigroups of semidirect products of a semilattice by a group (O'Carroll) or Cu-semigroups built over an inverse category acted upon by a group (Margolis and Pin). On the other hand, every inverse semigroup is known to have an E-unitary inverse cover (McAlister). The aim of this paper is to develop a similar theory for proper weakly left ample semigroups, a class with properties echoing those of inverse semigroups. We show how the structure of semigroups in this class is based on constructing semigroups from unipotent monoids and semilattices. The results corresponding to those of McAlister, O'Carroll and Margolis and Pin are obtained.

We investigate the topological structure of a cellular decomposition of the fixed locus of a unipotent operator of regular Jordan type acting on the wonderful compactification of the variety of complete quadrics and the variety of complete skew forms. The Poincaré polynomial is computed in each case and the poset of cell closures under inclusion is described in the complete quadrics case.

We present a generalized version of classical geometric invariant theory à la Mumford where we consider an affine algebraic group G acting on a specific affine algebraic variety X. We define the notions of linearly reductive and of unipotent action in terms of the G fixed point functor in the category of (G, 𝕜[X])-modules. In the case that X = {⋆} we recuperate the concept of linearly reductive and of unipotent group. We prove in our "relative" context some of the classical results of GIT such as: existence of quotients, finite generation of invariants, Kostant–Rosenlicht's theorem and Matsushima's criterion. We also present a partial description of the geometry of such linearly reductive actions.

In this paper, we study Conjecture II.1.9 of [A. B. Buan, O. Iyama, I. Reiten and J. Scott, Cluster structures for 2-Calabi–Yau categories and unipotent groups, Compos. Math. 145(4) (2009) 1035–1079], which said that any maximal rigid object without loops or 2-cycles in its quiver is a cluster-tilting object in a connected Hom-finite triangulated 2-CY category [Formula: see text]. We obtain some conditions equivalent to the conjecture, and by using them we prove the conjecture.

We find a pair of finite groups, one nilpotent and the other perfect, with the same set of character degrees.

Let R be a commutative ring with identity. The nilpotent graph of R, denoted by ΓN(R), is a graph with vertex set [Formula: see text], and two vertices x and y are adjacent if and only if xy is nilpotent, where [Formula: see text] is nilpotent, for some y ∈ R*}. In this paper, we determine all isomorphism classes of finite commutative rings with identity whose ΓN(R) has genus one.

We study a natural map from representations of a free (resp. free abelian) group of rank g in GL_r(C), to holomorphic vector bundles of degree zero over a compact Riemann surface X of genus g (resp. complex torus X of dimension g). This map defines what is called a Schottky functor. Our main result is that this functor induces an equivalence between the category of unipotent representations of Schottky groups and the category of unipotent vector bundles on X. We also show that, over a complex torus, any vector or principal bundle with a flat holomorphic connection is Schottky., Comment: In v3, the results are approached using standard cohomology and derived functor arguments, avoiding extensive use of Yoneda Extensions