1,614 research outputs found

### Torsion functors with monomial support

The dependence of torsion functors on their supporting ideals is
investigated, especially in the case of monomial ideals of certain subrings of
polynomial algebras over not necessarily Noetherian rings. As an application it
is shown how flatness of quasicoherent sheaves on toric schemes is related to
graded local cohomology.Comment: updated reference

### Invariant Subspaces of Nilpotent Linear Operators. I

Let $k$ be a field. We consider triples $(V,U,T)$, where $V$ is a finite
dimensional $k$-space, $U$ a subspace of $V$ and $T \:V \to V$ a linear
operator with $T^n = 0$ for some $n$, and such that $T(U) \subseteq U$. Thus,
$T$ is a nilpotent operator on $V$, and $U$ is an invariant subspace with
respect to $T$.
We will discuss the question whether it is possible to classify these
triples. These triples $(V,U,T)$ are the objects of a category with the
Krull-Remak-Schmidt property, thus it will be sufficient to deal with
indecomposable triples. Obviously, the classification problem depends on $n$,
and it will turn out that the decisive case is $n=6.$ For $n < 6$, there are
only finitely many isomorphism classes of indecomposables triples, whereas for
$n > 6$ we deal with what is called ``wild'' representation type, so no
complete classification can be expected.
For $n=6$, we will exhibit a complete description of all the indecomposable
triples.Comment: 55 pages, minor modification in (0.1.3), to appear in: Journal fuer
die reine und angewandte Mathemati

### Algebraic methods in the theory of generalized Harish-Chandra modules

This paper is a review of results on generalized Harish-Chandra modules in
the framework of cohomological induction. The main results, obtained during the
last 10 years, concern the structure of the fundamental series of
$(\mathfrak{g},\mathfrak{k})-$modules, where $\mathfrak{g}$ is a semisimple Lie
algebra and $\mathfrak{k}$ is an arbitrary algebraic reductive in
$\mathfrak{g}$ subalgebra. These results lead to a classification of simple
$(\mathfrak{g},\mathfrak{k})-$modules of finite type with generic minimal
$\mathfrak{k}-$types, which we state. We establish a new result about the
Fernando-Kac subalgebra of a fundamental series module. In addition, we pay
special attention to the case when $\mathfrak{k}$ is an eligible $r-$subalgebra
(see the definition in section 4) in which we prove stronger versions of our
main results. If $\mathfrak{k}$ is eligible, the fundamental series of
$(\mathfrak{g},\mathfrak{k})-$modules yields a natural algebraic generalization
of Harish-Chandra's discrete series modules.Comment: Keywords : generalized Harish-Chandra module, (g,k)-module of finite
type, minimal k-type, Fernando-Kac subalgebra, eligible subalgebra; Pages no.
: 13; Bibliography : 21 item

### Interval structure of the Pieri formula for Grothendieck polynomials

We give a combinatorial interpretation of a Pieri formula for double
Grothendieck polynomials in terms of an interval of the Bruhat order. Another
description had been given by Lenart and Postnikov in terms of chain
enumerations. We use Lascoux's interpretation of a product of Grothendieck
polynomials as a product of two kinds of generators of the 0-Hecke algebra, or
sorting operators. In this way we obtain a direct proof of the result of Lenart
and Postnikov and then prove that the set of permutations occuring in the
result is actually an interval of the Bruhat order.Comment: 27 page

### The lambda-dimension of commutative arithmetic rings

It is shown that every commutative arithmetic ring $R$ has $lambda$-dimension
$leq 3$. An example of a commutative Kaplansky ring with $lambda$-dimension 3
is given. If $R$ satisfies an additional condition then $lambda$-dim($R$

### Pointed Hopf Algebras with classical Weyl Groups

We prove that Nichols algebras of irreducible Yetter-Drinfeld modules over
classical Weyl groups $A \rtimes \mathbb S_n$ supported by $\mathbb S_n$ are
infinite dimensional, except in three cases. We give necessary and sufficient
conditions for Nichols algebras of Yetter-Drinfeld modules over classical Weyl
groups $A \rtimes \mathbb S_n$ supported by $A$ to be finite dimensional.Comment: Combined with arXiv:0902.4748 plus substantial changes. To appear
International Journal of Mathematic

### $S$-duality in Vafa-Witten theory for non-simply laced gauge groups

Vafa-Witten theory is a twisted N=4 supersymmetric gauge theory whose
partition functions are the generating functions of the Euler number of
instanton moduli spaces. In this paper, we recall quantum gauge theory with
discrete electric and magnetic fluxes and review the main results of
Vafa-Witten theory when the gauge group is simply laced. Based on the
transformations of theta functions and their appearance in the blow-up
formulae, we propose explicit transformations of the partition functions under
the Hecke group when the gauge group is non-simply laced. We provide various
evidences and consistency checks.Comment: 14 page

### Computing the Rank Profile Matrix

The row (resp. column) rank profile of a matrix describes the staircase shape
of its row (resp. column) echelon form. In an ISSAC'13 paper, we proposed a
recursive Gaussian elimination that can compute simultaneously the row and
column rank profiles of a matrix as well as those of all of its leading
sub-matrices, in the same time as state of the art Gaussian elimination
algorithms. Here we first study the conditions making a Gaus-sian elimination
algorithm reveal this information. Therefore, we propose the definition of a
new matrix invariant, the rank profile matrix, summarizing all information on
the row and column rank profiles of all the leading sub-matrices. We also
explore the conditions for a Gaussian elimination algorithm to compute all or
part of this invariant, through the corresponding PLUQ decomposition. As a
consequence, we show that the classical iterative CUP decomposition algorithm
can actually be adapted to compute the rank profile matrix. Used, in a Crout
variant, as a base-case to our ISSAC'13 implementation, it delivers a
significant improvement in efficiency. Second, the row (resp. column) echelon
form of a matrix are usually computed via different dedicated triangular
decompositions. We show here that, from some PLUQ decompositions, it is
possible to recover the row and column echelon forms of a matrix and of any of
its leading sub-matrices thanks to an elementary post-processing algorithm

### A coproduct structure on the formal affine Demazure algebra

In the present paper we generalize the coproduct structure on nil Hecke rings
introduced and studied by Kostant-Kumar to the context of an arbitrary
algebraic oriented cohomology theory and its associated formal group law. We
then construct an algebraic model of the T-equivariant oriented cohomology of
the variety of complete flags.Comment: 28 pages; minor revision of the previous versio

### Classification of finite dimensional uniserial representations of conformal Galilei algebras

With the aid of the $6j$-symbol, we classify all uniserial modules of
$\mathfrak{sl}(2)\ltimes \mathfrak{h}_{n}$, where $\mathfrak{h}_{n}$ is the
Heisenberg Lie algebra of dimension $2n+1$.Comment: Some references added, introduction expanded, title change

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