Fredman's reciprocity, invariants of abelian groups, and the permanent of the Cayley table

Let $R$ be the regular representation of a finite abelian group $G$ and let $C_n$ denote the cyclic group of order $n$. For $G=C_n$, we compute the Poincare series of all $C_n$-isotypic components in $S^{\cdot} R\otimes \wedge^{\cdot} R$ (the symmetric tensor exterior algebra of $R$). From this we derive a general reciprocity and some number-theoretic identities. This generalises results of Fredman and Elashvili-Jibladze. Then we consider the Cayley table, $M_G$, of $G$ and some generalisations of it. In particular, we prove that the number of formally different terms in the permanent of $M_G$ equals $(S^n R)^G$, where $n$ is the order of $G$.Comment: 15 pages, to appear in Journal of Algebraic Combinatoric

Short antichains in root systems, semi-Catalan arrangements, and B-stable subspaces

Let \be be a Borel subalgebra of a complex simple Lie algebra \g. An ideal of \be is called ad-nilpotent, if it is contained in [\be,\be]. The generators of an ad-nilpotent ideal give rise to an antichain in the poset of positive roots, and the whole theory can be expressed in a combinatorial fashion, in terms of antichains. The aim of this paper is to present a refinement of the enumerative theory of ad-nilpotent ideals for the case in which \g has roots of different length. An antichain is called short, if it consists of short roots. We obtain, for short antichains, analogues of all results known for the usual antichains.Comment: LaTeX2e, 20 page

Semi-direct products of Lie algebras and their invariants

The goal of this paper is to extend the standard invariant-theoretic design, well-developed in the reductive case, to the setting of representation of certain non-reductive groups. This concerns the following notions and results: the existence of generic stabilisers and generic isotropy groups for finite-dimensional representations; structure of the fields and algebras of invariants; quotient morphisms and structure of their fibres. One of the main tools for obtaining non-reductive Lie algebras is the semi-direct product construction. We observe that there are surprisingly many non-reductive Lie algebras whose adjoint representation has a polynomial algebra of invariants. We extend results of Takiff, Geoffriau, Rais-Tauvel, and Levasseur-Stafford concerning Takiff Lie algebras to a wider class of semi-direct products. This includes $Z_2$-contractions of simple Lie algebras and generalised Takiff algebras.Comment: 49 pages, title changed, section 11 is shortened, numerous minor corrections; accepted version, to appear in Publ. RIMS 43(2007

On divisible weighted Dynkin diagrams and reachable elements

Let D(e) denote the weighted Dynkin diagram of a nilpotent element $e$ in complex simple Lie algebra \g. We say that D(e) is divisible if D(e)/2 is again a weighted Dynkin diagram. (That is, a necessary condition for divisibility is that $e$ is even.) The corresponding pair of nilpotent orbits is said to be friendly. In this note, we classify the friendly pairs and describe some of their properties. We also observe that any subalgebra sl(3) in \g determines a friendly pair. Such pairs are called A2-pairs. It turns out that the centraliser of the lower orbit in an A2-pair has some remarkable properties. Let $Gx$ be such an orbit and $h$ a characteristic of $x$. Then $h$ determines the Z-grading of the centraliser $z=z(x)$. We prove that $z$ is generated by the Levi subalgebra $z(0)$ and two elements in $z(1)$. In particular, (1) the nilpotent radical of $z$ is generated by $z(1)$ and (2) $x\in [z,z]$. The nilpotent elements having the last property are called reachable.Comment: 17 pages; v2 minor corrrections; final version, to appear in Transformation Groups (2010

The poset of positive roots and its relatives

Let $\Delta$ be a root system with a subset of positive roots, $\Delta^+$. We consider edges of the Hasse diagrams of some posets associated with $\Delta^+$. For each edge one naturally defines its type, and we study the partition of the set of edges into types. For $\Delta^+$, the type is a simple root, and for the posets of ad-nilpotent and Abelian ideals the type is an affine simple roots. We give several descriptions of the set of edges of given type and uniform expressions for the number of edges. By a result of Peterson, the number of Abelian ideals is $2^n$, where $n$ is the rank of $\Delta$. We prove that the number of edges of the corresponding Hasse diagram is $(n+1)2^{n-2}$. For $\Delta^+$ and the Abelian ideals, we compute the number of edges of each type and prove that the number of edges of type $\alpha$ depends only on the length of $\alpha$.Comment: 24 page