Decomposition of Free Lie Algebras into Irreducible Components (original) (raw)

A new character formula for Lie algebras and Lie groups

Journal of Lie theory

The aim of this paper is to present a new character formula for finite-dimensional representations of finite-dimensional complex semisimple Lie Algebras and compact semisimple Lie Groups. Some applications of the new formula include the exact determination of the number of weights in a representation, new recursion formulas for multiplicities and, in some cases, closed formulas for the multiplicities themselves.

Lie elements in the group algebra

Given a representation V of a group G, there are two natural ways of defining a representation of the group algebra k[G] in the external power V^{\wedge m}. The set L(V) of elements of k[G] for which these two ways give the same result is a Lie algebra and a representation of G. For the case when G is a symmetric group and V = C^n, a permutation representation, these spaces L(C^n) are naturally embedded into one another. We describe L(C^n) for small n and formulate some questions and conjectures. This is a note on research in progress.

Two contributions to the representation theory of algebraic groups

2002

Sei V ein endlich-dimensionaler, komplexer Vektorraum. Eine Teilmenge X in V hat die Trennungseigenschaft, falls das Folgende gilt: Für je zwei linear unabhängige lineare Funktionen l, m auf V existiert ein Punkt x in X mit l(x) = 0 und m(x) = 0. Wir interessieren uns für den Fall V = C[x, y] n , d.h. V ist eine irreduzible Darstellung von SL 2 . Die Teilmengen, die wir untersuchen, sind Bahnabschlüsse von Elementen aus C[x, y] n . Wir beschreiben die Bahnen, die die Trennungseigenschaft erfüllen:

DECOMPOSITION OF TENSOR PRODUCTS OF MODULAR IRREDUCIBLE REPRESENTATIONS FOR SL3: THE p ≥ 5 CASE

We study the structure of the indecomposable direct summands of tensor products of two restricted rational simple modules for the algebraic group SL3(K), where K is an algebraically closed field of characteristic p ≥ 5. We also give a characteristicfree algorithm for the decomposition of such a tensor product into indecomposable direct summands. The p < 5 case was studied in the authors' earlier paper . We find that for characteristics p ≥ 5 all the indecomposable summands are rigid, in contrast to the characteristic 3 case.

On the irreducibility of irreducible characters of simple Lie algebras

Transactions of the American Mathematical Society, 2014

We establish an irreducibility property for the characters of finite dimensional, irreducible representations of simple Lie algebras (or simple algebraic groups) over the complex numbers, i.e., that the characters of irreducible representations are irreducible after dividing out by (generalized) Weyl denominator type factors. For SL(r) the irreducibility result is the following: let λ = (a 1 ≥ a 2 ≥ • • • a r−1 ≥ 0) be the highest weight of an irreducible rational representation V λ of SL(r). Assume that the integers a 1 + r − 1, a 2 + r − 2, • • • , a r−1 + 1 are relatively prime. Then the character χ λ of V λ is strongly irreducible in the following sense: for any natural number d, the function χ λ (g d), g ∈ SL(r, C) is irreducible in the ring of regular functions of SL(r, C).