in this article, we introduce monomial irreducible representations of the special linear lie algebra $sln$. we will show that this kind of representations have bases for which the action of the chevalley generators of the lie algebra on the basis elements can be given by a simple formula.

‎let $v$ be an $n$-dimensional complex inner product space‎. ‎suppose‎ ‎$h$ is a subgroup of the symmetric group of degree $m$‎, ‎and‎ ‎$chi‎ :‎hrightarrow mathbb{c} $ is an irreducible character (not‎ ‎necessarily linear)‎. ‎denote by $v_{chi}(h)$ the symmetry class‎ ‎of tensors associated with $h$ and $chi$‎. ‎let $k(t)in‎ (v_{chi}(h))$ be the operator induced by $tin‎ ‎text{end}(v)$‎. ‎the...

We show that the space of algebraic covariant derivative curvature tensors R is generated by Young symmetrized product tensors T ⊗ T̂ or T̂ ⊗ T , where T and T̂ are covariant tensors of order 2 and 3 whose symmetry classes are irreducible and characterized by the following pairs of partitions: {(2), (3)}, {(2), (2 1)} or {(1), (2 1)}. Each of the partitions (2), (3) and (1) describes exactly one s...

Considering prolongation of a Lie algebroid equipped with a spray, defining some classical tensors, we show that a Lie symmetry of a spray is a curvature collineation for these tensors.