Realise fundamental representation on multi-indexed total-antisymmetric tensors?

Question

Given a finite complex Lie algebra $\mathfrak{g}$ with simple roots $\alpha_i$. Denote $\Lambda_k$ the highest weight in the k-th fundamental representation so that $$(\check\alpha_i,\Lambda^k)={\delta_i}^k$$ If the rank of the Lie algebra is $n$ then there should be $n+1$ states in this representation. So to my understanding we may just represent states using a tensor with one index $T_i$ for example and ask $i$ to range from $1$ to $n+1$. Why then do we realise states in this representation by a $n-k$ anti-sytmmetric tensor $T^{i_1...i_{n-k}}$?

Answer 1

I think there are a number of misapprehensions here possibly due to conflicting naming from physics. Firstly, I'm not sure what "states" means in this context, perhaps a basis for the representation. Secondly "fundamental representation" in maths applies more broadly than in physics. There are in general several fundamental representations which will have different dimensions etc. Indeed there is one for each fundamental weight (resp. simple root) so as many as the rank of the Lie algebra.

What is true goes as follows:

The fundamental representations of $\mathfrak{sl}(n,\mathbb{C})$ (or equivalently $\mathfrak{su}(n)$, $\mathfrak{sl}(n,\mathbb{R})$ and other real forms) are the alternating tensor powers $\bigwedge^k\mathbb{C}^n$, $1\leq k \leq n-1$. These are the representations corresponding to the fundamental weights and have dimension $\begin{pmatrix} n \\ k \end{pmatrix}$. In physics, it is usual to call only $\mathbb{C}^n$ (i.e. $k=1$) the fundamental representation and for example $\bigwedge^{n-1}\mathbb{C}^n \cong (\mathbb{C}^n)^*$ the 'antifundamental' representation.

Fundamental representations of other semisimple Lie algebras are different however. For example, $\mathfrak{so}(n,\mathbb{C})$ has $\bigwedge^k\mathbb{C}^n$ as fundamental representations only up to $k<\frac{n}{2}$ (in fact $\bigwedge^k\mathbb{C}^n \cong \bigwedge^{n-k}\mathbb{C}^n$ in this case) and it has either one or two spin representations depending on the parity of $n$. For $\mathfrak{sp}(n,\mathbb{C})$, $\bigwedge^k\mathbb{C}^n$ is not even irreducible in general so the fundamental representations are subrepresentations of these and don't even get me started on the exceptional ones.

Note "fundamental representation" doesn't even really make sense outside of semisimple Lie algebras as it is firmly based on the weight system of a semisimple Lie algebra.