What are irreducible $n$-times antisymmetrized representations?

I came across the following statement while reading a string theory textbook:

if $\boldsymbol{16}_s$ and $\boldsymbol{16}_c$ are the are respectively spinor and conjugate spinor representationsof $Spin(10)$, we have the Clebsch-Gordan decomposition $$\boldsymbol{16}_s\otimes\boldsymbol{16}_c = [0]\oplus[2]\oplus[4]$$ where $[n]$ denotes the irreducible $n$-times antisymmetrized representation of $Spin(10)$, which from a field theoretic point of view correspond to $n$-forms.

I don't understand what exactly are those antisymmetrized representations. What is the general definition ? In what way does it corresponds to forms ?

Solution 1:

Let $SO(10) \to GL(\Bbb V)$ (or just $\Bbb V$) denote the defining (i.e., irreducible, $10$-dimensional) representation of $SO(10)$, and by slight abuse of notation also denote by $\Bbb V$ the representation $$Spin(10) \to SO(10) \to GL(\Bbb V)$$ of $Spin(10)$, where $Spin(10) \to SO(10)$ is the usual $2 : 1$ covering map. This representation is the first fundamental representation, i.e., the one with highest weight $[1, 0, 0, 0, 0]$.

The notation $[k]$ indicates the $k$th alternating power, $\bigwedge^k \Bbb V$ of $\Bbb V$: Explicitly, the underlying vector space is the space of $k$-vectors on $\Bbb V$, i.e., alternating (contravariant) $k$-tensors on $\Bbb V$ (thus $\dim [k] = \dim \bigwedge^k \,\Bbb V = {10 \choose k}$), and the (linear) action is characterized by the formula $$g \cdot (v_1 \wedge \cdots \wedge v_k) := (g \cdot v_1) \wedge \cdots \wedge (g \cdot v_k) .$$ (On the right-hand side, $\cdot$ denotes the action of the representation $Spin(10) \to GL(\Bbb V)$.)

We can use the inner product fixed on $\Bbb V$ fixed by $Spin(10)$ to raise and lower indices of tensors on $\Bbb V$, which in particular defines natural isomorphisms $\bigwedge^k \Bbb V \cong \bigwedge^k \Bbb V^*$. So we can just as well identify $[k]$ with the representation $\bigwedge^k \Bbb V^*$ of $k$-forms, i.e., alternating (covariant) $k$-tensors, on $\Bbb V$.

In summary, the representations $[k]$ are

$$\begin{array}{cccrl} \hline && \textrm{highest weight} & \dim & \textrm{name}\\ \hline [0] & \Bbb R & [0, 0, 0, 0, 0] & 1 & \textrm{trivial}\\ [1] & \Bbb V & [1, 0, 0, 0, 0] & 10 & \textrm{defining}\\ [2] & \bigwedge^2 \Bbb V & [0,1,0,0,0] & 45 & \textrm{alternating $2$-tensor}\\ [3] & \bigwedge^3 \Bbb V & [0,0,1,0,0] & 120 & \textrm{alternating $3$-tensor}\\ [4] & \bigwedge^4 \Bbb V & [0,0,0,1,1] & 210 & \textrm{alternating $4$-tensor}\\ [5] & \bigwedge^5 \Bbb V & \ast & 252 & \textrm{alternating $5$-tensor}\\ \hline \end{array} .$$

For $0 \leq k \leq 10$, the Hodge star operator $\ast : \bigwedge^{10 - k} \Bbb V \stackrel{\cong}{\to} \bigwedge^k \Bbb V$ defines an isomorphism $[10 - k] \cong [k]$, so the above table exhausts all of the representations $[k]$ up to isomorphism. In particular, for $k = 5$ the $\ast$ is a (nontrivial) involution of $\bigwedge^5 \Bbb V$, so that representation is not irreducible: It decomposes into the direct sum of the (in fact, irreducible) $\pm 1$-eigenspaces $\bigwedge^5_\pm \Bbb V$ of $\ast$, both of which have have dimension $126$. Here $\bigwedge^5_\pm \Bbb V$ are the representations of self-dual and anti-self-dual $5$-forms. In highest-weight notation they are $[0,0,0,2,0]$ and $[0,0,0,0,2]$, respectively.

As expected, $$\dim ([0] \oplus [2] \oplus [4]) = 1 + 45 + 210 = 256 = 16 \times 16 = \dim ({\bf 16}_s \otimes {\bf 16}_c) .$$

The above claims hold, mutatis mutandis, for the representations of alternating $k$-tensors for all groups $Spin(2 m)$, at least for $m \geq 3$; more adjustments need to be made for the groups $Spin(n)$ with $n$ odd.