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I'm studding Hitchin's Generalized Calabi-Yau Manifolds https://arxiv.org/abs/math/0209099 and I've stuck here:

Suppose that $V$ is a vector space and denote its dual by $V^*$. Now we know that the $\bigwedge^\bullet(V^*)$ that is the exterior algebra over the dual space is a representation for Clifford algebra $CL(V \oplus V^*)$ by the action $$(v,\xi).\varphi=i_v\varphi+\xi\wedge\varphi , (v,\xi)\in CL(V \oplus V^*)$$

we are mainly interested in those representations of Spin group $\operatorname{Spin}(V \oplus V^*)$ that is not a representation of $SO(V \oplus V^*)$ and we call them Spinor representation. I know that restriction of this representation to the subgroup $\operatorname{Spin}(V \oplus V^*)$ of $CL(V \oplus V^*)$ is one of these representations. but I don't get to understand how tensoring $\bigwedge^\bullet(V^*)$ in the space of top forms of $V$ $$\bigwedge^\bullet(V^*)\otimes (\bigwedge^n V)^\frac{1}{2}$$ will contruct another Spinor representation and in what aspects this will arise more useful constructions than $\bigwedge^\bullet(V^*)$ so Hitchin prefered this one.

Any help would be a lot appreciated.

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    $\begingroup$ I guess it is not written explicitly in Hitchin’s paper, but this description of the spin representation is as modules over the connected component of the identity of the natural $GL(V)$ subgroup of $SO(V \oplus V^*)$. This is spelled out in Chapter 2 of Marco Gualtieri’s thesis. $\endgroup$ – José Figueroa-O'Farrill Nov 9 '18 at 11:17

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