# Real representation of smallest dimension of Clifford Algebra with $d$ generators

I'm trying to understand the model described in this paper. I have a question about a claim they make. From page 2:

To describe the fermionic degrees of freedom let, as a preliminary

\begin{align*} \gamma^i &= (\gamma^i_{\alpha \beta})_{\alpha, \beta = 1, ... ,d}, \end{align*}

be the real representation of smallest dimension, called $$s_d$$, of the Clifford Algebra with $$d$$ generators: $$\{\gamma^s, \gamma^t\} = 2\delta^{st} \mathbb{1}_d$$. On the representation space, $$Spin(d)$$ is realized through the matrices $$R \in SO(s_d)$$, so that we may view

\begin{align*} Spin(d) \hookrightarrow SO(s_d) \end{align*} as a simply connected subgroup. We recall that

$$s_d = \left\{ \begin{array}{ll} 2^{\lfloor\frac{d}{2}\rfloor} &, d = 0, 1, 2, mod(8) \\ 2^{\lfloor\frac{d}{2}\rfloor+ 1} & else \\ \end{array} \right.$$

where $$\lfloor\cdot\rfloor$$ denotes the integer part.

I have checked the appendix in Polchinski's string theory book, and various more mathematical notes I could find online. I cannot find any resource that will explain this result for $$s_d$$. I would appreciate if someone can explain it, and hopefully give some reference for this.

EDIT: I don't see why this question was closed. I don't see how the linked post answers the question.

I'm asking about real irreducible representations with positive definite metric. The linked post seems to deal with complex and/or Lorentzian cases. And I am asking about both odd and even dimensional cases.
The unique complex irreducible representation of the Clifford algebra in $$d$$ dimensions has dimension $$2^{p}$$, where $$p = \lfloor \frac{d}{2}\rfloor$$, see also this answer of mine. This representation, if we forget the complex structure, is a real representation of dimension $$2^{p+1}$$.
It reduces to a real representation of dimension $$2^p$$ when Majorana spinors exist, and the question of when exactly the Clifford algebra of signature $$(p,q)$$ has Majoranas is a bit of a subtle question and depends on what exactly one means by this, see this question of mine. The claim in the source in the OP that such real representations exist for $$d = 0,1,2 \mod 8$$ fit with O'Farrill's convention in the linked question that Majoranas exist for $$p-q = 0,6,7 \mod 8$$, since for $$p = 0,q = d$$ we have that $$-d = 0,6,7\mod 8$$ implies $$d = 0,1,2 \mod 8$$.