Let's say we have $\gamma^{a}$ matrices $(a=1,2,...,D)$. They satisfy the following condition
$$\gamma^{a}\gamma^{b}+\gamma^{b}\gamma^{a}=2\delta^{ab}I^{N\times N}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$
all gamma matrices are $N\times N$. Then we denote
$$ \Gamma^{A} : \big\{I,\gamma^{a_{1}},\gamma^{a_{1}a_{2}},\gamma^{a_{1}a_{2}a_{3}},...,\gamma^{a_{1}a_{2}a_{3}...a_{D}} \big\} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)$$
where $\gamma^{a_{1}a_{2}...a_{n}}=\gamma^{a_{1}}\gamma^{a_{2}} ...\gamma^{a_{n}}$ and $a_{1}<a_{2}<...<a_{D}$. The total number of $\Gamma^{A}$ matrices is $2^{D}$. Also denote $\tilde{\gamma}=\gamma^{1}\gamma^{2}...\gamma^{D}$.
It's easy to derive the following formula: if $\Gamma^{A}=\gamma^{a_{1}a_{2}...a_{n}}$ then $\displaystyle\sum_{a=1}^{D}\gamma^{a}\Gamma^{A}\gamma^{a}=(-1)^{n}(D-2n)\Gamma^{A}$.
If we take the trace of this expression we get $[D-(-1)^{n}(D-2n)]Tr(\Gamma^{A})=0$ from where we conclude
$Tr(\Gamma^{A})=0$ if $\Gamma^{A}\neq I,\tilde{\gamma} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(3)$
$Tr(\tilde{\gamma})=0$ if $D$ is even $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(4)$
Questions
1)
Formulas (3) and (4) do not give us any information about the trace of $\tilde{\gamma}$ matrix in odd dimension. How to prove that $Tr(\tilde{\gamma})\neq 0$ if $D$ is odd?
2)
I found in many books that $\Gamma^{A}$ matrices are liner independent and they form basis. This means
$$\displaystyle\sum_{A=1}^{2^{D}}x_{A}\Gamma^{A}=0$$
holds if and only if $x_{A}=0$. We can prove that fact using $Tr(\Gamma^{A})=0$. But we know that if $D$ is odd $Tr(\tilde{\gamma})\neq 0$. So, here is my question: Is it true that $\Gamma^{A}$ matrices are linear independent? If it is true, how to prove that in odd dimension?
3)
We take $N\times N$ matrices $\gamma^{a}$, with $a=1,2,...,D$. How can we express $D$ with $N$? If we fix $D$ what is the minimal dimension of gamma matrices? And how to prove that?