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\begin{document}

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\section*{Fierz identity}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{spin_geometry}{}\paragraph*{{Spin geometry}}\label{spin_geometry}

[[!include higher spin geometry - contents]]

\hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory}

[[!include representation theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{in_terms_of_cochains_on_superminkowski_spacetimes}{In terms of cochains on super-Minkowski spacetimes}\dotfill \pageref*{in_terms_of_cochains_on_superminkowski_spacetimes} \linebreak
\noindent\hyperlink{BilinearFierzIdentities}{Bilinear Fierz identities}\dotfill \pageref*{BilinearFierzIdentities} \linebreak
\noindent\hyperlink{QuadraticFierzIdentities}{Quadrilinear Fierz identities}\dotfill \pageref*{QuadraticFierzIdentities} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

What are called \emph{Fierz identities} in [[physics]] are the relations that hold between [[multilinear map|multilinear]] expression in [[spinors]]. For example for all [[Majorana spinors]] $\psi$ in Lorentzian spacetime dimension 4,5,7,11, then the following identity holds (example \ref{TheM2andM5CocyclesAsFierzIdentities} below):

\begin{displaymath}
\left(\overline{\psi} \wedge \Gamma_{a b} \psi\right)
  \wedge
  \left(\overline{\psi} \wedge \Gamma^b \psi\right)
  \;=\;
  0  
  \,.
\end{displaymath}
(Here $\overline{(-)}$ denotes the Majorana conjugate, $\Gamma_a$ are a Clifford representations, the ``$\wedge$''-signs denotes symmetrization in the spinor components and summation over repeated indices is understood. The details of this are discussed \hyperlink{BilinearFierzIdentities}{below}.)

In \hyperlink{DAuriaFreMainaRegge82}{D'Auria-Fr\'e{}-Maina-Regge 82} it was pointed out that all Fierz identities may be understood as expressing the product operation in the [[representation ring]] of the [[spin group]] (in some given dimension): for $\{S_i\}_{i \in I}$ denoting [[isomorphism classes]] of [[irreducible representations|irreducible]] [[spin representations]], then, by definition of [[irreps]], their [[tensor product of representations]] decomposes again as a [[direct sum]] of [[irreducible representations]]

\begin{displaymath}
S_i \otimes S_j  = \underset{k}{\oplus} C_{i j}{}^k S_k
\end{displaymath}
with ``[[Clebsch-Gordan coefficients]]'' $C_{i j}{}^k$. These coefficients are effectively the Fierz identities.

For example for Lorentzian dimension 11 with $(\tfrac{1}{2})^5$ denoting the unique irreducible [[Majorana spinor]] representation, then one finds (\hyperlink{DAuriaFre82b}{D'Auria-Fr\'e{} 82b, section 3}) that the symmetric part in the quadruple tensor product of this representation with itself decomposes as a direct sum of irreps as follows

\begin{displaymath}
\left\{
    (\tfrac{1}{2})^5 
      \otimes
    (\tfrac{1}{2})^5 
      \otimes
    (\tfrac{1}{2})^5 
      \otimes
    (\tfrac{1}{2})^5 
  \right\}_{sym}
   \;\simeq\;
  (0)^5
    \;\oplus\;
  (1)^3 (0)^2
    \;\oplus\;
  (1)^4 (0)
    \;\oplus\;
  (1)^5 
    \;\oplus\;
  (2) (0)^4
    \;\oplus\;
  (2)(1)(0)^3
    \;\oplus\;
  (2)^2 (0)^3
    \;\oplus\;
  (2)^2 (1)^3
    \;\oplus\;
  (2)^5
\end{displaymath}
where the symbols refer to [[Young diagrams]] canonically labeling representations (details are in example \ref{11dQuadrilinearCGCoefficients} below).

The point is that the expression $\left(\overline{\psi} \wedge \Gamma_{a b} \psi\right) \wedge \left(\overline{\psi} \wedge \Gamma^b \psi\right)$ from above is a spinor quadrilinear which transforms in the vector representation $(1)(0)^4$ (due to its one free spacetime index). But that vector representation $(1)(0)^4$ is missing from the [[direct sum]] above, meaning that the spinor quadrilinear has vanishing components in this vector representation, hence that this expression vanishes identically.

\hypertarget{in_terms_of_cochains_on_superminkowski_spacetimes}{}\subsection*{{In terms of cochains on super-Minkowski spacetimes}}\label{in_terms_of_cochains_on_superminkowski_spacetimes}

We discuss Fierz identities as identities among multispinorial elements of the [[Chevalley-Eilenberg algebra]] $CE(\mathbb{R}^{d-1,1\vert N})$ of [[super-Minkowski spacetime]] $\mathbb{R}^{d-1,1\vert N}$, regarded as the super-translation [[supersymmetry]] [[super Lie algebra]]. In this form Fierz identities encode [[cocycles]] in the [[supersymmetry]] super-[[Lie algebra cohomology]], such as those which serve as [[higher WZW terms]] characterizing [[super p-branes]]. We follow \hyperlink{CDF}{Castellani-D'Auria-Fr\'e{} 82, section II.8}.

\hypertarget{BilinearFierzIdentities}{}\subsubsection*{{Bilinear Fierz identities}}\label{BilinearFierzIdentities}

Given a fixed [[real spin representation]] $N$, then the odd [[coordinates]] $\{\theta^\alpha\}_{\alpha = 1}^{dim_{\mathbb{R}}(N) }$ of the [[super Minkowski spacetime]] [[supermanifold]] $\mathbb{R}^{d-1,1\vert N}$ span, by construction, precisely that representation space, and hence so do the spinorial components of the [[super vielbein]] form

\begin{displaymath}
\psi^\alpha = \mathbf{d}\theta^\alpha
   \;\;\;
   \in \Omega^{\bullet}_{li}(\mathbb{R}^{d-1,1\vert N})
   \simeq
    CE(\mathbb{R}^{d-1,1\vert N})
  \,,
\end{displaymath}
since in the construction of [[super differential forms]] on $\mathbb{R}^{d-1,1\vert N}$, the de Rham operator $\mathbf{d}$ acts on the odd coordinates just formally, by sending the generator $\theta^\alpha$ to the new generator named $\mathbf{d} \theta^\alpha$.

Therefore we may identify the [[spin representation]] $N$ with the [[linear span]] (over $\mathbb{R}$) of these elements

\begin{displaymath}
N  \simeq \langle \mathbf{d}\theta^\alpha \rangle_{\alpha = 1}^{dim_{\mathbb{R}}(N) }
  \,,
\end{displaymath}
were the [[spin group]] acts on the elements on the right in the defining way (see at \emph{[[geometry of physics -- supersymmetry]]}): a spinorial rotation in a plane $\omega = \{\omega^{a b}\}$ by an angle $\alpha$ acts by

\begin{displaymath}
R_\omega(\psi) \coloneqq \exp(\tfrac{\alpha}{4} \omega^{a b} \Gamma_{a b} ) \psi
  \,.
\end{displaymath}
We may build new [[spin representations]] from this one by forming multilinear expressions in the [[super vielbein]]. For example the elements in $CE(\mathbb{R}^{d-1,1\vert N})$ of the form

\begin{displaymath}
\begin{aligned}
    \overline{\psi} \wedge \Gamma_a \psi
    &=
    \left(C_{\alpha \alpha'} \Gamma_a{}^{\alpha'}_{\beta}\right) \, \psi^\alpha \wedge \psi^{\beta}
    \\
    & =
    \left(C_{\alpha \alpha'} \Gamma_a{}^{\alpha'}_{\beta}\right) \, \mathbf{d}\theta^\alpha \wedge \mathbf{d}\theta^\beta
  \end{aligned}
\end{displaymath}
span, as the spacetime index $a$ ranges in $\{0, 1, \cdots, d-1\}$, a $d$-dimensional [[real vector space]]

\begin{displaymath}
\left\langle
     \,\overline{\psi} \wedge \Gamma_a \psi\,
  \right\rangle_{a = 0}^{d-1}
\end{displaymath}
which still carries a linear [[action]] of the [[spin group]], induced from the spin action on the $\psi$-s:

\begin{displaymath}
\begin{aligned}
    R_\omega(\overline{\psi} \wedge \Gamma_a \psi)
    & =
    \overline{\left( \exp(\tfrac{\alpha}{4}\omega^{a b}\Gamma_{a b} ) \psi \right)}
      \wedge \Gamma_a
    \left( \exp(\tfrac{\alpha}{4}\omega^{a b}\Gamma_{a b} \psi ) \right)
    \\
    & =
    \overline{\psi} \wedge \exp(-\tfrac{\alpha}{4} \omega^{a b} \Gamma_{a b}) \Gamma_a \exp(\tfrac{\alpha}{2}\omega^{a b} \Gamma_{a b})
     \psi
    \\
    & =
    \overline{\psi}
      \wedge
      (R_\omega(\Gamma_a))
    \psi
    \end{aligned}
  \,.
\end{displaymath}
Of course similarly we obtain elements

\begin{displaymath}
\overline{\psi} \Gamma_{a_1 \cdots a_p} \psi
\end{displaymath}
which, if they are non-vanishing at all, span the representation

\begin{displaymath}
\wedge^p \mathbb{R}^d
\end{displaymath}
Now observe that we may say all this more abstractly as follows:

\begin{enumerate}%
\item the elements $(\psi \wedge \overline{\psi})^{\alpha \beta}$ span the symmetrized [[tensor product of representations]]

\begin{displaymath}
\{N \otimes N\}_{sym}
    \;\simeq\;
  \langle 
    \,  (\psi \wedge \overline{\psi})^\alpha{}_\beta \,
  \rangle_{\alpha,\beta = 1}^{dim_{\mathbb{R}}(N)}
\end{displaymath}

\item for given $p \in \mathbb{N}$, then the elements of the form $\overline{\psi} \wedge \Gamma_{a_1 \cdots a_p} \psi$ form a [[subrepresentation]] thereof, equivalent to the vector representation $\wedge^p\mathbb{R}^{d}$


\item hence there is a [[direct sum]] decomposition

\begin{displaymath}
\left\{N \otimes N\right\}_{sym}
    \;\simeq\;
   \underset{p \in \mathbb{N}}{\bigoplus}
   c_p \left(\wedge^p \mathbb{R}^d\right)
\end{displaymath}
in the [[category of representations]] of the [[spin group]], which expresses the (symmetrized) [[tensor product of representations]] of the [[Majorana spinor]] representation as a [[direct sum]] of skew-symmetrized tensor products of the vector representation.



\end{enumerate}
Indeed this direct sum decomposition is exhaustive:

\begin{prop}
\label{BilinearFierzDecomposition}\hypertarget{BilinearFierzDecomposition}{}
For $d \in \mathbb{N}$ and $N$ a [[Majorana spinor]] representation of $Spin(d-1,1)$, then the following identity holds:

\begin{displaymath}
(\psi \wedge \overline{\psi})^\alpha{}_\beta
   \;=\;
  \tfrac{1}{dim_{\mathbb{R}}(N)}
  \left(
     \left(
       \overline{\psi}\psi
     \right)
     +
     \left(
       \overline{\psi} \Gamma_a \psi
     \right)
     (\Gamma^a)^\alpha{}_\beta
     + 
     \tfrac{1}{2!}
     \left(
       \overline{\psi} \Gamma_{a_1 a_2} \psi
     \right)
     (\Gamma^{a_1 a_2})^\alpha{}_\beta
      + 
     \cdots
  \right)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By the discussion there, the [[Majorana spinor]] representation is a real sub-representation of a complex \emph{Dirac representation} $\mathbb{C}^{(2^\nu)}$. The latter has the special property that

\begin{enumerate}%
\item the [[Clifford algebra]] contains the full [[matrix algebra]];


\item for $p \geq 1$ the Clifford elements $\Gamma_{a_1 \cdots a_p}$ have vanishing [[trace]].



\end{enumerate}
The first point implies that there exists coefficients $X^{a_1 \cdots a_p} \in \mathbb{C}$ for $p \in \mathbb{N}$ such that

\begin{displaymath}
\psi \wedge \overline{\psi}
    =
  \tfrac{1}{dim_{\mathbb{R}}(N)}
  \left(
    X
    +
    X^a \Gamma_a 
    +
    X^{a b} \Gamma_{a b} 
    +
    \cdots
  \right)
  \,.
\end{displaymath}
The second condition then implies that multiplying this expression with $\Gamma^{a_1 \cdots a_p}$ and taking the trace projects out the coefficient $X^{a_1 \cdots a_p}$:

\begin{displaymath}
\begin{aligned}
    X^{a_1 \cdots a_p}
      & = 
    \frac{1}{p! dim_{\mathbb{R}}(N)}
    tr_N
    \left(
      \left(
        X
        +
        X^a \Gamma_a 
        +
        X^{a b} \Gamma_{a b} 
        +
        \cdots
      \right)
      \Gamma^{a_1 \cdots a_p}
   \right)
   \\
   & = 
   \tfrac{1}{p!}
   tr_N
   \left(
     \psi \wedge \overline{\psi} \, \Gamma^{a_1 \cdots a_p}
   \right)
   \\
   & = 
   \tfrac{1}{p!}
   \left(
     \overline \psi \wedge \Gamma^{a_1 \cdots a_p} \psi
   \right)
  \end{aligned}
  \,.
\end{displaymath}
Notice that it is the last step, identifying the trace over $\psi \wedge \overline{\psi} \Gamma^{a_1 \cdots a_p}$ with the $\psi$-$\psi$ component of the matrix $\Gamma^{a_1 \cdots a_p}$, where we use the \emph{symmetrization} of the spinor tensor product, namely the identity $\psi^\alpha \wedge \overline{\psi}_\beta = \overline{\psi}_\beta \wedge \psi^\alpha$.

\end{proof}
Some of the coefficients in prop. \ref{BilinearFierzDecomposition} may vanish identically. These are the \emph{bilinear Fierz identities}, of the form

\begin{displaymath}
\overline{\psi} \Gamma_{a_1 \cdots a_p} \psi = 0
  \,.
\end{displaymath}
\begin{example}
\label{}\hypertarget{}{}
Let $d = 11$. Write $\mathbf{32}$ or $(\tfrac{1}{2})^5$ for the [[Majorana spinor]] representation of $Spin(d-1,1)$. Then

\begin{displaymath}
\left\{
    (\tfrac{1}{2})^5
      \otimes
    (\tfrac{1}{2})^5
  \right\}_{sym}
     \;\simeq\;
  \underset{\simeq \mathbb{R}^d}{\underbrace{(1)^1 (0)^4}}
    \;\oplus\;
  \underset{\simeq \wedge^2 \mathbb{R}^d}{\underbrace{(1)^2 (0)^3}}
    \;\oplus\;
  \underset{\wedge^5 \mathbb{R}^d}{\underbrace{(1)^5}}
  \,.
\end{displaymath}
\end{example}
(\hyperlink{DAuriaFre82b}{D'Auria-Fr\'e{} 82b (3.1)})

\begin{proof}
Since we know from prop. \ref{BilinearFierzDecomposition} that the right hand side has to be some direct sum of representations of the form $\wedge^p \mathbb{R}^d$, it is sufficient to check that there is only one choice of sum such that [[dimensions]] match on both sides of the equation.

Now the dimension of $\{N \otimes N\}_{sym}$ is that of the space of symmetric $32 \times 32$ matrices:

\begin{displaymath}
dim_{\mathbb{R}} 
  \left(
     \{\mathbf{32} \otimes \mathbf{32}\}_{sym}
  \right)
   \;=\;
  \frac{1}{2}
  \left(
    32 \times 33 
  \right)
   = 
  528
\end{displaymath}
while the dimension of $\wedge^p \mathbb{R}^d$ is the [[binomial coefficient]]

\begin{displaymath}
dim_{\mathbb{R}}(\wedge^p \mathbb{R}^d)
   \;=\;
  \left(
    11 \atop p
  \right)
  \,.
\end{displaymath}
Hence the claim follows from the fact that

\begin{displaymath}
\begin{aligned}
    528 
     & = 11 + 55 + 462
    \\
    & =
  \left(11 \atop 1\right)
   +
  \left(11 \atop 2\right)
   +
  \left(11 \atop 5\right)
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\hypertarget{QuadraticFierzIdentities}{}\subsubsection*{{Quadrilinear Fierz identities}}\label{QuadraticFierzIdentities}

Now we consider the [[direct sum]] decomposition of the [[tensor product of representations]] of \emph{four} copies of a [[spin representation]]. This yields the quadrilinear Fierz identities.

\begin{example}
\label{11dQuadrilinearCGCoefficients}\hypertarget{11dQuadrilinearCGCoefficients}{}
The group $Spin(10,1)$ has [[rank of a Lie group|rank]] 5, and hence its [[irreducible representation|irreducible]] vector representations are labeled by [[Young diagrams]] consisting of five rows. For instance

\begin{displaymath}
(2)^2 (1)^2 (0)
\end{displaymath}
denotes the representation whose elements may be identified with tensors of the form

\begin{displaymath}
X_{\itexarray{ a_1 & a_2 \\ a_3 & a_4 \\ a_5 }}
\end{displaymath}
which are

\begin{enumerate}%
\item skew-symmetric in indices in the same column;


\item symmetric and trace-less in indices in the same row.



\end{enumerate}
Write again $(\tfrac{1}{2})^5$ for the [[Majorana spinor]] representation. Then the following identity holds in the [[representation ring]]:

\begin{displaymath}
\left\{
    (\tfrac{1}{2})^5 
      \otimes
    (\tfrac{1}{2})^5 
      \otimes
    (\tfrac{1}{2})^5 
      \otimes
    (\tfrac{1}{2})^5 
  \right\}_{sym}
   \;\simeq\;
  \left.
    \itexarray{
    (0)^5
    \\
    \oplus
    \\
    (2) (0)^4
    \\
    \oplus
    \\
    (1)^3 (0)^2 \oplus  (2)(1)(0)^3
    \\
    \oplus
    \\
    (1)^4 (0) \oplus (2)^2 (0)^3
    \\
    \oplus
     \\
    (1)^5 
     \\
     \oplus
     \\
    (2)^2 (1)^3
    \\
    \oplus
    \\
    (2)^5
   }
  \right.
\end{displaymath}
\end{example}
(\hyperlink{DAuriaFre82b}{D'Auria-Fr\'e{} 82b (3.3)})

\begin{proof}
As before, this is supposed to follow already by matching total dimensions on both sides

\begin{displaymath}
\frac{32 \times 33 \times 34 \times 35}{4 \times 3 \times 2}
   \;=\;
  \left.
    \itexarray{
    1
    \\
    + 
    \\
    65
    \\
    +
    \\
    165 +  429
    \\
    +
    \\
    330 + 1144
    \\
    +
     \\
    462
     \\
     +
     \\
    17160
    \\
    +
    \\
    32604
   }
  \right.
\end{displaymath}
\end{proof}
More in detail we have the following decompositions, in the notation from \hyperlink{QuadraticFierzIdentities}{above}.

\begin{equation}
\left(\overline{\psi} \wedge \Gamma_{a_1} \psi\right) 
   \wedge 
  \left( \overline{\psi} \wedge \Gamma_{a_2} \psi \right)
   \;=\;
  X^{(\mathbf{65})}_{\itexarray{a_1 \\ a_2}}
   + 
  \frac{1}{11} \delta_{\itexarray{a_1 a_2}}X^{(\mathbf{1})}
\label{Fierz11dA}\end{equation}
Here for instance the symbol $X^{(\mathbf{65})}_{\itexarray{a_1 \\ a_2}}$ denotes the projection of the term on the left into the direct summand given by the [[representation]] $(2)(0)^4$ of dimension $65$. Similarly:

\begin{equation}
\left(\overline{\psi} \wedge \Gamma_{a_1 a_2} \psi\right)
    \wedge 
  \left(\overline{\psi} \wedge \Gamma_{a_3}\right)
    \;=\;
  X^{(\mathbf{429})}_{\itexarray{ a_1 & a_2 \\ a_3}}
   +
  X^{(\mathbf{165})}_{\itexarray{a_1 a_2 a_3}}
\label{Fierz11dB}\end{equation}
\begin{equation}
\left(
    \overline{\psi}\Gamma_{a_1 a_2} \psi
  \right)
  \left(
    \overline{\psi} \Gamma_{a_3 a_4}
  \right)
   \;=\;
  X^{(\mathbf{1144})}_{\itexarray{a_1 a_2 \\ a_3 a_4}}
   + 
  X^{(\mathbf{330})}_{\itexarray{a_1 a_2 a_3 a_4}}
   +
  \tfrac{4}{9}\delta_{\itexarray{ [a_1 \\ [a_3} } X^{(\mathbf{65})}_{\itexarray{a_2] \\ a_4] } }
  -
  \tfrac{2}{11} \delta_{\itexarray{a_1 & a_2 \\ a_3 & a_4}} X^{(\mathbf{1})}
\label{Fierz11dC}\end{equation}
\begin{equation}
\left(
    \overline{\psi}
      \wedge
      \Gamma_{a_1 \cdots a_5}
    \psi
  \right)
  \wedge
  \left(
    \overline{\psi}
      \wedge
    \Gamma_{a_6}
   \psi
  \right)
  \;=\;
  \epsilon_{a_1 \cdots a_6}{}^{b_1 \cdots b_5}
  X^{(\mathbf{462})}_{b_1 \cdots b_5}
  + 
  X^{(\mathbf{4290})}_{\itexarray{a_1 & \cdots & a_5 \\ a_6}}
  +
  \frac{15}{7}
  \delta_{a_6 [ a_1} X^{(\mathbf{330})}_{\itexarray{a_2 & \cdots & a_5}}
\label{Fierz11dD}\end{equation}
and some more.

(\hyperlink{DAuriaFre82b}{D'Auria-Fr\'e{} 82b table 2})

As a corollary:

\begin{example}
\label{TheM2andM5CocyclesAsFierzIdentities}\hypertarget{TheM2andM5CocyclesAsFierzIdentities}{}
For $d = 11$ then

\begin{enumerate}%
\item the following Fierz identity holds:

\begin{displaymath}
\left(
    \overline{\psi} \wedge \Gamma_{a b} \psi  
  \right) 
  \wedge
  \left(
     \overline{\psi} \wedge \Gamma^b \psi
  \right)
  \;= \; 0
  \,.
\end{displaymath}
(this is the cocycle condition for the [[higher WZW term]] of the [[M2-brane]] (\hyperlink{BergshoeffSezginTownsend87}{Bergshoeff-Sezgin-Townsend 87}), \hyperlink{AETW87}{AETW 87})


\item the following Fierz identity holds:

\begin{displaymath}
\left(
    \overline{\psi}
      \wedge
      \Gamma_{a_1 \cdots a_4 b}
    \psi
  \right)
  \wedge
  \left(
    \overline{\psi}
     \wedge
     \Gamma^{b}
     \psi
  \right)
  \;=\;
  3
  \left(
  \overline{\psi}
     \Gamma_{[a_1 a_2}
  \psi
  \right)
  \wedge
  \left(
    \overline{\psi}
    \Gamma_{a_3 a_4]}
    \psi
  \right)
\end{displaymath}
(this is the cocycle condition for the [[higher WZW term]] of the [[M5-brane]] (\hyperlink{BLNPST97}{BLNPST 97}, \hyperlink{FSS15}{FSS 15})).



\end{enumerate}
\end{example}
(\hyperlink{DAuriaFre82b}{D'Auria-Fr\'e{} 82b (3.13) and (3.28)})

\begin{proof}
The first identity is the result of equation \eqref{Fierz11dB} after tracing over the indices $a_2$ and $a_3$. Under this trace both summands on the right of \eqref{Fierz11dB} vanish: $X^{(\mathbf{429})}_{\itexarray{ a_1 & a_2 \\ a_3}}$ because it is trace-free in indices in a column, and $X^{(\mathbf{165})}_{\itexarray{a_1 a_2 a_3}}$ because it is skew-symmetric in all indices.

The second identity follows from taking the trace over the indices $a_5 and a_6$ in \eqref{Fierz11dD} and of skew-symmetrizing over all indices in \eqref{Fierz11dC}. By the symmetry properties of the tensors on the right of both equations, in both cases all tensors vanish except, in both cases, the contribution proportional to $X^{(\mathbf{330})}_{[a_1 \cdots a_3]}$, which both identities share. So it only remains to check that the proportionality factor is 3, as claimed. By writing out the skew-symmetrization in the last term in \eqref{Fierz11dD} one finds:

\begin{displaymath}
\begin{aligned}
    \frac{15}{7}
    \delta^{a_1 a_6}
    \delta_{a_6 [a_1}
    X^{(\mathbf{330})}_{a_2 \cdots a_5]}
    & =
    \frac{15}{7}  
    \delta^{a_1}{}_{[a_1} X^{(\mathbf{330})}_{a_2 \cdots a_5]}
    \\
    & =
    \frac{15}{7}
    \frac{1}{5!}
    \sum_{ \left\{\sigma \atop { {\text{permutation of}} \atop {\{1,\cdots , 5\}} } \right\}}
    (-1)^{\vert \sigma\vert }
    \delta^{a_1}{}_{a_{\sigma(1)}}
    X_{a_{\sigma(2)} \cdots a_{(\sigma(5))}}
    \\
    & =
    \frac{15}{7}
    \frac{1}{5!}
    \sum_{\left\{\sigma \atop { {\text{permutation of}} \atop {\{1,\cdots , 4\}} } \right\} }
    (-1)^{\vert \sigma\vert } 
    \left(  
      \underset{= 11}{\underbrace{\delta^{a_1}_{a_1}}}
      X^{(\mathbf{330})}_{a_{\sigma(1)}\cdots a_{\sigma(4)}}
      -
      4
      \delta^{a_1}{}_{a_{\sigma(1)}}
      X_{a_1 a_{\sigma(2)} \cdots a_{\sigma(4)}}
    \right) 
   \\
   & =
    \frac{15}{7}
    (11-4)
    \frac{1}{5}
    \;
    \underset{X^{(\mathbf{330})}_{a_1\cdots a_4}}{
    \underbrace{
    \frac{1}{4!}
    \sum_{ \left\{ \sigma \atop { {\text{permutation of}} \atop {\{1,\cdots , 4\}} } \right\} }
    (-1)^{\vert \sigma\vert}
    X^{(\mathbf{330})}_{a_{\sigma(1)}\cdots a_{\sigma(4)}}
    }
    }
    \\
    & =
    3 \; X^{(\mathbf{330})}_{a_{\sigma(1)} \cdots a_{\sigma(4)}}
  \end{aligned}
\end{displaymath}
where we used that $X^{(\mathbf{330})}_{a_1 \cdots a_4}$ is already skew-symmetric in all indices.

\end{proof}
\begin{example}
\label{QuadraticFierzIdentitiesIn5d}\hypertarget{QuadraticFierzIdentitiesIn5d}{}
On D=5 [[number of supersymmetries|N = 2]] [[super Minkowski spacetime]] ([[5d supergravity]]) there are quadrilinear Fierz identities of this form:



(\hyperlink{CDF}{Castellani-D'Auria-Fré 91 (III.5.50)}).

\end{example}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Clebsch-Gordan coefficient]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Named after [[Markus Fierz]].

The interpretation of Fierz identities as relations satisfied by [[Clebsch-Gordan coefficients]] in the [[representation ring]] of the [[spin group]] originates in

\begin{itemize}%
\item [[Riccardo D'Auria]], [[Pietro Fré]], E. Maina, [[Tullio Regge]] \emph{A New Group Theoretical Technique for the Analysis of Bianchi Identities and Its Application to the Auxiliary Field Problem of $D=5$ Supergravity}, Annals Phys. 139 (1982) 93 (\href{http://dx.doi.org/10.1016/0003-4916(82}{doi:10.1016/0003-4916(82)90007-0}90007-0), \href{http://inspirehep.net/record/167640/}{spire})

\end{itemize}
where it was applied to $Spin(4,1)$ (relevant in [[5-dimensional supergravity]]).

By this method the Fierz identities for $Spin(9,1)$ (relevant in [[heterotic supergravity]] and [[type II supergravity]]) are discussed in

\begin{itemize}%
\item [[Riccardo D'Auria]], [[Pietro Fré]], \emph{Geometric Structure of $N=1, D=10$ and $N=4, D=4$ Super Yang-Mills Theory}, Nucl. Phys. B196 (1982) 205 (\href{http://inspirehep.net/record/167639}{spire})

\end{itemize}
see also appendix C of

\begin{itemize}%
\item [[José Figueroa-O'Farrill]], Emily Hackett-Jones, George Moutsopoulos, \emph{The Killing superalgebra of ten-dimensional supergravity backgrounds}, Class.Quant.Grav.24:3291-3308,2007 (\href{http://arxiv.org/abs/hep-th/0703192}{arXiv:hep-th/0703192})

\end{itemize}
and the Fierz identities for $Spin(10,1)$ (relevant in [[11-dimensional supergravity]]) were tabulated in

\begin{itemize}%
\item [[Riccardo D'Auria]], [[Pietro Fré]], pages 12, 13 of \emph{[[GeometricSupergravity.pdf:file]]}, Nuclear Physics B201 (1982) ([[GeometricSupergravityErrata.pdf:file]])

\end{itemize}
see also

\begin{itemize}%
\item S. Naito, K. Osada, T. Fukui, \emph{Fierz Identities and Invariance of Eleven-dimensional Supergravity Action}, Phys.Rev. D34 (1986) 536-552 (\href{http://inspirehep.net/record/236376/?ln=de}{spire})

\end{itemize}
A textbook account of the representation ring method and summary of these results is in

\begin{itemize}%
\item [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], chapter II.8 of \emph{[[Supergravity and Superstrings - A Geometric Perspective]]}, World Scientific (1991)

\end{itemize}
See also

\begin{itemize}%
\item C. C. Nishi, \emph{Simple derivation of general Fierz-type identities}, Am. J. Phys. 73 (2005) 1160-1163 (\href{http://arxiv.org/abs/hep-ph/0412245}{arXiv:hep-ph/0412245})


\item [[Calin Lazaroiu]], Elena-Mirela Babalic, Ioana-Alexandra Coman, \emph{The geometric algebra of Fierz identities in arbitrary dimensions and signatures}, JHEP09(2013)156 (\href{https://arxiv.org/abs/1304.4403}{arXiv:1304.4403})


\item Elena-Mirela Babalic, Ioana-Alexandra Coman, [[Calin Lazaroiu]], \emph{A unified approach to Fierz identities}, AIP Conf. Proc. 1564, 57 (2013) (\href{https://arxiv.org/abs/1303.1575}{arxiv:1303.1575})



\end{itemize}
From the point of view of [[division algebras and supersymmetry]] the Fierz identities that give the vanishing of trilinear and of quadratic terms in spinors in certain dimensions are discussed in

\begin{itemize}%
\item [[John Huerta]], section 2.4 \emph{Division Algebras, Supersymmetry and Higher Gauge Theory} (\href{http://arxiv.org/abs/1106.3385}{arXiv:1106.3385})

\end{itemize}
The recognition of some Fierz identities as cocycle conditions defining the [[higher WZW terms]] of the [[super p-branes]] is due to

\begin{itemize}%
\item [[Marc Henneaux]], Luca Mezincescu, \emph{A Sigma Model Interpretation of Green-Schwarz Covariant Superstring Action}, Phys.Lett. B152 (1985) 340 (\href{http://inspirehep.net/record/15922?ln=en}{web})


\item [[Eric Bergshoeff]], [[Ergin Sezgin]], [[Paul Townsend]], \emph{Supermembranes and eleven dimensional supergravity}, Phys.Lett. B189 (1987) 75-78, In [[Mike Duff]], (ed.), \emph{[[The World in Eleven Dimensions]]} 69-72 (\href{http://streaming.ictp.trieste.it/preprints/P/87/010.pdf}{pdf}, \href{http://inspirehep.net/record/248230?ln=en}{spire})


\item Anna Ach\'u{}carro, [[Jonathan Evans]], [[Paul Townsend]], [[David Wiltshire]], \emph{Super $p$-Branes}, Phys. Lett. B \textbf{198} (1987) 441 (\href{http://inspirehep.net/record/22286?ln=en}{spire})


\item [[Igor Bandos]], [[Kurt Lechner]], Alexei Nurmagambetov, [[Paolo Pasti]], [[Dmitri Sorokin]], Mario Tonin, \emph{Covariant Action for the Super-Five-Brane of M-Theory}, Phys. Rev. Lett. 78 (1997) 4332-4334 (\href{http://arxiv.org/abs/hep-th/9701149}{arXiv:hep-th/9701149})


\item [[nLab:Domenico Fiorenza]], [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]], \emph{[[schreiber:The WZW term of the M5-brane|The WZW term of the M5-brane and differential cohomotopy]]}, J. Math. Phys. 56, 102301 (2015) (\href{https://arxiv.org/abs/1506.07557}{arXiv:1506.07557})



\end{itemize}
[[!redirects Fierz identities]]



\end{document}