I have a space $V$ of $6 \times 6$ matrices whose basis is $\{1,S,M\}$; writing the components indices: $\{1_{ij},S_{ij},M_{ij}\}$ where $i,j=1,\cdots, 6$. The inner product on this space is \begin{equation} \langle A, B\rangle_{V} := \text{Tr} [ A^{\dagger}B ] =\sum_{i,j}(A^{\dagger})_{ij}(B)_{ji} \equiv (A^{\dagger})_{ij}(B)_{ji}. \end{equation} Besides, the tensor product space $V \otimes V$ has the basis $\{1\otimes 1, 1 \otimes S, \cdots, M \otimes M \}$; writing the components indices: $\{(1\otimes 1)_{ijmn} = 1_{ij}1_{mn},(1\otimes S)_{ijmn} =1_{ij}S_{mn},\cdots, (M\otimes M)_{ijmn} =M_{ij}M_{mn}\}$, and it's possible to define an inner product on this tensor product space as \begin{equation} \langle A \otimes B, C \otimes D \rangle_{V\otimes V} := \langle A,C \rangle_{V} \langle B,D \rangle_{V} = \text{Tr}[A^{\dagger}C] \text{Tr}[B^{\dagger}D] = (A^{\dagger})_{ij} (C)_{ji} (B^{\dagger})_{mn} (D)_{nm}. \end{equation} Now, doing some physics calculations in QFT, I have found a kind of new space, let's call it $W$, whose basis seems to be \begin{align} \mathcal{B}_{W}=&\{\mathbf{1}_{ijmn}=(1\otimes 1)_{ijmn}+(1\otimes 1)_{inmj} , \mathbf{S}_{ijmn}=(S\otimes S)_{ijmn}+(S\otimes S)_{inmj}, \mathbf{M}_{ijmn}=(M\otimes M)_{ijmn}+(M\otimes M)_{inmj} \} \\ =& \{\mathbf{1}_{ijmn}=1_{ij}1_{mn}+1_{in}1_{mj} , \mathbf{S}_{ijmn}=S_{ij}S_{mn}+S_{in}S_{mj}, \mathbf{M}_{ijmn}=M_{ij}M_{mn}+M_{in}M_{mj} \}. \end{align}
Observations:
The elements of this basis are made of two terms. Each term is a tensor product, but the second term has a pair of indices exchanged with respect to the first $j \leftrightarrow n$.
All the elements of the basis are of the form $\mathbf{X}_{ijmn}= (X\otimes X)_{ijmn}+(X\otimes X)_{inmj} = X_{ij}X_{mn}+X_{in}X_{mj}$; there are no cross terms like $(\mathbf{XY})_{ijmn}= (X\otimes Y)_{ijmn}+(X\otimes Y)_{inmj} = X_{ij}Y_{mn}+X_{in}Y_{mj}$
All the elements have the symmetry $\mathbf{A}_{ijmn}=\mathbf{A}_{mjin}=\mathbf{A}_{inmj}$.
This space seems to be a kind of symmetric (direct?) sum of two copies of $V \otimes V$, where the second copy has the indices $j \leftrightarrow n$ exchanged with respect the first copy.
My main question is:
- Is it possible to define an inner product on $W$? If so, how is it defined?
Any reference, idea or suggestion is welcome. By the way: I'm physicist (not mathematician), sorry for my ignorance.
