definition of antisymmetrization of tensor w.r.t. some arguments
Topics
About: vectors space
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of antisymmetrization of tensor with respect to some arguments.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( F\): \(\in \{\text{ the fields }\}\)
\( \{V_1, ..., V_k, W\}\): \(\subseteq \{\text{ the } F \text{ vectors spaces }\}\), where \(V_{j_1} = ... = V_{j_l} := V\) for some \(\{V_{j_1}, ..., V_{j_l}\} \subseteq \{V_1, ..., V_k\}\)
\( L (V_1, ..., V_k: W)\): \(= \text{ the tensors space }\)
\(P_{\{j_1, ..., j_l\}}\): \(= \text{ the group of the permutations of } (1, ..., k) \text{ that move only } (j_1, ..., j_l)\)
\(*Asym_{\{j_1, ..., j_l\}}\): \(: L (V_1, ..., V_k: W) \to L (V_1, ..., V_k: W)\)
//
Conditions:
\(\forall f \in L (V_1, ..., V_k: W), \forall (v_1, ..., v_k) \in V_1 \times ... \times V_k (Asym_{\{j_1, ..., j_l\}} (f) (v_1, ..., v_k) = 1 / l! \sum_{\sigma \in P_{\{j_1, ..., j_l\}}} sgn \sigma f (v_{\sigma_1}, ..., v_{\sigma_k}))\)
//
2: Note
\(V_{j_1} = ... = V_{j_l} := V\) is required because otherwise, putting \(v_{\sigma_{j_m}}\), which is in \(\{v_{j_1}, ..., v_{j_l}\}\), into the \(j_m\)-th argument would not make sense.
While it requires that \(V_{j_1} = ... = V_{j_l} := V\), another \(V_j\) is allowed to equal \(V\): we do not necessarily need to do the antisymmetrization with respect to all the vectors spaces that equal \(V\).
Let us see that \(Asym_{\{j_1, ..., j_l\}}\) is indeed into \(L (V_1, ..., V_k: W)\).
For each \(f \in L (V_1, ..., V_k: W)\), \(Asym_{\{j_1, ..., j_l\}} (f)\) is obviously \(: V_1 \times ... \times V_k \to F\).
Let us see the multi-linearity of \(Asym_{\{j_1, ..., j_l\}} (f)\): for \((v_1, ..., v_k) = (v_1, ..., r' v'_j + r'' v''_j, ..., v_k)\), \(Asym_{\{j_1, ..., j_l\}} (f) (..., r' v'_j + r'' v''_j, ...) = r' Asym_{\{j_1, ..., j_l\}} (f) (..., v'_j, ...) + r'' Asym_{\{j_1, ..., j_l\}} (f) (..., v''_j, ...)\)?
There are 2 cases: 1) \(j \notin \{j_1, ..., j_l\}\); 2) \(j \in \{j_1, ..., j_l\}\).
Let \(j \notin \{j_1, ..., j_l\}\).
\(Asym_{\{j_1, ..., j_l\}} (f) (v_1, ..., v_k) = 1 / l! \sum_{\sigma \in P_{\{j_1, ..., j_l\}}} sgn \sigma f (v_{\sigma_1}, ..., v_{\sigma_k})) = 1 / l! \sum_{\sigma \in P_{\{j_1, ..., j_l\}}} sgn \sigma f (v_{\sigma_1}, ..., r' v'_j + r'' v''_j, ..., v_{\sigma_k}))\), where \(r' v'_j + r'' v''_j\) is not moved by any \(\sigma\), \(= 1 / l! \sum_{\sigma \in P_{\{j_1, ..., j_l\}}} sgn \sigma (r' f (v_{\sigma_1}, ..., v'_j, ..., v_{\sigma_k})) + r'' f (v_{\sigma_1}, ..., v''_j, ..., v_{\sigma_k}))) = r' 1 / l! \sum_{\sigma \in P_{\{j_1, ..., j_l\}}} sgn \sigma f (v_{\sigma_1}, ..., v'_j, ..., v_{\sigma_k}) + r'' 1 / l! \sum_{\sigma \in P_{\{j_1, ..., j_l\}}} sgn \sigma f (v_{\sigma_1}, ..., v''_j, ..., v_{\sigma_k}) = r' Asym_{\{j_1, ..., j_l\}} (f) (..., v'_j, ...) + r'' Asym_{\{j_1, ..., j_l\}} (f) (..., v''_j, ...)\).
Let \(j \in \{j_1, ..., j_l\}\).
\(Asym_{\{j_1, ..., j_l\}} (f) (v_1, ..., v_k) = 1 / l! \sum_{\sigma \in P_{\{j_1, ..., j_l\}}} sgn \sigma f (v_{\sigma_1}, ..., v_{\sigma_k}))\), where \(v_j = r' v'_j + r'' v''_j\) is moved to the \(j_m\)-th argument as \(v_{\sigma_{j_m}}\), which means that \(\sigma_{j_m} = j\), \(= 1 / l! \sum_{\sigma \in P_{\{j_1, ..., j_l\}}} sgn \sigma f (v_{\sigma_1}, ..., v_{\sigma_{j_m}}, ..., v_{\sigma_k})) = 1 / l! \sum_{\sigma \in P_{\{j_1, ..., j_l\}}} sgn \sigma f (v_{\sigma_1}, ..., r' v'_j + r'' v''_j, ..., v_{\sigma_k})) = 1 / l! \sum_{\sigma \in P_{\{j_1, ..., j_l\}}} sgn \sigma (r' f (v_{\sigma_1}, ..., v'_j, ..., v_{\sigma_k})) + r'' f (v_{\sigma_1}, ..., v''_j, ..., v_{\sigma_k})))\), but \(v'_j\) and \(v''_j\) can be denoted as \(v'_{\sigma_{j_m}}\) and \(v''_{\sigma_{j_m}}\), \(= r' 1 / l! \sum_{\sigma \in P_{\{j_1, ..., j_l\}}} sgn \sigma f (v_{\sigma_1}, ..., v'_{\sigma_{j_m}}, ..., v_{\sigma_k})) + r'' 1 / l! \sum_{\sigma \in P_{\{j_1, ..., j_l\}}} sgn \sigma f (v_{\sigma_1}, ..., v''_{\sigma_{j_m}}, ..., v_{\sigma_k}))) = r' Asym_{\{j_1, ..., j_l\}} (f) (..., v'_j, ...) + r'' Asym_{\{j_1, ..., j_l\}} (f) (..., v''_j, ...)\).
So, yes, \(Asym_{\{j_1, ..., j_l\}} (f)\) is indeed multi-linear, and \(Asym_{\{j_1, ..., j_l\}} (f) \in L (V_1, ..., V_k: W)\).
Let us see that \(Asym_{\{j_1, ..., j_l\}} (f)\) is antisymmetric with respect to the \(\{j_1, ..., j_l\}\) arguments, which is the reason why \(Asym_{\{j_1, ..., j_l\}}\) is called "antisymmetrization".
Let \(\sigma' \in P_{\{j_1, ..., j_l\}}\) be any. What we need to see is that \(Asym_{\{j_1, ..., j_l\}} (f) (v_{\sigma'_1}, ..., v_{\sigma'_k}) = sgn \sigma' Asym_{\{j_1, ..., j_l\}} (f) (v_1, ..., v_k)\).
\(Asym_{\{j_1, ..., j_l\}} (f) (v_{\sigma'_1}, ..., v_{\sigma'_k}) = sgn \sigma' 1 / l! \sum_{\sigma \in P_{\{j_1, ..., j_l\}}} sgn \sigma sgn \sigma' f (v_{(\sigma \circ \sigma')_1}, ..., v_{(\sigma \circ \sigma')_k})) = sgn \sigma' 1 / l! \sum_{\sigma \in P_{\{j_1, ..., j_l\}}} sgn (\sigma \circ \sigma') f (v_{(\sigma \circ \sigma')_1}, ..., v_{(\sigma \circ \sigma')_k}))\), but by the proposition that for any group, the multiplication map with any fixed element from left or right is a bijection, \(\sigma \circ \sigma'\) visits each element of \(P_{\{j_1, ..., j_l\}}\) once, so, \(= sgn \sigma' 1 / l! \sum_{\sigma \circ \sigma' \in P_{\{j_1, ..., j_l\}}} sgn (\sigma \circ \sigma') f (v_{(\sigma \circ \sigma')_1}, ..., v_{(\sigma \circ \sigma')_k})) = sgn \sigma' Asym_{\{j_1, ..., j_l\}} (f) (v_1, ..., v_k)\).
