Conversely, symmetric tensors are defined as invariants : given the natural action of the symmetric group on the tensor algebra, the symmetric tensors are the subspace on which the symmetric group acts trivially.
22.
Natural actions resulting in sediment being deposited include alluvial ( water-related ) or jungles and other areas of lush plant growth, decomposed vegetative sediment can result in layers of soil deposited over remains.
23.
These are the vertex permutations ( " S " 4 in its natural action ) and the edge permutations ( " S " 4 acting on unordered pairs ) that they induce:
24.
If we let G be the set of all such permutations \ varphi _ { x, y }, then G is a group and X is a G-torsor under the natural action.
25.
It is never made clear in the book whether this is in fact true, and all of Shardik's behaviour can be explained as simply the natural actions of a very large and savage bear.
26.
It was already observed by Gromov that the natural action by translations of a word-hyperbolic group G on its boundary \ partial G is a uniform convergence action ( see for a formal proof ).
27.
The natural action of group " G " 1 above and its action on itself ( via left multiplication ) are not equivalent as the natural action has fixed points and the second action does not.
28.
The natural action of group " G " 1 above and its action on itself ( via left multiplication ) are not equivalent as the natural action has fixed points and the second action does not.
29.
The vector bundles associated to these principal bundles via the natural action of " G " on "'F " "'k " are just the tautological bundles over the Grassmannians.
30.
By construction the Outer space " X n " is a finite-dimensional simplicial complex equipped with a natural action of cohomological information about Out ( " F " " n " ).
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