Relation With Symmetric Tensors
Consider an n-dimensional vector space V and a linear operator with eigenvalues X1, X2,...,Xn. Denote by Symk(V) its k-th symmetric tensor power and Msym(k) induced operator .
Proposition:
The proof is easy: consider eigenbasis ei for M. The basis in Symk(V) can be indexed by sequences i1≤i2≤...≤ik, indeed, consider the symmetrizations of . All such vectors are eigenvectors for Msym(k) with eigenvalues, hence proposition is true.
Similar one can express elementary symmetric polynomials via traces over antisymmetric tensor powers. Both expressions are subsumed in expressions of Schur polynomials as traces over Schur functors. Which can be seen as Weyl character formula for GL(V).
Read more about this topic: Complete Homogeneous Symmetric Polynomial, Properties
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