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Results tagged with symmetric-functions
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user 4366
Symmetric functions are symmetric polynomials, in finitely many, or countably infinitely many variables. They arise in the representation theory of symmetric groups and in the polynomial representation theory of general linear groups. Bases of the ring of symmetric functions are indexed by integer partitions. Schur functions, elementary symmetric functions, complete symmetric functions, and power sum symmetric functions are the most commonly used bases.
1
vote
A particular specialization of symmetric polynomials: is it bijective?
This is not a complete answer, but maybe this will be helpful. Let
$$H_k(t)=\prod_{i=1}^k(1-x_it)^{-1}=\sum_{n\geq0}h_n(x_1,\ldots,x_k)t^n.$$
Applying the map $\mathcal{C}$ to this generating series …
- 1,472
2
votes
A class of matrix determinants between Wronskians and Vandermondes
Couple of quick observations for $\alpha_i(x)=x^{(d_i)}$ ($x^{(d)}=x^d/d!$ as usual).
Note that if $d_i>d_{i+1}$, for all $i$, then $G(x_1,\ldots,x_n)=0$ unless $d_i-d_{i+1}=1$. In particular, if th …
- 1,472