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Results tagged with symmetric-polynomials
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user 4366
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
1
vote
Reference request on symmetric polynomials
Not a complete answer, but note that
\begin{align}
(*) \;\;\;e_k =e_k(x_1,\ldots,x_n)=\sum x_{i_1}x_{i_2}\cdots x_{i_k},
\end{align}
where the sum is over $1 \leq i_1 < i_2 < \ldots < i_k \leq n$. Mo …
- 1,472
8
votes
Free $k[x_1, \dots, x_n]^{S_n}$-module?
Let $\mathcal{P}_n$ be the polynomial ring $k\left[x_1, x_2, \ldots, x_n\right]$. The symmetric group $S_n$ acts on $\mathcal{P}_n$ from the left by the formula
$${}^\pi f = f\left(x_{\pi\left(1\right …
- 1,472