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Results tagged with symmetric-functions
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user 40349
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
Evaluation of a complete homogeneous symmetric polynomial related to Stirling number of 2nd ...
Consider the fact that
$$
\prod_{i=1}^k \frac{1}{1-x_i t} = \sum_j h_j(x_1,\dots,x_k) t^j
$$
Writing $h_j = h_{j+k-k}$ we get from your first fact:
$$
\prod_{i=1}^k \frac{1}{1- i t} = \sum_j S(j+k,k) …
- 273
10
votes
1
answer
517
views
Cauchy identity in three sets of variables?
The Cauchy identity states that
$$
\prod_{i,j} \frac{1}{1-x_i y_j} = \sum_\lambda s_\lambda(x) s_\lambda(y),
$$
where $s_\lambda(x)$ is the Schur function.
Is there a known decomposition of the produ …
- 273