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Results tagged with symmetric-functions
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user 1306
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.
7
votes
Understanding a quip from Gian-Carlo Rota
I am not a native speaker of English and moreover belong to the ethnic group that is known to mess up the articles, but I certainly don't feel that the sentence "Behind these and several other attrac …
5
votes
Accepted
Show that sets are equal
Use the standard notations $e_k=\sum_{A\subset \{1,\dots,n\}, |A|=k} \prod_{i\in A} x_i$, with the conventions $e_0=1$ and $e_m=0$ for $m>n$; $p_k=\sum_{i=1}^n x_i^k$.
If $n=p$, the statement is true …
- 16.9k
4
votes
Isotypic components of the action of the symmetric group on polynomials
The answer depends on the structure that you want on the collection of generators you are looking for. If you really want just generators of $\mathbb{C}[x_1,\ldots,x_n]$ over $\mathbb{C}[x_1,\ldots,x_ …
- 16.9k
4
votes
Accepted
A nice generating set for the symmetric power of an algebra
Since your algebra is finitely generated, you only really need this result in the case when $A=\mathbb{C}[t_1,\ldots,t_n]$.
It seems that this result first appeared in
F. Junker, Über symmetrische …
- 16.9k
2
votes
Does the ring generated by the odd power sum symmetric functions have a name?
Darij gave a very good reference indeed. One other instance where I saw this before is the Maple package SF, where they talk about "signed class functions" (http://www.math.lsa.umich.edu/~jrs/software …
- 16.9k
2
votes
Show that sets are equal
Let me prove that if for two sets of $n$ distinct numbers $X=\{x_1,\ldots,x_n\}$ and $Y=\{y_1,\ldots,y_n\}$ the sums of $k$th powers for $k=1,\ldots,2n-1$ are the same, then $X=Y$, irrespectively of t …
- 16.9k
2
votes
Jack polynomials and the Witt algebra
There are two very famous instances of Jack polynomials in relationship to the Virasoro algebra (there are some others, but they very often seem to be related to one of these):
Katsuhisa Mimachi and …
- 16.9k
2
votes
Using Schur-Weyl duality
In your context, you want to think of the Schur-Weyl duality as a way to construct representations of $GL(V)$ out of representations of symmetric groups.
To give a precise answer along these lines tha …
- 16.9k