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Results tagged with symmetric-groups
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user 1306
The symmetric group $S_n$ is the group of permutations of the set of integers $\{1,\dots,n\}$. This has $n!$ elements and is generated by the $n-1$ involutions exchanging consecutive integers. The symmetric groups form the simplest family of Coxeter groups.
19
votes
Accepted
An n!-dimensional representation of the symmetric group S_{n+2}
Yes your series of representations looks (except for the first term - but there must be the law of small numbers lurking around) like the sign representation times the Whitehouse module, see, e.g. th …
- 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
3
votes
0
answers
103
views
working with symmetric groups presented via nonstandard generators
This is follow-up to my earlier question.
Suppose that we have elements $\sigma_1,\ldots,\sigma_k\in S_n$, and that we established that these elements actually generate $S_n$.
Since that previous qu …
- 16.9k
4
votes
Characters of permutation groups
For the reasons apparent below I shall use the notation $C_N(m,j)$, not $C(m,j)$. It is sufficient to prove
$$
\sum_{m=0}^N\sum_{j=0}^m jC_N(m,j)x^m=Nx\cdot x(x+1)\cdots (x+N-2),
$$
as this formula …
- 16.9k
7
votes
Accepted
Generalizing the Fundamental Theorem of Symmetric Polynomials
I heard of three relatively recent works in that direction, taking different routes and arriving to interesting information about diagonal invariants.
First, there is the paper of Vaccarino that Dar …
- 16.9k
2
votes
Schur Weyl duality for sl_n representations
Depends what you mean by "the same", for example:
For which representations $W$ we can find various reps as summands in $W^{\otimes n}$? A good idea of course is to look at faithful self-dual $W$.
F …
- 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
7
votes
factorization of the regular representation of the symmetric group
The fact that the Lie module (as proposed by Darij Grinberg) works, as well, as an explicit isomorphism of modules, follows from the theory of cyclic operads: see Corollary 6.9 in http://sites.math.no …
- 16.9k