All Questions
Tagged with symmetric-groups lie-algebras
8 questions
2
votes
0
answers
184
views
The canonical automorphism of the symmetric group
Let $S_n$ be the symmetric group of order $n$. Denoting simple transpositions by $\sigma_i$ the collection $\sigma_1, \dots, \sigma_{n-1}$ generates $S_n$ subject to the following relations:
$$
\sigma ...
- 1,144
10
votes
1
answer
339
views
What is the Lie superalgebra generated by permutations?
Consider the group algebra of the symmetric group $\mathbb{C}S_n$. Then there is a corresponding Lie algebra $\mathfrak{L}(S_n)$ defined by
$$[\sigma, \tau] = \sigma\circ\tau - \tau\circ\sigma,$$
...
- 593
15
votes
2
answers
762
views
Gelfand-Tsetlin algebras and "Jucys-Murphy elements" for $\mathfrak{gl}_n$
I'm trying to figure out/find in literature the details concerning Gelfand-Tsetlin algebras for $\mathfrak{gl}_n(\mathbb C)$ (Okounkov-Vershik style, if you wish).
Consider the chain $$\mathcal U(\...
- 3,513
6
votes
1
answer
341
views
Sum of Young symmetrisers of a given shape
Preliminaries and notation:
Let $n\in \mathbb{Z}_{>0}$ and $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_s)\vdash n$ be a partition. Given a Young diagram of shape $\lambda$, we can associate it ...
- 87
15
votes
2
answers
838
views
factorization of the regular representation of the symmetric group
Let $\mathbb{C}[S_n]$ be the regular representation of the symmetric group $S_n$, and let $\mathbb{C}^n$ be the vector representation.
Question: Does there exist a representation $V$ (of dimension $(...
- 2,537
2
votes
1
answer
736
views
Schur Weyl duality for sl_n representations
Consider a finite dimensional vector space $V$ and the general linear group $GL(V)$ acting on it. Both $GL(V)$ and the symmetric group $S_d$ act on the tensor product of $d$ copies of $V$, and by Weyl ...
- 596
3
votes
3
answers
2k
views
The Weyl group of $SL(2, F)$
Let $G= SL(2, F)$, given a torus $T$, the Weyl group with respect to $T$ is defined to be $W=N(T)/Z(T)$, the quotient of the normalizer $N(T)$ of the torus by the centralizer $Z(T)$ of the torus.
My ...
- 85
-2
votes
2
answers
734
views
Lie algabra of symmetric group
It's easy to see that the descending central series of a group induces a graded Lie algebra .(see for example Serre's Harvard lectures or Magnus-Solitar book). I think in general this can be ...
- 1,180