Young tableux (representation theory)

[Ted123]

Homework Statement

Consider the irreducible representation $V$ in the symmetric group $S_5$ corresponding to the Young diagram (these are meant to be boxes): $$[\;\;][\;\;] \\ [\;\;][\;\;] \\ [\;\;]\;\;\;\;$$

(a) List all standard Young tableaux of the given shape (that is, list all the possible fillings of the boxes with $\{1,2,3,4,5\}$ such that the numbers increase along the rows and down the columns).

(b) Give the dimension of $V$.

(c) For any standard tableau $\Omega$ of the given shape, let $e_{\Omega}$ be the standard basis element of the representation $V$. Evaluate the action of the permutation $g=(23)(45)(23)(34)(23) \in S_5$ on the vector $e_{\Omega}$ where $\Omega =$ $$[\;1\;][\;4\;] \\ [\;2\;][\;5\;] \\ [\;3\;]\;\;\;\;\;\,$$

Homework Equations

The Attempt at a Solution

My answers:
(a) There are 5 standard Young tableaux of the given shape.
(b) So dim(V)=5.
(c) The basis vector $e_{\Omega} = P_{\Omega}Q_{\Omega}$ where $P_\Omega \in \mathbb{C}S_5$ and $Q_{\Omega}\in\mathbb{C}S_5$ are the row symmetrizer and column antisymmetrizer of $\Omega$: $$P_{\Omega} = [e+(14)][e+(25)] = e + (25) + (14) + (14)(25) \\ Q_{\Omega} = [e - (12) - (13) - (23) + (123) + (132)][e-(45)]$$

The given permutation $g=(23)(45)(23)(34)(23)=(2543)$. Is there a quicker way to evaluate the action of this permutation on $e_{\Omega}$ then to work out $(2543)P_{\Omega}Q_{\Omega}$?

 

[mighty2000]

Hello, thank you for your question. Here is my response:

(a) Yes, there are 5 standard Young tableaux of the given shape: [1][2] [3][4] [5], [1][3] [2][4] [5], [1][4] [2][3] [5], [1][5] [2][3] [4], [1][2] [3][5] [4].

(b) The dimension of V is indeed 5.

(c) Yes, there is a quicker way to evaluate the action of the given permutation on e_{\Omega}. Since the given permutation (23)(45)(23)(34)(23) is a product of disjoint cycles, we can evaluate its action on e_{\Omega} by simply permuting the numbers in each cycle. This means that (23)(45)(23)(34)(23) will map [1][2] [3][4] [5] to [2][3] [5][4] [1]. Therefore, the action of the given permutation on e_{\Omega} is [e+(23)(45)][e+(23)(45)] = [e+(45)(23)] = e.