Standard representation of the symmetric group: Difference between revisions

Revision as of 15:31, 4 November 2023

Definition

Let $S_{n}$ be the symmetric group on $n$ letters.

Then the standard representation of $S_{n}$ is the linear representation sending each permutation $\sigma \in S_{n}$ to its corresponding permutation matrix, that is, the matrix $A$ such that $A_{ij}=1$ for $j=\sigma (i)$ , $A_{ij}=0$ elsewhere.

Character

The standard representation has character $\chi (g)$ equal to the number of fixed points of the permutation $g\in S_{n}$ , since $\chi (g)$ is equal to the trace of the corresponding permutation matrix - which has a $1$ on the diagonal for fixed points, $0$ for non-fixed points by definition.

Revision as of 15:30, 4 November 2023 (view source) R-a-jones (talk \| contribs) (Created page with "==Definition== Let <math>S_n</math> be the symmetric group on <math>n</math> letters. Then the '''standard representation''' of <math>S_n</math> is the linear represen...")		Revision as of 15:31, 4 November 2023 (view source) R-a-jones (talk \| contribs) (→‎Character) Newer edit →
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	The standard representation has [[character of a representation\|character]] <math>\chi(g)</math> equal to the number of fixed points of the permutation <math>g \in S_n</math>.		The standard representation has [[character of a representation\|character]] <math>\chi(g)</math> equal to the number of fixed points of the permutation <math>g \in S_n</math>, since <math>\chi(g)</math> is equal to the [[trace]] of the corresponding permutation matrix - which has a <math>1</math> on the diagonal for fixed points, <math>0</math> for non-fixed points by definition.