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Let $S_n$ be the symmetric group of $n$ points. I want to find references (or proofs) for the following statement (1).

(1). There does not exist any faithful orthogonal representation $$ S_n\longrightarrow O(n-2). $$

In order to prove (1), I want to use the result that any unitary representation of $S_n$ is over $\mathbb{Q}$ and find references or proofs for the following statement (2).

(2). There does not exist any faithful unitary representation $$ S_n\longrightarrow U(n-2). $$

Where to find references for (1)?

(if cannot, then find references for (2)? )

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1 Answer 1

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The smallest degree faithful representation of $S_n$ is $n-1$ in characteristic does not divide $n$. It is Theorem 22 of Chapter 19, Section 8 of Y. G. Berkovich, E. M. Zhmud; Characters of finite groups. Part 2. Translated from the Russian manuscript by P. Shumyatsky, V. Zobina and Berkovich. Translations of Mathematical Monographs, 181. American Mathematical Society, Providence, RI, 1999.

The result is due to Burnside. The proof is inductive.

Dickson proved of the characteristic divides n and n is at least 5, then n-2 is the smallest degree faithful representation.

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    $\begingroup$ That is the case in characteristic $0$. It can certainly be found in the book of G.D. James. $\endgroup$
    – Geoff Robinson
    Commented Oct 9, 2015 at 10:26

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