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We know that the alternating group of degree $n>7$ has an irreducible character of degree $n-1$. The latter number is the smallest nontrivial one for each the alternating group has an irreducible character of that degree.

Does the Alternating group of degree $n>7$ have exactly one irreducible character of degree $n-1$?

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Old answer: You know already that the answer is ``yes.'' For a reference, see result 2 of

Rasala, Richard On the minimal degrees of characters of $S_n$. J. Algebra 45 (1977), no. 1, 132–181.

This gives the answer for $n\geq 9$. The Atlas of Finite Groups then gives the result for $n=7,8,9$. (Note that the statement is also true for $n=7$, but not for $n=6$.)

I can email you copies of these sources should you need them.

New answer: I was unhappy with the previous answer, because one should be able to answer this using basic facts about the representation theory of the symmetric group. Here is such an answer: my reference is Fulton and Harris' Representation theory.

We know that irreducible reps of $S_n$ are associated to partitions $\lambda$ of $n$. Given such a partition $\lambda$, there are several formulae for the dimension of the associated irreducible rep - see (4.10), (4.11) and (4.12) of F&H. The latter is the hook length formula, which reads:

$$ \dim(V_\lambda)=\frac{d!}{\prod (\textrm{ Hook lengths})}.$$

Using this formula it is easy to work out that, for $n\geq 7$, there are only four reps of $S_n$ of degree less than $n$ - two of dim $1$ and two of dim $n-1$ (this is Exercise 4.14 of F&H). It's pretty obvious what they all are - each pair correspond to partitions that are conjugate to each other.

Now one uses Proposition 5.1 of F&H to see which of these remain as irreducibles when one restricts to the alternating group $A_n$ - the answer is that they remain irreducible so long as the associated partition is not self-conjugate. None of the partitions in question are self-conjugate so they remain irreducible; what is more representations corresponding to partitions that are conjugate yield isomorphic representations when one restricts to $A_n$ - this yields our irreducible reps for $A_n$ of dimension $1$ and $n-1$.

Now to complete the proof one needs to check that there are no self-conjugate partitions of $n$ for which the associated irreducible rep has dimension $\leq 2(n-1)$. (If there were these would split in half to yield irreducible representations of $A_n$.) This is easy.

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  • $\begingroup$ Thanks for your response. Result 2 of Rasala's paper is only saying that $n-1$ is the second least irreducible degree character of $S_n$ (not $A_n$). It can be shown that $n-1$ can be occured as the degree of an irreducible character of $A_n$. My question is about the uniqueness of such a character of degree $n-1$ for all $n>7$ for $A_n$. I know that the character is $\chi$ (see above, my comment for Dror Speiser). I would like to know if a implicit reference for the latter. Thanks to ban of Elsevier by some brave people, the paper can be freely downloaded from the publihser. $\endgroup$ Nov 11 '13 at 20:01
  • $\begingroup$ @Alireza, I'm just about to add a better answer. $\endgroup$
    – Nick Gill
    Nov 11 '13 at 20:05
  • $\begingroup$ I am grateful to you. $\endgroup$ Nov 11 '13 at 20:31
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As dear Derek Holt said, the answer is yes. These two references are related to this problem:

$1)$ "The Faithful Linear Representation of Least Degree of $S_n$ and $A_n$ over a Field of Characteristic 2" by $A.$ $Wagner$.

$2)$ "The Faithful Linear Representations of Least Degree of $S_n$ and $A_n$ over a Field of Odd Characteristic" by $A.$ $Wagner$.

The links for downloading these paper are:

"https://eudml.org/doc/172437"

"https://eudml.org/doc/172514"

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    $\begingroup$ Stefan, if there is a faithful linear representation of some degree d in characteristic 0, there also exist a faithful linear representation of the same degree in any sufficiently large characteristic. $\endgroup$
    – Joël
    Nov 9 '13 at 22:31
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    $\begingroup$ The answer of the question could not be obtained directly from these two papers. Could you please refer me to a reference that directly examines this question? $\endgroup$ Nov 10 '13 at 15:41
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    $\begingroup$ @Shahrooz: The author's name is actually "Ascher Wagner" (with "Ascher" as the first name). Also, you should spell out the connection with characteristic 0 representations. $\endgroup$ Nov 10 '13 at 18:14
  • $\begingroup$ @Shahrooz. After some thought I did not arrive to the conclusion that how you can derive the uniqueness of the irreducible character of degree $n-1$ From these two papers which are about nonzero characteristic. The symmetric group of degree $n$ has two irreducible character of degree $n-1$ corresponding to the partitions $n-1,1$ and $2, 1^{n-2}$. I think one must show that these two irreducible characters give only one character for the alternating. Actually I need an explicite reference as Derek Holt mentioned above it is widely known since it is quoted in the Wikipedia. $\endgroup$ Nov 10 '13 at 19:39
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    $\begingroup$ @Alireza: isn't one such representation necessarily the other times the sign? $\endgroup$ Nov 11 '13 at 0:59
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Uniqueness.

With respect to the proposition 20.13 provided in the book Representations and characters of groups by James and Liebeck, and the answer given above by Nick Gill, the uniqueness of the "irreducible character of degree $ n−1$" is satisfied.

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