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If $\Gamma$ is a compact simply-connected semisimple Lie group, then the Weyl Dimension Formula tells us exactly which dimensions it can act irreducibly on.

For certain $\Gamma$, it is easy to find pairs of nonisomorphic representations of the same dimension:

1). $A_n (n\geq 2)$, $C_n$ ($n$ = A116940(k)), $D_n (n\geq 4)$, and $E_6$ each have pairs of fundamental irreducible representations of the same dimension.

2). Additionally $G_2$ has two irreducible representations of dimension 77.

Furthermore, given that $\Gamma$ has one pair of nonisomorphic representations of the same dimension, it is easy to prove (using the Weyl Dimension Formula) that it has infinitely many such pairs.

Question 1: Among the remaining groups not mentioned above, which are known to have pairs of nonisomorphic irreducible representations of the same dimension?

Question 2: For obvious reasons $A_1$ cannot have such pairs, but are there any other cases where one can rule out the existence of such pairs?

It would seem that since the Dimension Formula so greatly restricts the possible dimensions for the other groups that in the long run there must be pairs of irreducibles of the same dimension; if anyone knows of any results along the lines of such heuristic forcing arguments those would be useful as well.

Edit: Robert's answer below reminded me that such pairs also occur for:

3). $B_n$ when $n$ = A116940(k) just as in the $C_n$ case because of a result I proved awhile back. More generally, one can show that if $B_n$ has such a pair then so does $C_n$ and conversely if $C_n$ has such a pair then so does $B_n$, so Robert's comment also shows that $C_4$ and $C_5$ have such pairs (the pairs arising from Robert's examples are in dimensions 11354112 and 38928384 for $C_4$ and 24741150720 for $C_5$).

4). $F_4$ has two irreducible representations of dimension 1053 that I had completely forgotten about in my list.

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for finite groups, one often have characters that are non-real, and so the conjugate character gives one a pair of irreducibles of the same dimension. It seems that as soon as a group, finite or not, has an element that is not conjugate to its inverse, one is bound to have such pairs of representations. –  Dima Pasechnik Feb 1 '11 at 7:17
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@Dima: all such pairs are already covered by the $A_n$, $D_n$ and $E_6$ examples cited in the question. –  ARupinski Feb 1 '11 at 7:24
    
Are you saying that in all the other examples it holds that each element is conjugate to its inverse? –  Dima Pasechnik Feb 1 '11 at 7:35
    
@Dima: Yes. The only groups which have irreducible representations of complex type are $A_n$ for $n\geq 2$, $D_n$ for $n$ odd, and $E_6$. The others only have irreducible representations of real or quaternionic type and hence every element of these groups is conjugate to its own inverse. –  ARupinski Feb 1 '11 at 8:45
    
@Dima: It is true that all irreducible representations of $A_1$, $B_n$, $C_n$, $D_{2n}$ $G_2$, $F_4$, $E_7$ and $E_8$ are either real or quaternionic, so all complex representations are self-conjugate, see mathoverflow.net/questions/47492. –  Andrei Moroianu Feb 1 '11 at 8:59
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2 Answers

up vote 6 down vote accepted

We give a table for irreducible representations up to dimension $2^{15}$ in the supplement of our preprint http://arxiv.org/abs/1012.5256v1

I immediately found the following examples with respect to the first question: (1) $B_3$ occurs twice in dimension 112, 168, etc. (2) $B_4$ occurs three times in dimension 2772, twice in dimension 9504, etc. (3) $B_5$ occurs twice in dimension 23595. (4) $F_4$ occurs twice in dimension 1053.

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Thanks for these examples. I can't believe I didn't remember the $F_4$ example considering how many times I've looked at its irreducible dimensions. As for the $B_3$ example, it reminded me of another class of examples in the $B_n$ series when $n = A116940(k)$ as with the $C_n$ series that I had totally overlooked when making my list above. Will edit the question to reflect this. –  ARupinski Feb 1 '11 at 17:35
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This also happens for $E_8$, see OEIS A121732 where you find the remark: Inequivalent representations can have the same dimension. For example, the highest weights 10100000 and 10000011 (with fundamental weights numbered as in Bourbaki) both correspond to irreducible representations of dimension 8634368000. –  Skip Feb 1 '11 at 20:05
    
After seeing Skip`s comment I ran my program for $E_7$. I found two irreducible representations of dimension 1903725824 with highest weights 0000023 and 0001100 (using the Bourbaki ordering). –  Robert Zeier Feb 1 '11 at 21:20
    
@Skip and Robert: Thanks for those $E_7$ and $E_8$ examples. I'm rather surprised that these examples occur for such small highest weight vectors. –  ARupinski Feb 2 '11 at 3:37
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Without offering a complete answer to the stated question, I'd first ask what significance the answer would have (one way or the other) in terms of Lie theory? I'd also want to extract the essential numerical problem, which only concerns the numerator polynomial in Weyl's dimension formula. (The denominator is constant and doesn't affect the outcome.)

Given an irreducible root system belonging to a simple Lie type of rank $r$, the polynomial in the numerator is a product of $n$ factors each of which is a $\mathbb{Z}$-linear combination of $r$ variables $x_1, \dots, x_r$ with positive coefficients depending on the root system. Here $n$ is the number of positive roots. The question is when if ever this polynomial can take the same value at two different $r$-tuples of strictly positive integers. If it does take the same value twice, it will take the same value infinitely many times: multiply each $r$-tuple by the same positive integer.

From the viewpoint of root systems (compact groups being far in the background now), types $A_r$ with $r>1$, $D_r$ with $r>3$, and $E_6$ have graph automorphisms which guarantee a positive answer to the question. Beyond this I can only see a possibility of accidental positive (or negative) answers as in the case of $G_2$. By now the question is only remotely connected to Lie theory, via the fixed linear combinations of the variables occurring as factors in the given polynomial.

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Actually, the question is partially based on Lie Theory and partly on number theory. On the number theory side, as you point out, the dimension formula is a polynomial in $n$ variables where $n$ is the rank of $\Gamma$ and we want to know when this polynomial achieves the same value on two different points of $(\mathbb{Z}_{\geq 0})^n$; essentially it is looking at when one can find a level set which contains more than one positive integer point. –  ARupinski Feb 1 '11 at 17:26
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On the Lie theory side, my interest is in zero sets of (possibly virtual) characters of $\Gamma$ as subvarieties of a maximal torus. One place I am looking at these zero sets is on 0-dimensional characters. Hence if I find two irreducibles of the same dimension, I can look at the zero set of their difference as such a character. –  ARupinski Feb 1 '11 at 17:28
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