Which groups have only real and quaternionic irreducible representations? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T12:45:25Z http://mathoverflow.net/feeds/question/47492 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47492/which-groups-have-only-real-and-quaternionic-irreducible-representations Which groups have only real and quaternionic irreducible representations? John Baez 2010-11-27T08:55:35Z 2010-11-28T22:30:35Z <p>Consider a continuous irreducible representation of a compact Lie group on a finite-dimensional complex Hilbert space. There are three mutually exclusive options:</p> <p>1) it's not isomorphic to its dual (in which case we call it 'complex')</p> <p>2) it has a nondegenerate symmetric bilinear form (in which case we call it 'real')</p> <p>3) it has a nondegenerate antisymmetric bilinear form (in which case we call it 'quaternionic')</p> <p>It's 'real' in this sense iff it's the complexification of a representation on a real vector space, and it's 'quaternionic' in this sense iff it's the underlying complex representation of a representation on a quaternionic vector space.</p> <p>Offhand, I know just <b>four compact Lie groups whose continuous irreducible representations on complex vector spaces are all either real or quaternionic in the above sense</b>:</p> <p>1) the group Z/2 </p> <p>2) the trivial group</p> <p>3) the group SU(2)</p> <p>4) the group SO(3)</p> <p>Note that I'm so desperate for examples that I'm including 0-dimensional compact Lie groups, i.e. finite groups! </p> <p>1) is the group of unit-norm real numbers, 2) is a group covered by that, 3) is the group of unit-norm quaternions, and 4) is a group covered by <i>that</i>. This probably explains why these are all the examples I know. For 1), 2) and 4), all the continuous irreducible representations are in fact real.</p> <p><b>What are all the examples?</b> </p> http://mathoverflow.net/questions/47492/which-groups-have-only-real-and-quaternionic-irreducible-representations/47500#47500 Answer by Torsten Ekedahl for Which groups have only real and quaternionic irreducible representations? Torsten Ekedahl 2010-11-27T11:34:03Z 2010-11-28T05:49:39Z <p>An irreducible representation is real or quaternionic precisely when its character is real-valued. By the Peter-Weyl theorem all characters are real-valued precisely when every element in the group is conjugate to its inverse. When the group is connected a more precise answer is as follows: The Weyl group (in its tautological representation) must contain multiplication by $-1$ and this is true precisely when all indecomposable root system factors have that property. I don't remember off hand which indecomposable root systems have this property but it is of course well known (type A is out, type B/C is in, type D depends on the parity of the rank).</p> <p><b>Addendum</b>: Found the relevant places in Bourbaki. All characters are real-valued precisely when the element he calls $w_0$ is $-1$ (Ch. VIII,Prop. 7.5.11) and one can also read off if a given representation is real or quaternionic (loc. cit. Prop 12). From the tables in Chapter 6 one gets that $w_0=-1$ precisely for $A_1$, B/C, D for even rank, $E_7$, $E_8$, $F_4$ and $G_2$.</p> http://mathoverflow.net/questions/47492/which-groups-have-only-real-and-quaternionic-irreducible-representations/47512#47512 Answer by Skip for Which groups have only real and quaternionic irreducible representations? Skip 2010-11-27T16:35:25Z 2010-11-27T16:35:25Z <p>Torsten answered this question perfectly for the definition of real/complex/quaternionic in John's original question. But this usage of real/complex/quaternionic is foreign to my experience. Specifically, if you look at an irreducible real representation of a group, then its endomorphism ring is (by Schur and Frobenius) R, C, or H. And this seems to give a natural meaning of the terms "real", "complex" and "quaternionic" for irreps. This definition does not agree with John's, as you can see by considering the spin reps of Spin(7,1).</p> <p>My definition is also what you find in Noah Snyder's answer <a href="http://mathoverflow.net/questions/17495/what-rings-groups-have-interesting-quaternionic-representations" rel="nofollow">here</a> and in Wikipedia's definition of quaternionic representation.</p> http://mathoverflow.net/questions/47492/which-groups-have-only-real-and-quaternionic-irreducible-representations/47513#47513 Answer by Faisal for Which groups have only real and quaternionic irreducible representations? Faisal 2010-11-27T16:37:49Z 2010-11-28T22:30:35Z <p>This was a comment on Torsten's answer, but it got too long.</p> <p>Suppose $G$ is connected and semisimple. Fixing a choice $\Phi^+$ of positive roots for $G$, we can describe $w_0$ as the unique element of the Weyl group of $G$ that takes $\Phi^+$ to the negative roots $\Phi^- = -\Phi^+$. Now, $-w_0$ is an involution of the Dynkin diagram of $G$. This involution is trivial when the components of the Dynkin diagram lack two-fold symmetry, and this happens precisely for components of type $A_1$, $B_n$, $C_n$, $D_{2n}$, $E_7$, $E_8$, $F_4$ and $G_2$, in which case $-w_0=1$. For type $A_n$ ($n>1$), the involution is given by $\alpha_i \leftrightarrow \alpha_{n-i+1}$, for $D_n$ it's given by $\alpha_i \leftrightarrow \alpha_{i-1}$, and for $E_6$ it's given by $\alpha_1 \leftrightarrow \alpha_6$ and $\alpha_2 \leftrightarrow \alpha_5$.</p> <p>Now if $V$ is an irrep of highest weight $\lambda$, then $V^\ast$ has highest weight $-w_0\lambda$. So $V \cong V^\ast$ whenever $-w_0=1$, and the above discussion tells us when this happens.</p> <p>Side note: There's a closely related <a href="http://mathoverflow.net/questions/38165/" rel="nofollow">MO question</a>, which was asked not too long ago, whose answers might be helpful.</p>