Timeline for Realizability of a real representation using real cyclotomic coefficients
Current License: CC BY-SA 4.0
23 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 22, 2021 at 13:31 | vote | accept | Denis Rosset | ||
| Mar 31, 2021 at 2:27 | comment | added | Noah Snyder | Sorry that was supposed to say $C_m$. The point is to just apply "Wigner's method of little groups." You take the representation of $C_m$ and the representation of $N$ and their external tensor product is a representation of $C_m \times N$ and then induce that to $G$. | |
| Mar 31, 2021 at 1:00 | comment | added | David E Speyer | I'm not sure what you mean by inducing "a tensor product of an irrep of $C_p$ with one of $N$". Do you mean to induce one irrep from $C_p$ to $G$, pull back another irrep from $N$ to $G$, and tensor the results? | |
| Mar 31, 2021 at 0:57 | comment | added | David E Speyer | @NoahSnyder Thanks for pointing me to that! Theorem 4.5.2 definitely implies that my guess was right, even without knowing anything about Brauer characters. | |
| Mar 30, 2021 at 16:37 | comment | added | Noah Snyder | I think if I understood Theorem 4.6 in that paper better then I could answer this question. The key case is where $C_m$ is a cyclic group of odd order, $P$ is a 2-group which acts on $C_m$ by inversion via $P/N \cong C_2$, we have $G = C_m \rtimes P$, and the representation that you're looking at is induced from a tensor product of an irrep of $C_p$ with one of N. Using Theorem 4.7, if we can show that the $\ell$-adic Schur indices vanish for odd $\ell$ then we'd be done, and I think Theorem 4.6 is supposed to let you figure out $\ell$-adic indices for $p$-quasi-elementary groups easily. | |
| Mar 30, 2021 at 16:34 | comment | added | Noah Snyder | @DavidESpeyer: I think you're right that the Schur index over an $\ell$-adic field where $\ell$ does not divide $|G|$ will always be 1. See the bottom of page 7 of arxiv.org/pdf/1712.00907.pdf (though I'm somewhat confused about the logic because I don't understand Brauer characters): "We see from the second part that the p-adic Schur indices that we need to consider are limited to those primes p dividing the order of the group." | |
| Mar 30, 2021 at 16:03 | comment | added | David E Speyer | @NoahSnyder Do you know: If $G$ is a finite group, $\ell$ is a prime not dividing $|G|$ and $V$ is an irrep of $G$ whose character takes values in $\mathbb{Q}_{\ell}$, does this mean that $V$ is defined over $\mathbb{Q}_{\ell}$? It seems like, even though $\mathbb{Q}_{\ell}$ has lots of division algebras, I can't get them to show up using groups with order prime to $\ell$. | |
| Mar 30, 2021 at 15:50 | vote | accept | Denis Rosset | ||
| Mar 30, 2021 at 15:53 | |||||
| Mar 30, 2021 at 3:11 | answer | added | David E Speyer | timeline score: 5 | |
| Mar 25, 2021 at 12:53 | answer | added | Dima Pasechnik | timeline score: 3 | |
| Mar 22, 2021 at 21:14 | comment | added | Noah Snyder | Is it true that if you have a monomial representation over $\mathbb{Q}(\zeta_n)$ and its induction is defined over some field $K$ then the induction is also defined over $K \cap \mathbb{Q}(\zeta_n)$? Or more generally if $V$ is defined over $L$ and $\mathrm{Ind}_H^G V$ is defined over $K$ is $\mathrm{Ind}_H^G V$ defined over $L \cap K$? | |
| Mar 22, 2021 at 16:52 | comment | added | Noah Snyder | That representation is induced from a faithful 1-dimensional representation of $C_4 \times A_3$. | |
| Mar 22, 2021 at 16:51 | comment | added | Noah Snyder | The most interesting example I've found is the 4-dimensional representation of $Q_8 \rtimes S_3$ where $S_3$ acts by an inner automorphism. I think that twice this representation has a $\mathbb{Q}$ form with quaternion algebra $(-1,3)_\mathbb{Q}$. Here the exponent is $12$, and indeed the original irrep is defined over the real subfield $\mathbb{Q}(\sqrt{3}) \subset \mathbb{Q}(\zeta_{12})$. But on the other hand, the rep is defined over the cyclotomic field $\mathbb{Q}(i)$ but not over its real subfield $\mathbb{Q}$. (Possible error, maybe $i$ should be $\sqrt{-3}$.) | |
| Mar 22, 2021 at 13:57 | comment | added | Noah Snyder | The thing that makes this problem tricky is that I’m not sure of an easy way to figure out which fields a monomial representation (ie induced from a 1-dimensional) is defined over. The usual cyclotomic argument only uses that if your 1-dimensional representation is defined over a field then so is its induction. For this question we need to understand when the field drops in size when you induce. | |
| Mar 22, 2021 at 11:57 | answer | added | Dima Pasechnik | timeline score: 3 | |
| Mar 21, 2021 at 8:53 | comment | added | Dima Pasechnik | depending on your application, you might be willing to settle for a bit less, i.e. for your invariant form to be given by a diagonal matrix D with real positive entries, so that the needed for D=I computation, involving doing square roots, is not done. | |
| Mar 19, 2021 at 0:06 | comment | added | Noah Snyder | On more thought I doubt that gives a counterexample. It may give an example of a similar phenomenon, namely I think it's defined over $\mathbb{Q}(\zeta_k)$ for some $k$ and over $\mathbb{R}$ but not over $\mathbb{Q}(\zeta_k) \cap \mathbb{R}$. But $k$ is a number smaller than the exponent, and if you go up to the exponent then I expect you're ok. | |
| Mar 18, 2021 at 21:02 | comment | added | Noah Snyder | I think I have a counterexample: the 4-dimensional irrep of $D_8 \rtimes S_3$ (i.e. the semidirect product of the dihedral group of order 8 with the group of permutations of three letters, where the latter acts via its quotient to the cyclic group $C_2$ acting by inversion). See this page and note the key words "orthogonal" (i.e. real) and "Schur index 2" (i.e. involves a generalized quaternion algebra). But I haven't worked out a proof that it's a counterexample. | |
| Mar 17, 2021 at 15:16 | comment | added | Noah Snyder | That is, I think it tells you that you can reduce this question to groups of the form $C \rtimes P$ where $P$ is a 2-group, $C$ is a cyclic group of odd order, and every element of $P$ acts on $C$ via the identity or inversion. This is only a little more general than the dihedral case. | |
| Mar 17, 2021 at 15:12 | comment | added | Noah Snyder | What you can't do is just say that if it's defined over $L$ and defined over $K$ then it's defined over $L \cap K$, see this question. So you're going to need to try to mimic the proof of the original result rather than just applying the original result. That said, I think once you get through the notation, the result from Serre I mention tells you pretty clearly where to look. | |
| Mar 17, 2021 at 13:57 | comment | added | Denis Rosset | Ouch, that's above my pay grade, though I appreciate Serre's little book immensely. I'm posting this to assess the difficulty of the question as well -- because then the next step is to find an algorithm to compute the images. | |
| Mar 17, 2021 at 2:58 | comment | added | Noah Snyder | I think one should be able to answer this question with the techniques of Serre's Linear Representations of Finite Groups Section 12.6 (the generalization of Brauer's theorem to representations defined over particular fields) applied to the real numbers. But I don't immediately see how to do it (or which way the answer will go!) without doing some calculations. | |
| Mar 16, 2021 at 23:22 | history | asked | Denis Rosset | CC BY-SA 4.0 |