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Problem with Schur index computation #103
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Dear Thomas, When I calculate the Schur index manually for this example, I am getting 1. The group is the semidirect product C_{17} by D_{16}. In both wedderga and Magma, the program first tries to reduce to a proper subgroup globally, but there are no subgroups that admit a global reduction. At this point in Magma's original algorithm it would look for the group in the list of Schur groups in [Peter Schmid, J. Algebra 169 (1994) 226-247], these are groups with no global reductions that are known to admit nontrivial local indices. When I checked Schmid's list this morning, I was not able to show that this group appears there, For this reason I am surprised Magma is not also giving 1 at this point - but I have yet to look at Unger's paper; so forgive my temporary ignorance I didn't look at his paper yet to see how he is getting a 2. At this point wedderga's algorithm would try to calculate the local indices. The 2 local and real Schur indices are 1. It then tries to compute the 17-local index directly; for this group the crossed product for a simple component corresponding to one of the faithful irreducibles is (F(\zeta_4, \zeta_{17}, C_2 x C_2). Since the 17-adic field Q_{17} contains a 16th root of 1, when we tensor this with Q_{17} the \zeta_4 part is absorbed and we get a 2x2 matrix ring over the cyclic algebra (K(\zeta_{17})/K,\sigma,\zeta_8) where K/ Q_{17} is 2-dimensional. Since \zeta_8 is the square of \zeta_{16} \in K, \zeta_8 is a norm in this extension, so this is why wedderga returns a 17-local index equal to 1. -Allen Herman, Jan 21,2025 |
Thanks @drallenherman. |
Yes @ThomasBreuer. Interesting. |
Dear @ThomasBreuer, I went ahead and calculated the 17-local index using Benard's Theorem, it again gives 1. This got me thinking to check the 2-local index. The Defect Group of this character turns out to be dihedral of order 8, which is not cyclic, so Benard's theorem does not apply. In wedderga we would have to identify the group as one of the exceptional Schur group to have a nontrivial local index. This group does not appear in [Schmid,1994] as an exceptional Schur group, so this is why wedderga is saying the 2-local index is 1. However, this was exactly the point Unger was making with this example. In [Riese and Schmid, JAlg 182 (1996) 183-200], the definition of dyadic Schur groups of type (QD,q) was extended slightly to include this sort of group, and there they prove faithful characters of these groups have 2-local index 2. This is an omission in Wedderga, we will need to upgrade wedderga's "IsDyadicSchurGroup" command so that it responds true to all of these additional (QD,q) groups. Thank you for bringing this issue to our attention. |
Fixes gap-packages#103 reported by Thomas Breuer. The fix provided by Allen Herman who wrote: "The cause of the problem is that my earlier implementation of this command does not recognize some dyadic Schur groups of type (QD,q) correctly, this causes our current Wedderga to return a 2-local index of 1 for these groups when it should be a 2. I believe the group SmallGroup(272,15) that Thomas Breuer mentioned is the smallest group with this issue. The new implementation of the command follows the description of the dyadic Schur groups of type (QD,q) in Riese and Schmid's 1996 paper very precisely, so I'm confident this will fix the problem."
See PR #104 by @olexandr-konovalov |
I get the following GAP session.
According to Magma, the Schur indices should be
2
.Here is a Magma session for that.
(This example occurs in Section 6.2 of a paper by Unger about an algorithm for computing Schur indices.)
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