Problem with Schur index computation

[ThomasBreuer]

I get the following GAP session.

gap> LoadPackage("wedderga");;
─────────────────────────────────────────────────────────────────────────────
Loading Wedderga 4.10.5 (Wedderga)
by Gurmeet Kaur Bakshi ([email protected]),
   Osnel Broche Cristo ([email protected]),
   Allen Herman ([email protected]),
   Olexandr Konovalov (https://olexandr-konovalov.github.io/),
   Sugandha Maheshwary ([email protected]),
   Gabriela Olteanu (http://math.ubbcluj.ro/~olteanu),
   Aurora Olivieri ([email protected]),
   Angel del Rio (http://www.um.es/adelrio), and
   Inneke Van Gelder (http://homepages.vub.ac.be/~ivgelder).
Homepage: https://gap-packages.github.io/wedderga
Report issues at https://github.com/gap-packages/wedderga/issues
─────────────────────────────────────────────────────────────────────────────
gap> G:= SmallGroup( 272, 15 );;
gap> filt:= Filtered( Irr( G ), x -> ClassPositionsOfKernel( x ) = [ 1 ] );;
gap> List( filt, x -> x[1] );
[ 4, 4, 4, 4, 4, 4, 4, 4 ]
gap> List( filt, x -> SchurIndexByCharacter( Rationals, G, x ) );
[ 1, 1, 1, 1, 1, 1, 1, 1 ]

According to Magma, the Schur indices should be 2.
Here is a Magma session for that.

> G:= SmallGroup( 272, 15 );
> tbl:= CharacterTable( G );
> chi:= tbl[40];
> chi[1];
4
> SchurIndex( chi );
2

(This example occurs in Section 6.2 of a paper by Unger about an algorithm for computing Schur indices.)

[drallenherman]

Dear Thomas,
Thank you for pointing this out to us. Wedderga's Schur index functions have been available since about 2011, our team has tweaked them on a couple of occasions to make corrections, the last time in 2020. But this is the first time anyone has ever reported a discrepancy in comparison to a calculation coming from Magma's Schur index functions! So it's about time these were tested against each other.

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

[ThomasBreuer]

Thanks @drallenherman.
Interesting.
I have just checked whether SmallGroup( 272, 15 ) really means the same group in GAP and Magma.
Yes, this is the case.

[drallenherman]

Yes @ThomasBreuer. Interesting.
We can apply Benard's Theorem to calculate the 17-local index in this case. I will try that to see if it gives a different answer.

[drallenherman]

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.

[fingolfin]

See PR #104 by @olexandr-konovalov