Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An Ennola duality for subgroups of groups of Lie type   We develop a theory of Ennola duality for subgroups of finite groups of Lie
type, relating subgroups of twisted and untwisted groups of the same type.
Roughly speaking, one finds that subgroups $H$ of $\mathrm{GU}_d(q)$ correspond
to subgroups of $\mathrm{GL}_d(-q)$, where $-q$ is interpreted modulo $|H|$.
Analogous results for types other than $\mathrm A$ are established, including
for exceptional types where the maximal subgroups are known, although the
result for type $\mathrm D$ is still conjectural. Let $M$ denote the Gram
matrix of a non-zero orthogonal form for a real, irreducible representation of
a finite group, and consider $\alpha=\sqrt{\det(M)}$. If the representation has
twice odd dimension, we conjecture that $\alpha$ lies in some cyclotomic field.
This does not hold for representations of dimension a multiple of $4$, with a
specific example of the Janko group $\mathrm J_1$ in dimension $56$ given.
(This tallies with Ennola duality for representations, where type $\mathrm
D_{2n}$ has no Ennola duality with ${}^2\mathrm D_{2n}$.)

31
The categories of this paper are: ao-sci
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
24
The categories of this paper are: dg-ga
20
The categories of this paper are: acc-phys
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
10
The categories of this paper are: q-bio
35
The categories of this paper are: mtrl-th
21
The categories of this paper are: adap-org
37
The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
11
The categories of this paper are: hep-lat
29
The categories of this paper are: funct-an
14
The categories of this paper are: math-ph
9
The categories of this paper are: nlin