Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The action of the Weyl group on the $E_8$ root system   Let $\Gamma$ be the graph on the roots of the $E_8$ root system, where any
two distinct vertices $e$ and $f$ are connected by an edge with color equal to
the inner product of $e$ and $f$. For any set $c$ of colors, let $\Gamma_c$ be
the subgraph of $\Gamma$ consisting of all the $240$ vertices, and all the
edges whose color lies in $c$. We consider cliques, i.e., complete subgraphs,
of $\Gamma$ that are either monochromatic, or of size at most $3$, or a maximal
clique in $\Gamma_c$ for some color set $c$, or whose vertices are the vertices
of a face of the $E_8$ root polytope. We prove that, apart from two exceptions,
two such cliques are conjugate under the automorphism group of $\Gamma$ if and
only if they are isomorphic as colored graphs. Moreover, for an isomorphism $f$
from one such clique $K$ to another, we give necessary and sufficient
conditions for $f$ to extend to an automorphism of $\Gamma$, in terms of the
restrictions of $f$ to certain special subgraphs of $K$ of size at most 7.

15
The categories of this paper are: nucl-ex
36
The categories of this paper are: cmp-lg
14
The categories of this paper are: math-ph
34
The categories of this paper are: chem-ph
31
The categories of this paper are: ao-sci
5
The categories of this paper are: gr-qc
11
The categories of this paper are: hep-lat
30
The categories of this paper are: q-alg
18
The categories of this paper are: econ
19
The categories of this paper are: eess
13
The categories of this paper are: quant-ph
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
3
The categories of this paper are: physics
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The categories of this paper are: dg-ga
10
The categories of this paper are: q-bio