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{\bf Sergei Evdokimov and Ilia Ponomarenko}
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{\bf Separability Number and Schurity Number of Coherent Configurations}
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To each coherent configuration (scheme) ${\cal C}$ and positive
integer~$m$
we associate a natural scheme~$\widehat{\cal C}^{(m)}$ on the $m$-fold Cartesian
product of the point set of~${\cal C}$ having the same automorphism
group as~${\cal C}$. Using this construction we define and study two
positive integers: the separability number~$s({\cal C})$ and the Schurity
number~$t({\cal C})$ of~${\cal C}$. It turns out that
$s({\cal C})\le m$ iff ${\cal C}$ is uniquely determined up to isomorphism
by the intersection numbers of the scheme~$\widehat{\cal C}^{(m)}$. Similarly,
$t({\cal C})\le m$ iff the diagonal subscheme of~$\widehat{\cal C}^{(m)}$ is an orbital
one.
In particular, if ${\cal C}$ is the scheme of a distance-regular
graph~$\Gamma$,
then $s({\cal C})=1$ iff $\Gamma$ is uniquely determined by its parameters
whereas $t({\cal C})=1$ iff $\Gamma$ is distance-transitive. We show that
if~${\cal C}$ is a Johnson, Hamming or Grassmann scheme, then $s({\cal C})\le
2$
and $t({\cal C})=1$. Moreover, we find the exact values of $s({\cal C})$ and
$t({\cal C})$ for the scheme~${\cal C}$ associated with any distance-regular
graph
having the same parameters as some Johnson or Hamming graph. In
particular, $s({\cal C})=t({\cal C})=2$ if~${\cal C}$ is the scheme of a Doob graph.
In addition, we prove that $s({\cal C})\le 2$ and $t({\cal C})\le 2$ for any
imprimitive 3/2-homogeneous scheme. Finally, we show that $s({\cal C})\le
4$,
whenever~${\cal C}$ is a cyclotomic scheme on a prime number of points.
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