Abstract
An (association) scheme is said to be Frobenius if it is the (orbital) scheme of a Frobenius group. A scheme which has the same tensor of intersection numbers as some Frobenius scheme is said to be pseudofrobenius. We establish a necessary and sufficient condition for an imprimitive pseudofrobenius scheme to be Frobenius. We also prove strong necessary conditions for existence of an imprimitive pseudofrobenius scheme which is not Frobenius. As a byproduct, we obtain a sufficient condition for an imprimitive Frobenius group G with abelian kernel to be determined up to isomorphism only by the character table of G. Finally, we prove that the Weisfeiler-Leman dimension of a circulant graph with n vertices and Frobenius automorphism group is equal to 2 unless \(n\in \{p,p^2,p^3,pq,p^2q\}\), where p and q are distinct primes.
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Notes
An imprimitivity system I is smaller than an imprimitivity system \(I'\) if each block of \(I'\) is contained in some block of I.
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The second author is supported by Leonard Euler International Mathematical Institute in Saint Petersburg under agreement No. 075-15-2019-1620 with the Ministry of Science and Higher Education of the Russian Federation.
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Ponomarenko, I., Ryabov, G. On pseudofrobenius imprimitive association schemes. J Algebr Comb 57, 385–402 (2023). https://doi.org/10.1007/s10801-022-01193-4
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DOI: https://doi.org/10.1007/s10801-022-01193-4