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  • On cocliques in commutative Schurian association schemes of the symmetric group [Elektronski vir]
    Maleki, Roghayeh ; Razafimahatratra, Andriaherimanana Sarobidy
    Given the symmetric group ▫$G = \sym(n)$▫ and a multiplicity-free subgroup ▫$H\leq G$▫, the orbitals of the action of ▫$G$▫ on ▫$G/H$▫ by left multiplication induce a commutative association scheme. ... The irreducible constituents of the permutation character of ▫$G$▫ acting on ▫$G/H$▫ are indexed by partitions of ▫$n$▫ and if ▫$\lambda \vdash n$▫ is the second largest partition in dominance ordering among these, then the Young subgroup ▫$\sym(\lambda)$▫ admits two orbits in its action on ▫$G/H$▫, which are ▫$\mathcal{S}_\lambda$▫ and its complement. In their monograph ▫$\cite[Problem~16.13.1]{godsil2016algebraic}$▫, Godsil and Meagher asked whether ▫$\mathcal{S}_\lambda$▫ is a coclique of a graph in the commutative association scheme arising from the action of ▫$G$▫ on ▫$G/H$▫. If such a graph exists, then they also asked whether its smallest eigenvalue is afforded by the ▫$\lambda$▫-module. In this paper, we initiate the study of this question by taking ▫$\lambda = [n-1,1]$▫. We show that the answer to this question is affirmative for the pair of groups ▫$\left(G,H\right)$▫, where ▫$G = \sym(2k+1)$▫ and ▫$H = \sym(2) \wr \sym(k)$▫, or ▫$G = \sym(n)$▫ and ▫$H$▫ is one of ▫$\alt(k) \times \sym(n-k),\ \alt(k) \times \alt(n-k)$, or $\left(\alt(k)\times \alt(n-k)\right) \cap \alt(n)$▫. For the pair ▫$(G,H) = \left(\sym(2k),\sym(k)\wr \sym(2)\right)$▫, we also prove that the answer to this question of Godsil and Meagher is negative.
    Vir: The Australasian journal of combinatorics [Elektronski vir]. - ISSN 2202-3518 (Vol. 89, iss. 3, 2024, str. 413-447)
    Vrsta gradiva - e-članek ; neleposlovje za odrasle
    Leto - 2024
    Jezik - angleški
    COBISS.SI-ID - 201865475

    Povezava(-e):

    https://ajc.maths.uq.edu.au/pdf/89/ajc_v89_p413.pdf

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