A partial order on the symmetric groups defined by 3-cycles

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We define a partial order on the symmetric group S_n of degree n by x ≥ y iff y = a_1 ··· a_kx with i(y) = i(x) + 2k where a_1, ··· , a_k are 3-cycles of increasing or decreasing consecutive three letters and i(*) is the number of inversions of the element * of S_n, on the analogy of the weak Bruhat order. Whether an even permutation is comparable to the identity or not in this ordering is considered. It is shown that all of the even permutations of degree n which map 1 to n or n - 1 are comparable to the identity.



  • Ryukyu mathematical journal

    Ryukyu mathematical journal 15 19-42, 2002-12-30

    Department of Mathematical Sciences, Faculty of Science, University of the Ryukyus

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