Permutations with Verma Multiplicities [M(p):L(q)] ≥ 2

2026/07/21 by Daiva Pucinskaite
Mathematics · #math.CO #math.RT

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Abstract

We consider permutations q in the symmetric group S_n whose Verma multiplicities in the principal block of O(\mathfraksl_n) satisfy [M(p):L(q)] ≥ 2. We present a construction along with a diagrammatic visualization, showing how permutations in S_n with this property generate a family of permutations in S_n+1 that share the same multiplicity property. While the method does not recover all such permutations in S_n+1, it systematically generates many new examples. In addition, we present all permutations in S₅ with [M(id):L(q)] ≥ 2 in a Bruhat diagram and describe all permutations in S₆ and S₇ with non-simple Verma multiplicities.

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