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Loop-augmented forests and a variant of Foulkes’s conjecture. (English) Zbl 06987759
Summary: A loop-augmented forest is a labeled rooted forest with loops on some of its roots. By exploiting an interplay between nilpotent partial functions and labeled rooted forests, we investigate the permutation action of the symmetric group on loop-augmented forests. Furthermore, we describe an extension of Foulkes’s conjecture and prove a special case. Among other important outcomes of our analysis are a complete description of the stabilizer subgroup of an idempotent in the semigroup of partial transformations and a generalization of the (Knuth-Sagan) hook length formula.
MSC:
20C30 Representations of finite symmetric groups
05E10 Combinatorial aspects of representation theory
16W22 Actions of groups and semigroups; invariant theory (associative rings and algebras)
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