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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.
20C30 Representations of finite symmetric groups
05E10 Combinatorial aspects of representation theory
16W22 Actions of groups and semigroups; invariant theory (associative rings and algebras)
Full Text: DOI
[1] Can, Mahir Bilen, A representation on labeled rooted forests, Commun. Algebra, 46, 10, 4273-4291, (2018) · Zbl 1395.05142
[2] Chen, Yao Min; Garsia, Adriano M.; Remmel, Jeffrey, Combinatorics and algebra (Boulder, Colo., 1983), 34, Algorithms for plethysm, 109-153, (1984), American Mathematical Society · Zbl 0556.20013
[3] Comtet, Louis, Advanced combinatorics. The art of finite and infinite expansions, xi+343 p. pp., (1974), Reidel Publishing Co. · Zbl 0283.05001
[4] Ganyushkin, Olexandr; Mazorchuk, Volodymyr, Classical finite transformation semigroups: an introduction, 9, xii+314 p. pp., (2009), Springer · Zbl 1166.20056
[5] Littlewood, Dudley E., The Theory of Group Characters and Matrix Representations of Groups, viii+292 p. pp., (1940), Oxford University Press · Zbl 0025.00901
[6] Loehr, Nicholas A.; Remmel, Jeffrey, A computational and combinatorial exposĂ© of plethystic calculus, J. Algebr. Comb., 33, 2, 163-198, (2011) · Zbl 1229.05275
[7] Macdonald, Ian Grant, Symmetric functions and Hall polynomials, x+475 p. pp., (1995), Clarendon Press · Zbl 0824.05059
[8] Remmel, Jeffrey; Whitney, Roger, Multiplying Schur functions, J. Algorithms, 5, 4, 471-487, (1984) · Zbl 0557.20008
[9] Thrall, Robert M., On symmetrized Kronecker powers and the structure of the free Lie ring, Am. J. Math., 64, 371-388, (1942) · Zbl 0061.04201
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