Set families and Foulkes modules
Rowena Paget
and Mark Wildon
DOI: 10.1007/s10801-011-0282-8
Abstract
We construct a new family of homomorphisms from Specht modules into Foulkes modules for the symmetric group. These homomorphisms are used to give a combinatorial description of the minimal partitions (in the dominance order) which label the irreducible characters appearing as summands of the characters of Foulkes modules. The homomorphisms are defined using certain families of subsets of the natural numbers. These families are of independent interest; we prove a number of combinatorial results concerning them.
Pages: 525–544
Keywords: keywords foulkes' conjecture; Specht module; foulkes module; module homomorphism; closed set family
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References
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2. Curtis, C.W., Reiner, I.: Representation Theory of Finite Groups and Associative Algebras. AMS Chelsea, Providence (2006). Reprint of the 1962 original
3. Dent, S.C., Siemons, J.: On a conjecture of Foulkes. J. Algebra 226(1), 236-249 (2000)
4. Foulkes, H.O.: Concomitants of the quintic and sextic up to degree four in the coefficients of the ground form. J. Lond. Math. Soc. 25, 205-209 (1950)
5. Fulton, W.: Young Tableaux. London Mathematical Society Student Texts, vol.
35. CUP, Cambridge (1997)
6. Howe, R.: (GLn, GLm)-duality and symmetric plethysm. Proc. Indian Acad. Sci. Math. Sci. 97, 85- 109 (1987)
7. James, G.D.: The representation theory of the symmetric groups. Lecture Notes in Mathematics, vol.
682. Springer, Berlin (1978)
8. McKay, T.: On plethysm conjectures of Stanley and Foulkes. J. Algebra 319(5), 2050-2071 (2008)
9. Müller, J., Neunhöffer, M.: Some computations regarding Foulkes' conjecture. Exp. Math. 14(3), 277-283 (2005)
10. Stanley, R.P.: Positivity problems and conjectures in algebraic combinatorics. In: Mathematics: Frontiers and Perspectives, pp. 295-319. AMS, Providence (2000)
11. Thrall, R.M.: On symmetrized Kronecker powers and the structure of the free Lie ring. Am. J. Math.
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