Plane Partitions and Characters of the Symmetric Group
Ernesto Vallejo
DOI: 10.1023/A:1008795704190
Abstract
In this paper we show that the existence of plane partitions, which are minimal in a sense to be defined, yields minimal irreducible summands in the Kronecker product 
of two irreducible characters of the symmetric group S(n). The minimality of the summands refers to the dominance order of partitions of n. The multiplicity of a minimal summand 
equals the number of pairs of Littlewood-Richardson multitableaux of shape ( , ), conjugate content and type . We also give lower and upper bounds for these numbers.
of two irreducible characters of the symmetric group S(n). The minimality of the summands refers to the dominance order of partitions of n. The multiplicity of a minimal summand equals the number of pairs of Littlewood-Richardson multitableaux of shape ( , ), conjugate content and type . We also give lower and upper bounds for these numbers.Pages: 79–88
Keywords: Kronecker product; character of symmetric group; dominance order of partition; tableau
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References
1. T. Brylawski, “The lattice of integer partitions,” Discrete Math. 6 (1973), 201-219.
2. M. Clausen and H. Meier, “Extreme irreduzible Konstituenten in Tensordarstellungen symmetrischer Gruppen,” Bayreuther Math. Schriften 45 (1993), 1-17.
3. A.J. Coleman, “Induced Representations with applications to Sn and GL(n),” Queen's papers in pure and applied mathematics, No. 4, Queens University, Kingston, Ontario, 1966.
4. P. Doubilet, J. Fox, and G.R. Rota, “The elementary theory of the symmetric group,” in Combinatorics, representation theory and statistical methods in groups, T.V. Narayama, R.M. Mathsen and J.G. Williams (eds.), Lecture Notes in Pure and Applied Mathematics, Vol. 57, Marcel Decker, New York, 1980.
5. Y. Dvir, “On the Kronecker product of Sn characters,” J. Algebra 154 (1993), 125-140.
6. Y. Dvir, “A family of Z bases for the ring of S(n) characters and applications to the decomposition of Kronecker products,” Europ. J. Combinatorics 15 (1994), 449-457.
7. \ddot O. E\~gecio\~glu and J.B. Remmel, “A combinatorial interpretation of the inverse Kostka matrix,” Linear and Multilinear Algebra 26 (1990), 59-84.
8. P.C. Fishburn, J.C. Lagarias, J.A. Reeds, and L.A. Shepp, “Sets uniquely determined by projections on axes II. Discrete case,” Discrete Math. 91 (1991), 149-159.
9. D. Gale, “A theorem of flows in networks,” Pacif. J. Math. 7 (1957), 1073-1082. VALLEJO
10. A. Garsia and J. B. Remmel, “Shuffles of permutations and the Kronecker product,” Graphs Combin. 1 (1985), 217-263.
11. G.D. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of mathematics and its applications, Vol. 16, Addison-Wesley, Reading, Massachusetts, 1981.
12. T.Y. Lam, Young diagrams, “Schur functions, the Gale-Ryser theorem and a conjecture of Snapper,” J. Pure Appl. Algebra 10 (1977), 81-94.
13. D.E. Littlewood, “The Kronecker product of symmetric group representations,” J. London Math. Soc. 31 (1956), 89-93.
14. I.G. Macdonald, Symmetric functions and Hall polynomials, 2nd. edition, Oxford Univ. Press, Oxford, 1995.
15. P.A. MacMahon, Combinatorial Analysis, Vol. II. Cambridge Univ. Press, London-New York 1916; reprinted by Chelsea, New York, 1960.
16. F.D. Murnaghan, “The analysis of the Kronecker product of irreducible representations of the symmetric group,” Amer. J. Math. 60 (1938), 761-784.
17. J.B. Remmel, “A formula for the Kronecker products of Schur functions of hook shapes,” J. Algebra 120 (1989), 100-118.
18. J.B. Remmel, “Formulas for the expansion of Kronecker products S(m,n) \otimes S(1p - r ,r) and S(1k,2l) \otimes S(1p - r ,r),” Discrete Math. 99 (1992), 265-287.
19. J.B. Remmel and T. Whitehead, “On the Kronecker product of Schur functions of two row shapes,” Bull. Belg. Math. Soc. 1 (1994), 649-683.
20. H.J. Ryser, “Combinatorial properties of matrices of zeros and ones,” Canad. J. Math. 9 (1957), 371-377.
21. H.J. Ryser, Combinatorial mathematics, Carus monograph No. 14, Math. Association of America, John Wiley and Sons, 1963.
22. B.E. Sagan, The symmetric group, Wadsworth & Brooks/Cole, Pacific Grove, California, 1991.
23. J. Saxl, “The complex characters of the symmetric groups that remain irreducible on subgroups, ” J. Algebra 111 (1987), 210-219.
24. E. Snapper, “Group characters and nonnegative integral matrices,” J. Algebra 19 (1971), 520-535.
25. A. Torres-Cházaro and E. Vallejo, “Sets of uniqueness and minimal matrices,” J. Algebra 208 (1998), 444-451.
26. E. Vallejo, “Reductions of additive sets, sets of uniqueness and pyramids,” Discrete Math. 173 (1997), 257- 267.
27. E. Vallejo, “On the Kronecker product of the irreducible characters of the symmetric group,” Pub. Prel. Inst. Mat. UNAM 526, (1997).
28. I. Zisser, “The character covering numbers of the alternating groups,” J. Algebra 153 (1992), 357-372.
2. M. Clausen and H. Meier, “Extreme irreduzible Konstituenten in Tensordarstellungen symmetrischer Gruppen,” Bayreuther Math. Schriften 45 (1993), 1-17.
3. A.J. Coleman, “Induced Representations with applications to Sn and GL(n),” Queen's papers in pure and applied mathematics, No. 4, Queens University, Kingston, Ontario, 1966.
4. P. Doubilet, J. Fox, and G.R. Rota, “The elementary theory of the symmetric group,” in Combinatorics, representation theory and statistical methods in groups, T.V. Narayama, R.M. Mathsen and J.G. Williams (eds.), Lecture Notes in Pure and Applied Mathematics, Vol. 57, Marcel Decker, New York, 1980.
5. Y. Dvir, “On the Kronecker product of Sn characters,” J. Algebra 154 (1993), 125-140.
6. Y. Dvir, “A family of Z bases for the ring of S(n) characters and applications to the decomposition of Kronecker products,” Europ. J. Combinatorics 15 (1994), 449-457.
7. \ddot O. E\~gecio\~glu and J.B. Remmel, “A combinatorial interpretation of the inverse Kostka matrix,” Linear and Multilinear Algebra 26 (1990), 59-84.
8. P.C. Fishburn, J.C. Lagarias, J.A. Reeds, and L.A. Shepp, “Sets uniquely determined by projections on axes II. Discrete case,” Discrete Math. 91 (1991), 149-159.
9. D. Gale, “A theorem of flows in networks,” Pacif. J. Math. 7 (1957), 1073-1082. VALLEJO
10. A. Garsia and J. B. Remmel, “Shuffles of permutations and the Kronecker product,” Graphs Combin. 1 (1985), 217-263.
11. G.D. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of mathematics and its applications, Vol. 16, Addison-Wesley, Reading, Massachusetts, 1981.
12. T.Y. Lam, Young diagrams, “Schur functions, the Gale-Ryser theorem and a conjecture of Snapper,” J. Pure Appl. Algebra 10 (1977), 81-94.
13. D.E. Littlewood, “The Kronecker product of symmetric group representations,” J. London Math. Soc. 31 (1956), 89-93.
14. I.G. Macdonald, Symmetric functions and Hall polynomials, 2nd. edition, Oxford Univ. Press, Oxford, 1995.
15. P.A. MacMahon, Combinatorial Analysis, Vol. II. Cambridge Univ. Press, London-New York 1916; reprinted by Chelsea, New York, 1960.
16. F.D. Murnaghan, “The analysis of the Kronecker product of irreducible representations of the symmetric group,” Amer. J. Math. 60 (1938), 761-784.
17. J.B. Remmel, “A formula for the Kronecker products of Schur functions of hook shapes,” J. Algebra 120 (1989), 100-118.
18. J.B. Remmel, “Formulas for the expansion of Kronecker products S(m,n) \otimes S(1p - r ,r) and S(1k,2l) \otimes S(1p - r ,r),” Discrete Math. 99 (1992), 265-287.
19. J.B. Remmel and T. Whitehead, “On the Kronecker product of Schur functions of two row shapes,” Bull. Belg. Math. Soc. 1 (1994), 649-683.
20. H.J. Ryser, “Combinatorial properties of matrices of zeros and ones,” Canad. J. Math. 9 (1957), 371-377.
21. H.J. Ryser, Combinatorial mathematics, Carus monograph No. 14, Math. Association of America, John Wiley and Sons, 1963.
22. B.E. Sagan, The symmetric group, Wadsworth & Brooks/Cole, Pacific Grove, California, 1991.
23. J. Saxl, “The complex characters of the symmetric groups that remain irreducible on subgroups, ” J. Algebra 111 (1987), 210-219.
24. E. Snapper, “Group characters and nonnegative integral matrices,” J. Algebra 19 (1971), 520-535.
25. A. Torres-Cházaro and E. Vallejo, “Sets of uniqueness and minimal matrices,” J. Algebra 208 (1998), 444-451.
26. E. Vallejo, “Reductions of additive sets, sets of uniqueness and pyramids,” Discrete Math. 173 (1997), 257- 267.
27. E. Vallejo, “On the Kronecker product of the irreducible characters of the symmetric group,” Pub. Prel. Inst. Mat. UNAM 526, (1997).
28. I. Zisser, “The character covering numbers of the alternating groups,” J. Algebra 153 (1992), 357-372.