Donin, I. F. Multiplicities of \(S_ n\)-modules and the index and the charge of tableaux. (English. Russian original) Zbl 0827.20015 Funct. Anal. Appl. 27, No. 4, 280-282 (1993); translation from Funkts. Anal. Prilozh. 27, No. 4, 71-74 (1993). Let \(A\) and \(B\) be skew diagrams of the same order \(n\), and let \(T_A\) and \(T_B\) be the corresponding representations of the symmetric group \(S_n\) (over a fixed field of characteristic zero). There are several combinatorial interpretations of the intertwining number \(\nu(A,B)=\dim\text{Hom}_{S_n}(T_A,T_B)\). We give a new combinatorial interpretation of the number \(\nu(A,B)\) as the set of integral solutions of a system of linear inequalities, the inequalities being symmetric with respect to \(A\) and \(B\) and the relation \(\nu(A,B)=\nu(B,A)\) thus being self-evident. Furthermore, it turns out that the deviations from these inequalities can be characterized by numbers which are generalizations of the index and the charge of a tableau. MSC: 20C30 Representations of finite symmetric groups 05E10 Combinatorial aspects of representation theory Keywords:skew diagrams; representations of symmetric groups; intertwining numbers; index; charge; tableaux × Cite Format Result Cite Review PDF References: [1] V. I. Arnold, ”Remarks on perturbation theory for problems of Mathieu type,” Usp. Mat. Nauk,38, No. 4, 189–203 (1983). [2] V. I. Arnold, ”Small denominators I. Mappings of the circumference onto itself,” Trans. Amer. Math. Soc.,46, 213–284 (1965). · Zbl 0152.41905 [3] O. G. Galkin, ”Phase-locking for Mathieu-type vector fields on the torus,” Funkts. Anal. Prilozhen.,26, No. 1, 1–8 (1992). · Zbl 0828.20025 · doi:10.1007/BF01077066 [4] A. Khinchin, Continued fractions, Groningen, Nordhorff (1963). · Zbl 0117.28601 [5] C. Baesens, J. Guckenheimer, S. Kim, and R. S. MacKay, ”Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos,” Phys. D,49, 387–475 (1991). · Zbl 0734.58036 · doi:10.1016/0167-2789(91)90155-3 [6] R. E. Ecke, J. D. Farmer, and D. K. Umberger, ”Scaling of the Arnold tongues,” Nonlinearity,2, 175–196 (1989). · Zbl 0689.58017 · doi:10.1088/0951-7715/2/2/001 [7] J. Franks and M. Misiurewicz, ”Rotation sets of toral flows,” Proc. Amer. Math. Soc.,109, 243–249 (1990). · Zbl 0701.57016 · doi:10.1090/S0002-9939-1990-1021217-5 [8] O. G. Galkin, ”Resonance regions for Mathieu type dynamical systems on a torus,” Phys. D,39, 287–298 (1989). · Zbl 0695.58025 · doi:10.1016/0167-2789(89)90011-0 [9] C. Grebogi, E. Ott, and J. A. Yorke, ”Attractors on ann-torus: quasiperiodicity versus chaos,” Phys. D,15, 354–373 (1985). · Zbl 0577.58023 · doi:10.1016/S0167-2789(85)80004-X [10] G. R. Hall, ”Resonance zones in two-parameter families of circle homeomorphisms,” SIAM J. Math. Anal.,15, 1075–1081 (1984). · Zbl 0554.58040 · doi:10.1137/0515083 [11] S. Kim, R. S. MacKay, and J. Guckenheimer, ”Resonance regions for families of torus maps,” Nonlinearity,2, 391–404 (1989). · Zbl 0678.58034 · doi:10.1088/0951-7715/2/3/001 [12] M. Misiurewicz and K. Ziemian, ”Rotation sets for maps of tori,” J. London Math. Soc.,40, 490–506 (1989). · Zbl 0663.58022 · doi:10.1112/jlms/s2-40.3.490 [13] S. Newhouse, J. Palis, and F. Takens, ”Bifurcations and stability of families of diffeomorphisms,” Publ. Math. IHES,57, 5–72 (1983). · Zbl 0518.58031 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.