zbMATH — the first resource for mathematics

A combinatorial interpretation of the inverse \(t\)-Kostka matrix. (English) Zbl 1061.05504
Summary: In this paper we use tournament matrices to give a combinatorial interpretation for the entries of the inverse t-Kostka matrix, which is the transition matrix between the Hall-Littlewood polynomials and the Schur functions.

05E05 Symmetric functions and generalizations
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05E10 Combinatorial aspects of representation theory
Full Text: DOI
[1] Carbonara, J.O.; Remmel, J.B.; Yang, M., S-series and plethysm of hook-shaped Schur functions with power sum symmetric functions, () · Zbl 0844.05097
[2] Carlitz, L., Sequences and inversions, Duke math. J., 37, 193-198, (1970) · Zbl 0206.02302
[3] Chen, Y.M., Combinatorial algorithms for plethysm, () · Zbl 0556.20013
[4] Chen, Y.M.; Garsia, A.M.; Remmel, J.B., Algorithms for plethysm, Contemp. math., 34, (1984) · Zbl 0556.20013
[5] Duncan, D.G., D.E. Littlewood’s algebra of S-functions, Can. J. math., 4, 504-512, (1952) · Zbl 0048.01103
[6] Egecioglu, O.; Remmel, J.B., A combinatorial interpretation of the inverse kostka matrix, Linear and multilinear algebra, 26, 59-84, (1990) · Zbl 0735.05013
[7] Garsia, A.M.; Procesi, C., On certain graded Sn-modules and the q-kostka polynomials, Adv. math., 94, 1, (1992) · Zbl 0797.20012
[8] Gessel, I., Tournaments and Vandermonde’s determinant, J. graph theory, 3, 305-307, (1979) · Zbl 0433.05008
[9] James, G.D.; Kerber, A., The representation theory of the symmetric group, ()
[10] A. Lascoux, M.P. Sch├╝tzenberger, Sur une conjecture de H.O. Foulkes, C.R. Acad. Sci. Paris 286A, 323-324.
[11] Littlewood, D.E., Invariant theory, tensors, and group characters, Philos. trans. roy. soc. London, ser., A239, 305-355, (1944) · Zbl 0060.04402
[12] Littlewood, D.E., The theory of group characters, (1950), Oxford University Press Oxford · Zbl 0011.25001
[13] Macdonald, I.G., ()
[14] Moon, J.W., Topics on tournaments, (1968), Holt New York · Zbl 0191.22701
[15] Remmel, J.B.; Whitney, R., Multiplying Schur functions, J. algorithms, 5, 471-487, (1984) · Zbl 0557.20008
[16] Stanton, D.; White, D., Constructive combinatorics, (1986), UTM Springer Berlin · Zbl 0595.05002
[17] Yaglom, A.M.; Yaglom, I.M., Challenging mathematical problems with elementary solutions, (1964), Holden-Day San Francisco, CA · Zbl 0123.24201
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.