Splitting the Square of a Schur Function into its Symmetric and Antisymmetric Parts Article DOI:
Cite this article as: Carré, C. & Leclerc, B. Journal of Algebraic Combinatorics (1995) 4: 201. doi:10.1023/A:1022475927626 Abstract
We propose a new combinatorial description of the product of two Schur functions. In the particular case of the square of a Schur function
S I, it allows to discriminate in a very natural way between the symmetric and antisymmetric parts of the square. In other words, it describes at the same time the expansion on the basis of Schur functions of the plethysms S 2( S I) and Λ 2( S I). More generally our combinatorial interpretation of the multiplicities c = S IJ K IS J, S Kleads to interesting q-analogues c (q) of these multiplicities. The combinatorial objects that we use are domino tableaux, namely tableaux made up of 1 × 2 rectangular boxes filled with integers weakly increasing along the rows and strictly increasing along the columns. Standard domino tableaux have already been considered by many authors , , , , , but, to the best of our knowledge, the expression of the Littlewood-Richardson coefficients in terms of Yamanouchi domino tableaux is new, as well as the bijection described in Section 7, and the notion of the diagonal class of a domino tableau, defined in Section 8. This construction leads to the definition of a new family of symmetric functions ( IJ K H-functions), whose relevant properties are summarized in Section 9. symmetric function domino tableaux plethysm Littlewood-Richardson rule References
D. Barbasch and D. Vogan, “Primitive ideals and orbital integrals in complex classical groups,”
A.D. Berenstein and A.V. Zelevinsky, “Triple multiplicities of
+1) and the spectrum of the exterior algebra of the adjoint representation,”
J. Alg. Combin.
C. Carré, “The rule of Littlewood-Richardson in a construction of Berenstein-Zelevinsky,”
Int. J. of Algebra and Computation
(4) (1991), 473–491.
C. Carré, “Plethysm of elementary functions,”
Bayreuther Mathematische Schriften
C. Carré, “Le pléthysme,”
Thèse, Université Paris 7 (1991).
S. Fomin and D. Stanton, “Rim hook lattices,” Mittag-Leffler institute, preprint No.
F. Grosshans, “The symbolic method and representation theory,”
Advances in Math.
D. Garfinkle, “On the classification of primitive ideals for complex classical Lie algebras, I,”
(1990) 2, 135–169.
F. Grosshans, G.C. Rota, and J. Stein, “Invariant theory and superalgebras,”
Amer. Math. Soc. Providence, RI, 1987.
I.M. Gel'fand and A.V. Zelevinsky, “Polyhedra in the scheme space and the canonical basis for irreducible representations of
Funct. Anal. Appl.
(1985), No. 2, 72–75.
I.M. Gel'fand and A.V. Zelevinsky, “Multiplicities and good bases for
,” n Proc. III. Int. Semin. on Group-Theoretical Methods in Physics, Yurmala, 1985.
G.D. James and A. Kerber,
The representation theory of the symmetric group, Addison-Wesley, 1981.
G.D. James and M.H. Peel, “Specht series for skew representations of symmetric groups,”
A.N. Kirillov, “On the Kostka-Green-Foulkes polynomials and Clebsch-Gordan numbers,”
J. of Geometry and Physics
(3), (1988), 365–389.
A. Kerber, A. Kohnert, and A. Lascoux, “SYMMETRICA, an object oriented computer-algebra system for the symmetric group,”
J. Symbolic Computation
A.N. Kirillov, A. Lascoux, B. Leclerc, and J.Y. Thibon, “Séries génératrices pour les tableaux de dominos,”
C. R. Acad. Sci. Paris, t.
, (1994), 395–400.
D. Knuth, “Permutations, matrices and generalized Young tableaux,”
Pacific J. Math.
A.N. Kirillov and N.Yu. Reshetikhin, “Bethe ansatz and the combinatorics of Young tableaux,”
Zap. Nauch. Semin. LOMI
A. Lascoux, “Cyclic permutations on words, tableaux and harmonic polynomials,”
Proc. of the Hyderabad conference on algebraic groups,
1989, Manoj Prakashan, Madras (1991), 323–347.
D.E. Littlewood, “Polynomial concomitants and invariant matrices,”
Proc. London Math. Soc. (2)
D.E. Littlewood, “Invariant theory, tensors and group characters,”
Phil. Trans. Roy. Soc. A
The theory of group characters and matrix representations of groups, Oxford, 1950 (second edition).
D.E. Littlewood, “Modular representations of symmetric groups,”
Proc. Roy. Soc. A.
A. Lascoux, B. Leclerc, and J.Y. Thibon, “Une nouvelle expression des fonctions P de Schur,”
C.R. Acad. Sci. Paris, t.
316, Série I, (1993), 221–224.
A. Lascoux, B. Leclerc, and J.Y. Thibon, “Fonctions de Hall-Littlewood et polynômes de Kostka-Foulkes aux racines de 1'unité,”
C.R. Acad. Sci. Paris, t.
316, Série I (1993), 1–6.
A. Lascoux and M.P. Schützenberger,
Formulaire raisonné de fonctions symétriques, Publ. Math. Univ. Paris 7, 1985.
A. Lascoux and M.P. Schützenberger, “Le monoïde plaxique,” in
Noncommutative structures in algebra and geometric combinatorics
(A. de Luca Ed.), Quaderni della Ricerca Scientifica del C.N.R., Roma, 1981.
A. Lascoux and M.P. Schützenberger, “Schubert polynomials and the Littlewood-Richardson rule,”
Letters in Math. Physics
Symmetric functions and Hall polynomials, Oxford, 1979.
A.O. Morris and N. Sultana, “Hall-Littlewood polynomials at roots of 1 and modular representations of the symmetric group,”
Math. Proc. Cambridge Phil. Soc.
G. de B. Robinson, “On the representation theory of the symmetric group,”
Amer. J. Math.
G. de B. Robinson,
Representation theory of the symmetric group, Edinburgh, 1961.
D. Stanton and D. White, “A Schensted algorithm for rim-hook tableaux,”
J. Comb. Theory A
M. van Leeuwen, “A Robinson-Schensted algorithm in the geometry of flags for Classical Groups”, Thesis, 1989.
A.V. Zelevinsky, “A generalization of the Littlewood-Richardson rule and the Robinson-Schensted-Knuth correspondence,”
Google Scholar Copyright information
© Kluwer Academic Publishers 1995