The Littlewood-Richardson rule, and related combinatorics

An introduction is given to the Littlewood-Richardson rule, and various combinatorial constructions related to it. We present a proof based on tableau switching, dual equivalence, and coplactic operations. We conclude with a section relating these fairly modern techniques to earlier work on the Littlewood-Richardson rule.

[1]  B. G. Wybourne,et al.  Symmetry principles and atomic spectroscopy , 1970 .

[2]  Jeffrey B. Remmel,et al.  A simple proof of the Littlewood-Richardson rule and applications , 1998, Discret. Math..

[3]  Donald E. Knuth,et al.  PERMUTATIONS, MATRICES, AND GENERALIZED YOUNG TABLEAUX , 1970 .

[4]  Jeffrey B. Remmel,et al.  Multiplying Schur functions , 1984, J. Algorithms.

[5]  de Ng Dick Bruijn,et al.  On the set of divisors of a number , 1951 .

[6]  Mark D. Haiman.  On mixed insertion, symmetry, and shifted young tableaux , 1989, J. Comb. Theory, Ser. A.

[7]  G. de B. Robinson,et al.  On the Representations of the Symmetric Group , 1938 .

[8]  Marcel P. Schützenberger.  Quelques remarques sur une Construction de Schensted. , 1963 .

[9]  Mark D. Haiman,et al.  Dual equivalence with applications, including a conjecture of Proctor , 1992, Discret. Math..

[10]  Glânffrwd P Thomas.  On Schensted's construction and the multiplication of schur functions , 1978 .

[11]  Curtis Greene,et al.  An Extension of Schensted's Theorem , 1974 .

[12]  Peter Littelmann,et al.  Paths and root operators in representation theory , 1995 .

[13]  Marc A. A. van Leeuwen.  Tableau algorithms defined naturally for pictures , 1996, Discret. Math..

[14]  C. Schensted.  Longest Increasing and Decreasing Subsequences , 1961, Canadian Journal of Mathematics.

[15]  D. E. Littlewood,et al.  Group Characters and Algebra , 1934 .

[16]  Marcel Paul Schützenberger,et al.  Promotion des morphismes d'ensembles ordonnes , 1972, Discret. Math..

[17]  Sergey Fomin,et al.  A Littlewood-Richardson Miscellany , 1993, Eur. J. Comb..

[18]  Edward A. Bender,et al.  Enumeration of Plane Partitions , 1972, J. Comb. Theory A.

[19]  A. Zelevinsky,et al.  A generalization of the Littlewood-Richardson rule and the Robinson-Schensted-Knuth correspondence , 1981 .

[20]  Frank Sottile,et al.  Tableau Switching: Algorithms and Applications , 1996, J. Comb. Theory, Ser. A.

[21]  An analogue of Jeu de taquin for Littelmann's crystal paths. , 1998 .

[22]  Alfred Young.  On Quantitative Substitutional Analysis , 1930 .

[23]  Daniel J. Kleitman,et al.  Strong Versions of Sperner's Theorem , 1976, J. Comb. Theory, Ser. A.

[24]  Masaki Kashiwara,et al.  Crystal Graphs for Representations of the q-Analogue of Classical Lie Algebras , 1994 .

[25]  Andrei Zelevinsky,et al.  Triple Multiplicities for sl(r + 1) and the Spectrum of the Exterior Algebra of the Adjoint Representation , 1992 .

[26]  A. Morris.  REPRESENTATIONS OF FINITE CLASSICAL GROUPS: A Hopf Algebra Approach (Lecture Notes in Mathematics, 869) , 1982 .

[27]  Vesselin Gasharov.  A Short Proof of the Littlewood-Richardson Rule , 1998, Eur. J. Comb..

[28]  Glânffrwd P. Thomas,et al.  On a construction of schützenberger , 1977, Discret. Math..

[29]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[30]  A. V. Zelevinsky,et al.  Representations of Finite Classical Groups: A Hopf Algebra Approach , 1981 .

[31]  Marc van Leeuwen,et al.  The Robinson-Schensted and Schützenberger algorithms, an elementary approach , 1995, Electron. J. Comb..

[32]  Marcel Paul Schützenberger,et al.  La correspondance de Robinson , 1977 .

[33]  J. Thibon,et al.  The Robinson-Schensted correspondence, crystal bases, and the quantum straightening at q = 0 , 1995 .

[34]  Peter Littelmann,et al.  A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras , 1994 .