Plactification
Victor Reiner
and Mark Shimozono
DOI: 10.1023/A:1022434000967
Abstract
We study a map called plactification from reduced words to words. This map takes Coxeter-Knuth equivalence to Knuth equivalence, and has applications to the enumeration of reduced words, Schubert polynomials and certain Specht modules.
Pages: 331–351
Keywords: reduced words; knuth equivalence; Specht module; Schubert polynomial
Full Text: PDF
References
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2. P.H. Edelman and C. Greene, "Balanced tableaux," Adv. Math. 63 (1987), 42-99.
3. P. Hanlon and S. Sundaram, "On a Bijection between Littlewood-Richardson Fillings of Conjugate Shape," J. Comb. Th. Ser. A 60 (1992), 1-18.
4. G.D. James, The Representation Theory of Symmetric Groups, Springer Lecture Notes in Math, Vol. 682, Springer- Verlag 1978.
5. G.D. James and M.H. Peel, "Specht series for skew representations of symmetric groups," /. Algebra 56 (1979), 343-364.
6. D.E. Knuth, "Permutations, matrices, and generalized Young tableaux," Pac. J. Math 34 (1970), 709-727.
7. W. Kraslciewicz and P. Pragacz, "Schubert functors and Schubert polynomials," unpublished preprint.
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9. A. Lascoux and M.-P. Schiitzenberger, "Structure de Hopf de 1'anneau de cohomologie et de I'anneau de Grothendieck d'une variety drapeaux," C. R. Acad. Sci. Paris Ser. I Math. 295 (1982), 629-633.
10. A. Lascoux and M.-P. Schutzenberger, "Le monoide plaxique, (Proc. Colloqu. Naples 1978)," Quad. Ricerca Sci. 109(1981).
11. A. Lascoux and M.-P. Schutzenberger, "Polynomes de Schubert," C. R. Acad. Sci. Paris 294 (1982), 447-50.
12. A. Lascoux and M.-P. Schutzenberger, "Keys and standard bases," Tableaux and Invariant Theory, IMA Volumes in Math and its Applications (D. Stanton, Ed.), Vol. 19, pp. 125-144,1990.
13. A. Lascoux and M.-P. Schutzenberger, "Tableaux and non-commutative Schubert polynomials," Func. Anal. Appl. 23 (1989), 63-64. 14. l.C. Macdonald, Nates an Schubert polynomials, LACIM, Montreal, 1991.
15. P. Magyar, "A Borel-Weil construction for Schur modules," preprint, 1994.
16. V. Reiner and M. Shimozono, "Key polynomials and a flagged Littlewood-Richardson rule," to appear in J. Comb. Th. Ser. A. 70 (1995), 107-143.
17. V. Reiner and M. Shimozono, "Specht series for column convex diagrams," to appear in J. Alg.
18. B. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Wadsworth & Brooks/Cole, Pacific Grove, 1991.
19. R.P. Stanley, "On the number of reduced decompositions of elements of Coxeter groups," Ear. J. Comb. 5 (1984), 359-372.
20. D. White, personal communication.
2. P.H. Edelman and C. Greene, "Balanced tableaux," Adv. Math. 63 (1987), 42-99.
3. P. Hanlon and S. Sundaram, "On a Bijection between Littlewood-Richardson Fillings of Conjugate Shape," J. Comb. Th. Ser. A 60 (1992), 1-18.
4. G.D. James, The Representation Theory of Symmetric Groups, Springer Lecture Notes in Math, Vol. 682, Springer- Verlag 1978.
5. G.D. James and M.H. Peel, "Specht series for skew representations of symmetric groups," /. Algebra 56 (1979), 343-364.
6. D.E. Knuth, "Permutations, matrices, and generalized Young tableaux," Pac. J. Math 34 (1970), 709-727.
7. W. Kraslciewicz and P. Pragacz, "Schubert functors and Schubert polynomials," unpublished preprint.
8. A. Lascoux, Personal communication.
9. A. Lascoux and M.-P. Schiitzenberger, "Structure de Hopf de 1'anneau de cohomologie et de I'anneau de Grothendieck d'une variety drapeaux," C. R. Acad. Sci. Paris Ser. I Math. 295 (1982), 629-633.
10. A. Lascoux and M.-P. Schutzenberger, "Le monoide plaxique, (Proc. Colloqu. Naples 1978)," Quad. Ricerca Sci. 109(1981).
11. A. Lascoux and M.-P. Schutzenberger, "Polynomes de Schubert," C. R. Acad. Sci. Paris 294 (1982), 447-50.
12. A. Lascoux and M.-P. Schutzenberger, "Keys and standard bases," Tableaux and Invariant Theory, IMA Volumes in Math and its Applications (D. Stanton, Ed.), Vol. 19, pp. 125-144,1990.
13. A. Lascoux and M.-P. Schutzenberger, "Tableaux and non-commutative Schubert polynomials," Func. Anal. Appl. 23 (1989), 63-64. 14. l.C. Macdonald, Nates an Schubert polynomials, LACIM, Montreal, 1991.
15. P. Magyar, "A Borel-Weil construction for Schur modules," preprint, 1994.
16. V. Reiner and M. Shimozono, "Key polynomials and a flagged Littlewood-Richardson rule," to appear in J. Comb. Th. Ser. A. 70 (1995), 107-143.
17. V. Reiner and M. Shimozono, "Specht series for column convex diagrams," to appear in J. Alg.
18. B. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Wadsworth & Brooks/Cole, Pacific Grove, 1991.
19. R.P. Stanley, "On the number of reduced decompositions of elements of Coxeter groups," Ear. J. Comb. 5 (1984), 359-372.
20. D. White, personal communication.