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Symmetric Functions

  • Anthony Mendes
  • Jeffrey Remmel
Chapter
  • 1.1k Downloads
Part of the Developments in Mathematics book series (DEVM, volume 43)

Abstract

The ring of symmetric functions is introduced. The six standard bases for symmetric functions; namely, the monomial, elementary, homogeneous, power, forgotten, and Schur symmetric functions, are defined. Numerous relationships between these functions are proved.

Keywords

Schur Symmetric Functions Macdonald Polynomials Combinatorial Interpretation Jacobi-Trudi Identity Solving Counting Problems 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 9.
    Beck, D.A., Remmel, J.B., Whitehead, T.: The combinatorics of transition matrices between the bases of the symmetric functions and the B n analogues. In: Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics, Florence, 1993, vol. 153, pp. 3–27 (1996)Google Scholar
  2. 33.
    Eğecioğlu, Ö., Remmel, J.B.: Brick tabloids and the connection matrices between bases of symmetric functions. Discret. Appl. Math. 34(1–3), 107–120 (1991) [Combinatorics and Theoretical Computer Science (Washington, DC, 1989)]Google Scholar
  3. 55.
    Haglund, J.: The q,t-Catalan Numbers and the Space of Diagonal Harmonics. University Lecture Series, vol. 41. American Mathematical Society, Providence (2008) [With an appendix on the combinatorics of Macdonald polynomials]Google Scholar
  4. 73.
    Knuth, D.E.: Permutations, matrices, and generalized Young tableaux. Pac. J. Math. 34, 709–727 (1970)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 80.
    LoBue Tiefenbruck, J., Remmel, J.B.: A Murnaghan-Nakayama rule for generalized Demazure atoms. In: 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013). Discrete Mathematics and Theoretical Computer Science Proceedings, AS, pp. 969–980. Association of Discrete Mathematics and Theoretical Computer Science, Nancy (2013)Google Scholar
  6. 81.
    Loehr, N.A.: Abacus proofs of Schur function identities. SIAM J. Discret. Math. 24(4), 1356–1370 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 82.
    Loehr, N.A.: Bijective Combinatorics. Discrete Mathematics and Its Applications (Boca Raton). CRC Press, Boca Raton (2011)Google Scholar
  8. 104.
    Robinson, G.d.B.: On the representations of the symmetric group. Am. J. Math. 60(3), 745–760 (1938). doi:10.2307/2371609. http://www.dx.doi.org/10.2307/2371609
  9. 108.
    Stanley, R.P.: Binomial posets, Möbius inversion, and permutation enumeration. J. Comb. Theory Ser. A 20(3), 336–356 (1976)CrossRefzbMATHGoogle Scholar
  10. 115.
    Wagner, J.: The permutation enumeration of wreath products C k§S n of cyclic and symmetric groups. Adv. Appl. Math. 30(1–2), 343–368 (2003) [Formal power series and algebraic combinatorics (Scottsdale, AZ, 2001)]Google Scholar
  11. 116.
    White, D.: Orthogonality of the characters of S n. J. Comb. Theory Ser. A 40(2), 265–275 (1985)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Anthony Mendes
    • 1
  • Jeffrey Remmel
    • 2
  1. 1.Mathematics DepartmentCalifornia Polytechnic State UniversitySan Luis ObispoUSA
  2. 2.Department of MathematicsUniversity of California at San DiegoLa JollaUSA

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