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Exploiting Symmetries Alternating Sign Matrices and the Weyl Character Formulas

  • David M. BressoudEmail author
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 17)

Summary

This paper illustrates some of the power and beauty of determinantevaluations, beginning with Cauchy’s proof of the Vandermondedeterminant evaluation, continuing through the Weyldenominator formulas and some open conjectures on alternating-sign{matrices}, and ending with the Izergin–Korepin determinantexpansion for the six-vertex model with domain wall boundaryconditions.

Vandermonde product Weyl denominator formula alternating-sign matrix six-vertex model. 

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References

  1. 1.
    Rodney J. Baxter. Eight-vertex model in lattice statistics. Physical Review Letters, 26:832–833, 1971.CrossRefGoogle Scholar
  2. 2.
    Rodney J. Baxter. Exactly Solved Models in StatisticalMechanics. Academic Press, London, 1982.Google Scholar
  3. 3.
    David M. Bressoud. Proofs and Confirmations: The Story of theAlternating-Sign Matrix Conjecture. Cambridge University Press,Cambridge, UK, 1999.Google Scholar
  4. 4.
    David M. Bressoud and James Propp. How the alternating-sign matrixconjecture was solved. Notices Amer. Math. Soc.,46:637–646, 1999.MathSciNetGoogle Scholar
  5. 5.
    5. Augustin-Louis Cauchy. Mémoire sur les fonctions qui nepeuvent obtenir que deux valeurs égales et de signescontraires par suite des transpositions opérées entre lesvariables qu’elles renferment. Journal de l’ÉcolePolytechnique}, 10(17):29–112, 1815. Reprinted in Œ uvrescomplètes d’Augustin Cauchy series 2, Vol. 1, 91–161.Paris: Gauthier-Villars, 1899.Google Scholar
  6. 6.
    P. Desnanot. Complément de la théorie des équations du premier degr’. Paris, 1819. Quoted in ThomasMuir, The Theory of Determinants in the Historical Order of Development, vol. 1, pp. 136–148. London: MacMillan andCo. 1906.Google Scholar
  7. 7.
    Anatoli G. Izergin. Partition function of a six-vertex model in afinite volume. Dokl. Akad. Nauk SSSR, 297:331–333, 1987.MathSciNetGoogle Scholar
  8. 8.
    8. C. G. J. Jacobi. De binis quibuslibet functionibus homogeneissecundi ordinis per {substitutiones} lineares in alias binastransformandis. Journal für die Reine und AngewandteMathematik, 2:247–257, 1833. Reprinted in C. G. J. Jacobi:Gesammelte Werke. Vol. 3, pp. 191–268. Berlin: Georg Reimer,1884.Google Scholar
  9. 9.
    Vladimir E. Korepin, Nikolai M. Bogoliubov, and Anatoli G.Izergin. Quantum inverse scattering method and correlationfunctions. Cambridge University Press, Cambridge, UK, 1993.Google Scholar
  10. 10.
    Greg Kuperberg. Another proof of the alternating-sign matrixconjecture. International Mathematics Research Notes,1996:139–150, 1996.Google Scholar
  11. 11.
    Greg Kuperberg. Symmetry classes of alternating-sign matricesunder one roof. Ann. of Math., 156:835–866, 2002.Google Scholar
  12. 12.
    W. H. Mills, D. P. Robbins, and H. Rumsey. Alternating-signmatrices and descending plane partitions. Journal ofCombinatorial Theory, 34:340–359, 1983.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Soichi Okada. Alternating-sign matrices and some deformations of W eyl’s denominator formulas. Journal of AlgebraicCombinatorics, 2:155–176, 1993.zbMATHGoogle Scholar
  14. 14.
    Soichi Okada. Enumeration of symmetry classes of alternating-signmatrices and characters of classical groups. Journal ofAlgebraic Combinatorics, 23:43–69, 2006.zbMATHCrossRefGoogle Scholar
  15. 15.
    David P. Robbins. The story of 1, 2, 7, 42, 429, 7436, The Mathematical Intelligencer, 13:12–19, 1991.zbMATHCrossRefGoogle Scholar
  16. 16.
    David P. Robbins and Howard Rumsey. Determinants andalternating-sign matrices. Advances in Mathematics,62:169–184, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    17. I. J. Schur. Ein beitrag zur additiven Zahlentheorie und zurTheorie der Kettenbrüche. S.-B. Preuss. Akad.Wiss. Phys.-Math. Kl., pages 302–321, 1917. Reprinted in Gesammelte Abhandlungen\/. Vol. 2, pp. 117–136. Berlin:Springer-Verlag, 1973.Google Scholar
  18. 18.
    Hermann Weyl. The Classical Groups: Their Invariants andRepresentations. Princeton University Press, Princeton, NewJersey, 1939.Google Scholar
  19. 19.
    Doron Zeilberger. Proof of the alternating-sign matrix conjecture.Electronic Journal of Combinatorics, 3, 1996. R13.Google Scholar
  20. 20.
    Doron Zeilberger. Proof of the refined alternating-sign matrixconjecture. New York Journal of Mathematics,2:59–68, 1996.Google Scholar

Copyright information

© Springer-Verlag New York 2008

Authors and Affiliations

  1. 1.Macalester CollegeSt. Paul

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