Journal of Algebraic Combinatorics

, Volume 27, Issue 3, pp 263–273 | Cite as

Degrees of stretched Kostka coefficients

Open Access


Given a partition λ and a composition β, the stretched Kostka coefficient \(\mathcal {K}_{\lambda \beta}(n)\) is the map n K n λ,n β sending each positive integer n to the Kostka coefficient indexed by n λ and n β. Kirillov and Reshetikhin (J. Soviet Math. 41(2), 925–955, 1988) have shown that stretched Kostka coefficients are polynomial functions of n. King, Tollu, and Toumazet have conjectured that these polynomials always have nonnegative coefficients (CRM Proc. Lecture Notes 34, 99–112, 2004), and they have given a conjectural expression for their degrees (Séminaire Lotharingien de Combinatoire 54A, 2006).

We prove the values conjectured by King, Tollu, and Toumazet for the degrees of stretched Kostka coefficients. Our proof depends upon the polyhedral geometry of Gelfand–Tsetlin polytopes and uses tilings of GT-patterns, a combinatorial structure introduced in De Loera and McAllister, (Discret. Comput. Geom. 32(4), 459–470, 2004).


Kostka coefficient Representation theory Gelfand–Tsetlin polytope 

Mathematics Subject Classification (2000)

17B10 52B12 


  1. 1.
    Berenstein, A. D., & Zelevinsky, A. V. (1990). When is the multiplicity of a weight equal to 1?. Funktsional’nyi Analiz i ego Prilozheniya, 24(4), 1–13, 96. MathSciNetGoogle Scholar
  2. 2.
    Billey, S., Guillemin, V., & Rassart, E. (2004). A vector partition function for the multiplicities of  \(\mathfrak{sl}\sb k\mathbb{C}\) . Journal of Algebra, 278(1), 251–293. MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    De Loera, J. A., & McAllister, T. B. (2004). Vertices of Gelfand-Tsetlin polytopes. Discrete and Computational Geometry, 32(4), 459–470. arXiv:math.CO/0309329. MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Fulton, W. (1997). Young tableaux. London mathematical society student texts (Vol. 35). Cambridge: Cambridge University Press. With applications to representation theory and geometry. MATHGoogle Scholar
  5. 5.
    Fulton, W., & Harris, J. (1991). Representation theory. Graduate texts in mathematics (Vol. 129). New York: Springer. A first course, Readings in Mathematics. MATHGoogle Scholar
  6. 6.
    Gelfand, I. M., & Tsetlin, M. L. (1950). Finite-dimensional representations of the group of unimodular matrices. Doklady Akademii Nauk SSSR (N.S.), 71, 825–828. English translation in Collected papers (Vol. II, pp. 653–656). Springer, Berlin, 1988. Google Scholar
  7. 7.
    King, R. C., Tollu, C., & Toumazet, F. (2004). Stretched Littlewood-Richardson and Kostka coefficients. In Symmetry in physics. CRM proc. lecture notes (Vol. 34, pp. 99–112). Providence: Am. Math. Soc. Google Scholar
  8. 8.
    King, R. C., Tollu, C., & Toumazet, F. (2006). The hive model and the factorisation of Kostka coefficients. Séminaire Lotharingien de Combinatoire, 54A, Article B54Ah. Google Scholar
  9. 9.
    Kirillov, A. N. (2001). Ubiquity of Kostka polynomials. In Physics and combinatorics, Nagoya, 1999 (pp. 85–200). River Edge: World Sci. Publishing. arXiv:math.QA/9912094. Google Scholar
  10. 10.
    Kirillov, A. N., & Berenstein, A. D. (1995). Groups generated by involutions, Gel’fand-Tsetlin patterns, and combinatorics of Young tableaux. Algebra i Analiz, 7(1), 92–152. Translation in St. Petersburg Mathematical Journal, 7(1) (1996), 77–127. MATHMathSciNetGoogle Scholar
  11. 11.
    Kirillov, A. N., & Reshetikhin, N. Y. (1986). The Bethe ansatz and the combinatorics of Young tableaux. Zapiski Nauchnykh Seminarov Leningradskogo Otdelenia Matematicheskogo Instituta im. V.A. Steklova (LOMI) (Differentsialnaya Geometriya, Gruppy Li i Mekhanika VIII), 155 65–115, 194. Translation in Journal of Soviet Mathematics, 41(2) (1988), 925–955. Google Scholar
  12. 12.
    Stanley, R. P. (1977). Some combinatorial aspects of the Schubert calculus. In Lecture notes in mathematics: Vol. 579. Combinatoire et représentation du groupe symétrique (pp. 217–251). Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976. Berlin: Springer. CrossRefGoogle Scholar
  13. 13.
    Stanley, R. P. (1997). Enumerative combinatorics, Vol. 1. Cambridge studies in advanced mathematics (Vol. 49). Cambridge: Cambridge University Press. With a foreword by Gian-Carlo Rota, Corrected reprint of the 1986 original. MATHGoogle Scholar
  14. 14.
    Stanley, R. P. (1999). Enumerative combinatorics, Vol. 2. Cambridge studies in advanced mathematics (Vol. 62). Cambridge: Cambridge University Press. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands

Personalised recommendations