Degrees of stretched Kostka coefficients
Tyrrell B. McAllister
Eindhoven University of Technology Department of Mathematics and Computer Science P.O. Box 513 5600 MB Eindhoven The Netherlands
DOI: 10.1007/s10801-007-0083-2
Abstract
Given a partition λ and a composition β , the stretched Kostka coefficient K l b( n) \mathcal {K}_{λβ}(n) is the map n \rightarrowtail 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).
Pages: 263–273
Keywords: keywords Kostka coefficient; representation theory; Gelfand-tsetlin polytope
Full Text: PDF
References
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194. Translation in Journal of Soviet Mathematics, 41(2) (1988), 925-955.
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579. Combinatoire et représentation du groupe symétrique (pp. 217-251). Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg,
1976. Berlin: Springer.
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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.
2. Billey, S., Guillemin, V., & Rassart, E. (2004). A vector partition function for the multiplicities of slkC. Journal of Algebra, 278(1), 251-293.
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.
4. Fulton, W. (1997). Young tableaux. London mathematical society student texts (Vol. 35). Cambridge: Cambridge University Press. With applications to representation theory and geometry.
5. Fulton, W., & Harris, J. (1991). Representation theory. Graduate texts in mathematics (Vol. 129). New York: Springer. A first course, Readings in Mathematics.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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