Geodesic
The Littlewood-Richardson rule and~Gelfand-Tsetlin patterns
Algebra and discrete mathematics, Tome 22 (2016) no. 1, pp. 21-47.

Voir la notice de l'article provenant de la source Math-Net.Ru

We give a survey on the Littlewood-Richardson rule. Using Gelfand-Tsetlin patterns as the main machinery of our analysis, we study the interrelationship of various combinatorial descriptions of the Littlewood-Richardson rule.
Keywords: Littlewood-Richardson rule, Gelfand-Tsetlin patterns. 
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Patrick Doolan; Sangjib Kim. The Littlewood-Richardson rule and~Gelfand-Tsetlin patterns. Algebra and discrete mathematics, Tome 22 (2016) no. 1, pp. 21-47. http://geodesic.mathdoc.fr/item/ADM_2016_22_1_a2/

[1] St. Petersburg Math. J., 7:1 (1996), 77–127 | MR | Zbl

[2] A. D. Berenstein and A. V. Zelevinsky, “Tensor product multiplicities and convex polytopes in partition space”, J. Geom. Phys., 5:3 (1988), 453–472 | DOI | MR | Zbl

[3] Soviet Math. Dokl., 37:3 (1988), 799–802 | MR | MR

[4] A. D. Berenstein and A. V. Zelevinsky, “Triple multiplicities for $\mathrm{sl}(r+1)$ and the spectrum of the exterior algebra of the adjoint representation”, J. Algebraic Combin., 1:1 (1992), 7–22 | DOI | MR | Zbl

[5] A. S. Buch, “The saturation conjecture (after A. Knutson and T. Tao)”, With an appendix by William Fulton, Enseign. Math. (2), 46:1–2 (2000), 43–60 | MR | Zbl

[6] V. I. Danilov and G. A. Koshevoi, “Arrays and the combinatorics of Young tableaux”, Russian Mathematical Surveys, 60:2 (2005), 269–334 | DOI | MR | Zbl

[7] W. Fulton, Young tableaux. With applications to representation theory and geometry, London Mathematical Society Student Texts, 35, Cambridge University Press, Cambridge, 1997 | MR | Zbl

[8] W. Fulton, “Eigenvalues, invariant factors, highest weights, and Schubert calculus”, Bull. Amer. Math. Soc. (N.S.), 37:3 (2000), 209–249 | DOI | MR | Zbl

[9] W. Fulton and J. Harris, Representation theory. A first course, Graduate Texts in Mathematics, 129, Readings in Mathematics, Springer-Verlag, New York, 1991 | MR | Zbl

[10] I. M. Gelfand and M. L. Tsetlin, “Finite-dimensional representations of the group of unimodular matrices”, Doklady Akad. Nauk SSSR (N.S.), 71 (1950), 825–828 | MR | Zbl

[11] I. M. Gelfand and A. V. Zelevinskiĭ, “Polyhedra in a space of diagrams and the canonical basis in irreducible representations of $\mathfrak{gl}_{n}$”, Functional Anal. Appl., 19:2 (1985), 141–144 | DOI | MR | Zbl

[12] I. M. Gelfand and A. V. Zelevinskiĭ, “Multiplicities and regular bases for $\mathfrak{gl}_n$”, Group-theoretic methods in physics (Yurmala, 1985), v. 2, Nauka, Moscow, 1986, 22–31 (Russian) | MR

[13] N. Gonciulea and V. Lakshmibai, “Degenerations of flag and Schubert varieties to toric varieties”, Transform. Groups, 1:3 (1996), 215–248 | DOI | MR | Zbl

[14] R. Goodman and N. R. Wallach, Symmetry, representations, and invariants, Graduate Texts in Mathematics, 255, Springer, Dordrecht, 2009 | DOI | MR | Zbl

[15] R. Howe and S. T. Lee, “Why should the Littlewood-Richardson rules be true?”, Bull. Amer. Math. Soc., 49 (2012), 187–236 | DOI | MR | Zbl

[16] R. Howe, E.-C. Tan, and J. F. Willenbring, “A basis for the $GL_n$ tensor product algebra”, Adv. Math., 196:2 (2005), 531–564 | DOI | MR | Zbl

[17] R. Howe, S. Jackson, S. T. Lee, E.-C. Tan, J. Willenbring, “Toric degeneration of branching algebras”, Adv. Math., 220:6 (2009), 1809–1841 | DOI | MR | Zbl

[18] R. C. King, C. Tollu, and F. Toumazet, “The hive model and the polynomial nature of stretched Littlewood-Richardson coefficients”, Sem. Lothar. Combin., 54A (2005/07), B54Ad, 19 pp. | MR | Zbl

[19] R. C. King, C. Tollu, and F. Toumazet, “The hive model and the factorisation of Kostka coefficients”, Sem. Lothar. Combin., 54A (2005/07), B54Ah, 22 pp. | MR | Zbl

[20] S. Kim, “Standard monomial theory for flag algebras of $\mathrm{GL}(n)$ and $\mathrm{Sp}(2n)$”, J. Algebra, 320:2 (2008), 534–568 | DOI | MR | Zbl

[21] A. Knutson and T. Tao, “The honeycomb model of $GL_n(\mathbb{C})$ tensor products. I. Proof of the saturation conjecture”, J. Amer. Math. Soc., 12:4 (1999), 1055–1090 | DOI | MR | Zbl

[22] A. Knutson, T. Tao, and C. Woodward, “The honeycomb model of $GL_n(\mathbb{C})$ tensor products. II. Puzzles determine facets of the Littlewood-Richardson cone”, J. Amer. Math. Soc., 17:1 (2004), 19–48 | DOI | MR | Zbl

[23] M. Kogan and E. Miller, “Toric degeneration of Schubert varieties and Gelfand-Tsetlin polytopes”, Adv. Math., 193:1 (2005), 1–17 | DOI | MR | Zbl

[24] I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, Oxford Science Publications, Second edition, The Clarendon Press, New York; Oxford University Press, 1995 | MR

[25] I. Pak and E. Vallejo, “Combinatorics and geometry of Littlewood-Richardson cones”, European J. Combin., 26:6 (2005), 995–1008 | DOI | MR | Zbl

[26] R. P. Stanley, Enumerative combinatorics, v. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999 | MR | Zbl

[27] H. Tamvakis, “The connection between representation theory and Schubert calculus”, Enseign. Math. (2), 50:3–4 (2004), 267–286 | MR | Zbl

[28] H. Thomas and A. Yong, “An $S_3$-symmetric Littlewood-Richardson rule”, Math. Res. Lett., 15:5 (2008), 1027–1037 | DOI | MR | Zbl

[29] M. A. A. van Leeuwen, “The Littlewood-Richardson rule, and related combinatorics”, Interaction of combinatorics and representation theory, MSJ Mem., 11, Math. Soc. Japan, Tokyo, 2001, 95–145 | MR | Zbl