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On the structures of hive algebras and tensor product algebras for general linear groups of low rank. (English) Zbl 07127350
MSC:
20G05 Representation theory for linear algebraic groups
13A50 Actions of groups on commutative rings; invariant theory
05E15 Combinatorial aspects of groups and algebras (MSC2010)
Software:
Macaulay2; Normaliz
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References:
[1] W. Bruns, B. Ichim, T. Römer and C. Söger, Normaliz. Algorithms for rational cones and affine monoids, https://www.normaliz.uni-osnabrueck.de/. · Zbl 1203.13033
[2] Buch, A. S., The saturation conjecture (after A. Knutson and T. Tao), Enseign. Math. (2)46(1-2) (2000) 43-60. With an appendix by William Fulton. · Zbl 0979.20041
[3] Dodgson, C. L., Condensation of determinants, being a new and brief method for computing their arithmetical values, Proc. Roy. Soc.15 (1866) 150-155.
[4] Dolgachev, I., Weighted projective varieties, in Group Actions and Vector Fields, , Vol. 956 (Springer, Berlin, 1982), pp. 34-71. · Zbl 0516.14014
[5] Doolan, P. and Kim, S., The Littlewood-Richardson rule and Gelfand-Tsetlin patterns, Algebra Discrete Math.22(1) (2016) 21-47. · Zbl 1368.05011
[6] Fulton, W., Young Tableaux: With Applications to Representation Theory and Geometry, , Vol. 35 (Cambridge University Press, Cambridge, 1997). · Zbl 0878.14034
[7] D. R. Grayson and M. E. Stillman, Macaulay 2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/.
[8] Grosshans, F. D., Invariants on \(G / U \times G / U \times G / U, G = \text{SL}(4, C)\), in Invariant Methods in Discrete and Computational Geometry (Kluwer Academic Publishers, Dordrecht, 1995), pp. 257-277. · Zbl 0984.13006
[9] Howe, R., Jackson, S., Lee, S. T., Tan, E.-C. and Willenbring, J., Toric degeneration of branching algebras, Adv. Math.220(6) (2009) 1809-1841. · Zbl 1179.22012
[10] Howe, R. and Lee, S. T., Why should the Littlewood-Richardson rule be true?Bull. Amer. Math. Soc. (N.S.)49(2) (2012) 187-236. · Zbl 1300.20053
[11] Howe, R., Tan, E.-C. and Willenbring, J. F., A basis for the \(G L_n\) tensor product algebra, Adv. Math.196(2) (2005) 531-564. · Zbl 1072.22007
[12] Kim, S., A presentation of the double Pieri algebra, J. Pure Appl. Algebra222(2) (2018) 368-381. · Zbl 1378.20055
[13] Kim, S. and Yoo, S., Pieri and Littlewood-Richardson rules for two rows and cluster algebra structure, J. Algebraic Combin.45(3) (2017) 887-909. · Zbl 1362.05133
[14] King, R. C., Tollu, C. and Toumazet, F., The hive model and the polynomial nature of stretched Littlewood-Richardson coefficients, Sém. Lothar. Combin.54A (2005/2007), Art. B54Ad, 19 pp. · Zbl 1178.05100
[15] Knutson, A. and Tao, T., The honeycomb model of \(\text{GL}_n(C)\) tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc.12(4) (1999) 1055-1090. · Zbl 0944.05097
[16] Knutson, A., Tao, T. and Woodward, C., A positive proof of the Littlewood-Richardson rule using the octahedron recurrence, Electron. J. Combin.11(1) (2004), Research Paper 61, 18 pp. · Zbl 1053.05119
[17] S. T. Lee, Branching rules and branching algebras for the complex classical groups, COE Lecture Note, Vol. 47, Math-for-Industry (MI) Lecture Note Series. Kyushu University, Faculty of Mathematics, Fukuoka (2013). · Zbl 1270.05001
[18] Macdonald, I. G., Symmetric functions and Hall polynomials, 2nd edn., (Oxford University Press, New York, 1995). · Zbl 0899.05068
[19] Miller, E. and Sturmfels, B., Combinatorial Commutative Algebra, , Vol. 227 (Springer-Verlag, New York, 2005). · Zbl 1090.13001
[20] Pak, I. and Vallejo, E., Combinatorics and geometry of Littlewood-Richardson cones, European J. Combin.26(6) (2005) 995-1008. · Zbl 1063.05133
[21] Purbhoo, K., Puzzles, tableaux, and mosaics, J. Algebraic Combin.28(4) (2008) 461-480. · Zbl 1171.14033
[22] Stanley, R. P., Enumerative Combinatorics, Vol. 2, (Cambridge University Press, Cambridge, 1999).
[23] Thomas, H. and Yong, A., An \(S_3\)-symmetric Littlewood-Richardson rule, Math. Res. Lett.15(5) (2008) 1027-1037. · Zbl 1194.05164
[24] van Leeuwen, M. A. A., The Littlewood-Richardson rule, and related combinatorics, in Interaction of Combinatorics and Representation Theory, , Vol. 11 (Mathematical Society of Japan, Tokyo, 2001), pp. 95-145. · Zbl 0991.05101
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