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Liu, Shiyuan

Fermionic formula for double Kostka polynomials. (English) Zbl 1429.17016

J. Math. Soc. Japan 70, No. 1, 283-324 (2018).
Summary: The \(X=M\) conjecture asserts that the 1D sum and the fermionic formula coincide up to some constant power. In the case of type \(A\), both the 1D sum and the fermionic formula are closely related to Kostka polynomials. Double Kostka polynomials \(K_{\lambda,\mu}(t)\), indexed by two double partitions \(\lambda,\mu,\) are polynomials in \(t\) introduced as a generalization of Kostka polynomials.
In the present paper, we consider \(K_{\lambda,\mu}(t)\) in the special case where \(\mu=(-,\mu'')\). We formulate a 1D sum and a fermionic formula for \(K_{\lambda,\mu}(t)\), as a generalization of the case of ordinary Kostka polynomials. Then we prove an analogue of the \(X=M\) conjecture.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
05E10 Combinatorial aspects of representation theory
82B23 Exactly solvable models; Bethe ansatz
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
05A30 \(q\)-calculus and related topics

Keywords:

crystals; double Kostka polynomials; fermionic formulas; rigged configurations
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\textit{S. Liu}, J. Math. Soc. Japan 70, No. 1, 283--324 (2018; Zbl 1429.17016)
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References:

[1] L. M. Butler, Subgroup Lattices and Symmetric Functions, \doihref10.1090/memo/0539Mem. Amer. Math. Soc., 112 (1994), no. 539.
[2] L. Deka and A. Schilling, New fermionic formula for unrestricted Kostka polynomials, \doihref10.1016/j.jcta.2006.01.003J. Combinatorial Theory, Series A, 113 (2006), 1435-1461. · Zbl 1153.17006
[3] S. Fomin, Knuth Equivalence, Jeu de Taquin, and the Littlewood-Richardson Rule, preliminary version of Appendix 1 to Chapter 7, In: Enumerative Combinatorics, (ed. R. P. Stanley), 2, Cambridge University Press.
[4] J. Hong and S.-J. Kang, Introduction to quantum groups and crystal bases, Amer. Math. Soc., 2002. · Zbl 1134.17007
[5] G. Hatayama, A. Kuniba, M. Okado, T. Takagi and Z. Tsuboi, Paths, crystals and fermionic formula, \doihref10.1007/978-1-4612-0087-1_9Prog. Math. Phys., 23 (2001), 205-272. · Zbl 1016.17011
[6] G. Hatayama, A. Kuniba, M. Okado, T. Takagi and Y. Yamada, Remarks on fermionic formula, \doihref10.1090/conm/248Contemporary Math., 248 (1999), 243-291. · Zbl 1032.81015
[7] V. G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, 1990. · Zbl 0716.17022
[8] M. Kashiwara, On crystal bases of the \(Q\)-analogue of universal enveloping algebras, \doihref10.1215/S0012-7094-91-06321-0Duke Math. J., 63 (1991), 465-516.
[9] M. Kashiwara, The crystal base and Littelmann’s refined Demazure character formula, \doihref10.1215/S0012-7094-93-07131-1Duke Math. J., 71 (1993), 839-858. · Zbl 0794.17008
[10] S. V. Kerov, A. N. Kirillov and N. Y. Reshetikhin, Combinatorics, the Bethe Ansatz and representations of the symmetric group, J. Soviet Math., 41 (1988), 916-924. · Zbl 0639.20028
[11] A. Kuniba, M. Okado, R. Sakamoto, T. Takagi and Y. Yamada, Crystal interpretation of Kerov-Kirillov-Reshetikhin bijection, \doihref10.1016/j.nuclphysb.2006.02.005Nuclear Phys. B, 740 (2006), 299-327. · Zbl 1109.81043
[12] A. N. Kirillov and N. Yu. Reshetikin, The Bethe ansatz and the combinatorics of Young tableaux, J. Soviet Math., 41 (1988), 925-955. · Zbl 0639.20029
[13] A. N. Kirillov and N. Yu. Reshetikin, Representations of Yangians and multiplicities of the inclusions of the irreducible components of the tensor product of representations of simple Lie algebras, J. Soviet Math., 52 (1990), 3156-3164.
[14] A. N. Kirillov, A. Schilling and M. Shimozono, A bijection between Littlewood-Richardson tableaux and rigged configurations, \doihref10.1007/s00029-002-8102-6Selecta Math., New Ser., 8 (2002), 67-135. · Zbl 0986.05013
[15] S. Liu and T. Shoji, Double Kostka polynomials and Hall bimodule, \doihref10.3836/tjm/1475723088Tokyo J. Math., 39 (2017), 743-776. · Zbl 1364.05082
[16] A. Lascoux, M.-P. Schützenberger, Sur une Conjecture de H.O. Foulkes, C.R. Acad. Sci. Pairs, 288A (1978), 323-324.
[17] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Second Edition, Clarendon Press, Oxford, 1995. · Zbl 0824.05059
[18] A. Nakayashiki and Y. Yamada, Kostka polynomials and energy functions in solvable lattice models, \doihref10.1007/s000290050020Selecta Math., New ser., 3 (1997), 547-599. · Zbl 0915.17016
[19] M. Okado, \(X=M\) conjecture, Combinatorial Aspect of Integrable Systems, \doihref10.2969/msjmemoirs/01701C030MSJ Memoirs, 17, Math. Soc. Japan, 2007, 43-73.
[20] R. Sakamoto, Rigged Configurations and Kashiwara Operators, \doihref10.3842/SIGMA.2014.028SIGMA, 10 (2014), paper 028, 88pp. · Zbl 1376.17024
[21] A. Schilling, Crystal structure on rigged configurations, \doihref10.1155/IMRN/2006/97376Int. Math. Res. Not., 2006 (2006), Art. ID 97376, 27pp.
[22] A. Schilling, \(X=M\) conjecture: Fermionic formulas and rigged configurations under review, Combinatorial Aspect of Integrable Systems, \doihref10.2969/msjmemoirs/01701C040MSJ Memoirs, 17, Math. Soc. Japan, 2007, 75-104.
[23] T. Shoji, Green functions associated to complex reflection groups, \doihref10.1006/jabr.2001.8898J. Algebra, 245 (2001), 650-694. · Zbl 0997.20044
[24] T. Shoji, Green functions attached to limit symbols, Representation theory of algebraic groups and quantum groups, Adv. Stud. Pure Math., 40, Math. Soc. Japan, Tokyo, 2004, 443-467. · Zbl 1065.05096
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