Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 10 (2014), 058, 9 pages      arXiv:1403.4750      https://doi.org/10.3842/SIGMA.2014.058
Contribution to the Special Issue on New Directions in Lie Theory

Schur Positivity and Kirillov-Reshetikhin Modules

Ghislain Fourier a and David Hernandez b
a) School of Mathematics and Statistics, University of Glasgow, UK
b) Sorbonne Paris Cité, Univ Paris Diderot-Paris 7, Institut de Mathématiques de Jussieu - Paris Rive Gauche CNRS UMR 7586, Bât. Sophie Germain, Case 7012,75205 Paris, France

Received April 04, 2014, in final form May 29, 2014; Published online June 04, 2014

Abstract
In this note, inspired by the proof of the Kirillov-Reshetikhin conjecture, we consider tensor products of Kirillov-Reshetikhin modules of a fixed node and various level. We fix a positive integer and attach to each of its partitions such a tensor product. We show that there exists an embedding of the tensor products, with respect to the classical structure, along with the reverse dominance relation on the set of partitions.

Key words: Kirillov-Reshetikhin modules; $Q$-systems; Schur positivity.

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References

  1. Chari V., Minimal affinizations of representations of quantum groups: the rank 2 case, Publ. Res. Inst. Math. Sci. 31 (1995), 873-911, hep-th/9410022.
  2. Chari V., On the fermionic formula and the Kirillov-Reshetikhin conjecture, Int. Math. Res. Not. 2001 (2001), 629-654, math.QA/0006090.
  3. Chari V., Fourier G., Khandai T., A categorical approach to Weyl modules, Transform. Groups 15 (2010), 517-549, arXiv:0906.2014.
  4. Chari V., Hernandez D., Beyond Kirillov-Reshetikhin modules, in Quantum Affine Algebras, Extended Affine Lie Algebras, and their Applications, Contemp. Math., Vol. 506, Amer. Math. Soc., Providence, RI, 2010, 49-81, arXiv:0812.1716.
  5. Chari V., Pressley A., Minimal affinizations of representations of quantum groups: the simply laced case, J. Algebra 184 (1996), 1-30, hep-th/9410036.
  6. Chari V., Venkatesh R., Demazure modules, fusion products and $Q$-systems, arXiv:1305.2523.
  7. Dobrovolska G., Pylyavskyy P., On products of ${\mathfrak{sl}}_{\mathfrak{n}}$ characters and support containment, J. Algebra 316 (2007), 706-714, math.CO/0608134.
  8. Feigin B., Loktev S., On generalized Kostka polynomials and the quantum Verlinde rule, in Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Amer. Math. Soc. Transl. Ser. 2, Vol. 194, Amer. Math. Soc., Providence, RI, 1999, 61-79, math.QA/9812093.
  9. Fourier G., New homogeneous ideals for current algebras: filtrations, fusion products and Pieri rules, arXiv:1303.4437.
  10. Fourier G., Extended partial order and applications to tensor products, Australas. J. Combin. 58 (2014), 178-196, arXiv:1211.2923.
  11. Frenkel E., Mukhin E., Combinatorics of $q$-characters of finite-dimensional representations of quantum affine algebras, Comm. Math. Phys. 216 (2001), 23-57, math.QA/9911112.
  12. Frenkel E., Reshetikhin N., The $q$-characters of representations of quantum affine algebras and deformations of $\mathcal W$-algebras, in Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), Contemp. Math., Vol. 248, Amer. Math. Soc., Providence, RI, 1999, 163-205, math.QA/9810055.
  13. Hatayama G., Kuniba A., Okado M., Takagi T., Tsuboi Z., Paths, crystals and fermionic formulae, in MathPhys Odyssey, 2001, Prog. Math. Phys., Vol. 23, Birkhäuser Boston, Boston, MA, 2002, 205-272, math.QA/0102113.
  14. Hernandez D., The Kirillov-Reshetikhin conjecture and solutions of $T$-systems, J. Reine Angew. Math. 596 (2006), 63-87, math.QA/0501202.
  15. Kedem R., A pentagon of identities, graded tensor products, and the Kirillov-Reshetikhin conjecture, in New Trends in Quantum Integrable Systems, World Sci. Publ., Hackensack, NJ, 2011, 173-193, arXiv:1008.0980.
  16. Lam T., Postnikov A., Pylyavskyy P., Schur positivity and Schur log-concavity, Amer. J. Math. 129 (2007), 1611-1622, math.CO/0502446.
  17. Mukhin E., Young C.A.S., Extended $T$-systems, Selecta Math. (N.S.) 18 (2012), 591-631, arXiv:1104.3094.
  18. Nakajima H., $t$-analogs of $q$-characters of Kirillov-Reshetikhin modules of quantum affine algebras, Represent. Theory 7 (2003), 259-274, math.QA/0009231.

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