Geodesic
Schur Positivity and Kirillov–Reshetikhin Modules
Symmetry, integrability and geometry: methods and applications, Tome 10 (2014) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
Keywords: Kirillov–Reshetikhin modules; $Q$-systems; Schur positivity. 
@article{SIGMA_2014_10_a57,
     author = {Ghislain Fourier and David Hernandez},
     title = {Schur {Positivity} and {Kirillov{\textendash}Reshetikhin} {Modules}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2014},
     volume = {10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a57/}
}
TY  - JOUR
AU  - Ghislain Fourier
AU  - David Hernandez
TI  - Schur Positivity and Kirillov–Reshetikhin Modules
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2014
VL  - 10
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a57/
LA  - en
ID  - SIGMA_2014_10_a57
ER  - 
%0 Journal Article
%A Ghislain Fourier
%A David Hernandez
%T Schur Positivity and Kirillov–Reshetikhin Modules
%J Symmetry, integrability and geometry: methods and applications
%D 2014
%V 10
%U http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a57/
%G en
%F SIGMA_2014_10_a57
Ghislain Fourier; David Hernandez. Schur Positivity and Kirillov–Reshetikhin Modules. Symmetry, integrability and geometry: methods and applications, Tome 10 (2014). http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a57/

[1] Chari V., “Minimal affinizations of representations of quantum groups: the rank {$2$} case”, Publ. Res. Inst. Math. Sci., 31 (1995), 873–911, arXiv: hep-th/9410022 | DOI | MR | Zbl

[2] Chari V., “On the fermionic formula and the {K}irillov–{R}eshetikhin conjecture”, Int. Math. Res. Not., 2001 (2001), 629–654, arXiv: math.QA/0006090 | DOI | MR | Zbl

[3] Chari V., Fourier G., Khandai T., “A categorical approach to {W}eyl modules”, Transform. Groups, 15 (2010), 517–549, arXiv: 0906.2014 | DOI | MR | Zbl

[4] Chari V., Hernandez D., “Beyond {K}irillov–{R}eshetikhin modules”, Quantum Affine Algebras, Extended Affine {L}ie Algebras, and their Applications, Contemp. Math., 506, Amer. Math. Soc., Providence, RI, 2010, 49–81, arXiv: 0812.1716 | DOI | MR | Zbl

[5] Chari V., Pressley A., J. Algebra, 184 (1996), Minimal affinizations of representations of quantum groups: the simply laced case, arXiv: hep-th/9410036 | DOI | MR

[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, arXiv: math.CO/0608134 | DOI | MR | Zbl

[8] Feigin B., Loktev S., “On generalized {K}ostka polynomials and the quantum {V}erlinde rule”, Differential Topology, Infinite-Dimensional {L}ie Algebras, and Applications, Amer. Math. Soc. Transl. Ser. 2, 194, Amer. Math. Soc., Providence, RI, 1999, 61–79, arXiv: math.QA/9812093 | MR | Zbl

[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, arXiv: math.QA/9911112 | DOI | MR | Zbl

[12] Frenkel E., Reshetikhin N., “The {$q$}-characters of representations of quantum affine algebras and deformations of {$\mathcal W$}-algebras”, Recent Developments in Quantum Affine Algebras and Related Topics ({R}aleigh, {NC}, 1998), Contemp. Math., 248, Amer. Math. Soc., Providence, RI, 1999, 163–205, arXiv: math.QA/9810055 | DOI | MR | Zbl

[13] Hatayama G., Kuniba A., Okado M., Takagi T., Tsuboi Z., “Paths, crystals and fermionic formulae”, Math{P}hys Odyssey, 2001, Prog. Math. Phys., 23, Birkhäuser Boston, Boston, MA, 2002, 205–272, arXiv: math.QA/0102113 | MR | Zbl

[14] Hernandez D., “The {K}irillov–{R}eshetikhin conjecture and solutions of {$T$}-systems”, J. Reine Angew. Math., 596 (2006), 63–87, arXiv: math.QA/0501202 | DOI | MR | Zbl

[15] Kedem R., “A pentagon of identities, graded tensor products, and the {K}irillov–{R}eshetikhin conjecture”, New Trends in Quantum Integrable Systems, World Sci. Publ., Hackensack, NJ, 2011, 173–193, arXiv: 1008.0980 | DOI | MR | Zbl

[16] Lam T., Postnikov A., Pylyavskyy P., “Schur positivity and {S}chur log-concavity”, Amer. J. Math., 129 (2007), 1611–1622, arXiv: math.CO/0502446 | DOI | MR | Zbl

[17] Mukhin E., Young C. A. S., “Extended {$T$}-systems”, Selecta Math. (N.S.), 18 (2012), 591–631, arXiv: 1104.3094 | DOI | MR | Zbl

[18] Nakajima H., “{$t$}-analogs of {$q$}-characters of {K}irillov–{R}eshetikhin modules of quantum affine algebras”, Represent. Theory, 7 (2003), 259–274, arXiv: math.QA/0009231 | DOI | MR | Zbl