FFLV-type monomial bases for type B
Algebraic Combinatorics, Volume 2 (2019) no. 2, p. 305-322
We present a combinatorial monomial basis (or, more precisely, a family of monomial bases) in every finite-dimensional irreducible 𝔰𝔬 2n+1 -module. These bases are in many ways similar to the FFLV bases for types A and C. They are also defined combinatorially via sums over Dyck paths in certain triangular grids. Our sums, however, involve weights depending on the length of the corresponding root. Accordingly, our bases also induce bases in certain degenerations of the modules but these degenerations are obtained not from the filtration by PBW degree but by a weighted version thereof.
Received : 2017-09-11
Revised : 2018-08-01
Accepted : 2018-08-21
Published online : 2019-03-05
DOI : https://doi.org/10.5802/alco.41
Classification:  17B10,  17B20,  05E10
Keywords: Lie algebras, type B, monomial bases, FFLV bases, FFLV polytopes, PBW degenerations
@article{ALCO_2019__2_2_305_0,
     author = {Makhlin, Igor},
     title = {FFLV-type monomial bases for type $B$},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {2},
     number = {2},
     year = {2019},
     pages = {305-322},
     doi = {10.5802/alco.41},
     language = {en},
     url = {https://alco.centre-mersenne.org/item/ALCO_2019__2_2_305_0}
}
FFLV-type monomial bases for type $B$. Algebraic Combinatorics, Volume 2 (2019) no. 2, pp. 305-322. doi : 10.5802/alco.41. https://alco.centre-mersenne.org/item/ALCO_2019__2_2_305_0/

[1] Ardila, Federico; Bliem, Thomas; Salazar, Dido Gelfand–Tsetlin polytopes and Feigin–Fourier–Littelmann–Vinberg polytopes as marked poset polytopes, J. Comb. Theory, Ser. A, Volume 118 (2011) no. 8, pp. 2454-2462 | Article | MR 2834187 | Zbl 1234.52009

[2] Backhaus, Teodor; Kus, Deniz The PBW filtration and convex polytopes in type B, J. Pure Appl. Algebra, Volume 223 (2019) no. 1, pp. 245-276 | Article | MR 3833459 | Zbl 06934116

[3] Carter, Roger Lie Algebras of Finite and Affine Type, Cambridge University Press, Cambridge Studies in Advanced Mathematics, Volume 96 (2005), xvii+632 pages | MR 2188930 | Zbl 1110.17001

[4] Cerulli Irelli, Giovanni; Feigin, Evgeny; Reineke, Markus Quiver Grassmannians and degenerate flag varieties, Algebra Number Theory, Volume 6 (2012) no. 1, pp. 165-194 | MR 2950163 | Zbl 1282.14083

[5] Cherednik, Ivan; Feigin, Evgeny Extremal part of the PBW-filtration and nonsymmetric Macdonald polynomials, Adv. Math., Volume 282 (2015), pp. 220-264 | Article | MR 3374526 | Zbl 1378.17040

[6] Feigin, Evgeny The PBW filtration, Represent. Theory, Volume 13 (2009), pp. 165-181 | Article | MR 2506263 | Zbl 1229.17026

[7] Feigin, Evgeny đ”Ÿ a M degeneration of flag varieties, Sel. Math., New Ser., Volume 18 (2012) no. 3, pp. 513-537 | Article | MR 2960025 | Zbl 1267.14064

[8] Feigin, Evgeny; Fourier, Ghislain; Littelmann, Peter PBW filtration and bases for irreducible modules in type A n , Transform. Groups, Volume 16 (2011) no. 1, pp. 71-89 | Article | MR 2785495 | Zbl 1237.17011

[9] Feigin, Evgeny; Fourier, Ghislain; Littelmann, Peter PBW-filtration and bases for symplectic Lie algebras, Int. Math. Res. Not., Volume 2011 (2011) no. 24, pp. 5760-5784 | Article | MR 2863380 | Zbl 1233.17007

[10] Feigin, Evgeny; Makhlin, Igor Vertices of FFLV polytopes, J. Algebr. Comb., Volume 45 (2017) no. 4, pp. 1083-1110 | MR 3641978 | Zbl 1370.05218

[11] Fourier, Ghislain Marked poset polytopes: Minkowski sums, indecomposables, and unimodular equivalence, J. Pure Appl. Algebra, Volume 220 (2016) no. 2, pp. 606-620 | Article | MR 3399380 | Zbl 1328.52007

[12] Hague, Chuck Degenerate coordinate rings of flag varieties and Frobenius splitting, Sel. Math., New Ser., Volume 20 (2014) no. 3, pp. 823-838 | Article | MR 3217462 | Zbl 1328.14081

[13] Kiritchenko, Valentina Newton–Okounkov polytopes of flag varieties, Transform. Groups, Volume 22 (2017) no. 2, pp. 387-402 | Article | MR 3649460 | Zbl 1396.14047

[14] Kus, Deniz Realization of affine type A Kirillov–Reshetikhin crystals via polytopes, J. Comb. Theory, Ser. A, Volume 120 (2013) no. 8, pp. 2093-2117 | MR 3102176 | Zbl 1300.17012

[15] Molev, Alexander Weight bases of Gelfand–Tsetlin type for representations of classical Lie algebras, J. Phys. A, Math. Gen., Volume 33 (1999) no. 22, pp. 4143-4168 | MR 1766625 | Zbl 0988.17005

[16] Stanley, Richard P. Two poset polytopes, Discrete Comput. Geom., Volume 1 (1986), pp. 9-23 | Article | MR 824105 | Zbl 0595.52008

[17] Vinberg, Èrnest On some canonical bases of representation spaces of simple Lie algebras (2005) (conference talk, Bielefeld)