We present a combinatorial monomial basis (or, more precisely, a family of monomial bases) in every finite-dimensional irreducible -module. These bases are in many ways similar to the FFLV bases for types and . 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.
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DOI: 10.5802/alco.41
Keywords: Lie algebras, type B, monomial bases, FFLV bases, FFLV polytopes, PBW degenerations
Makhlin, Igor 1
@article{ALCO_2019__2_2_305_0, author = {Makhlin, Igor}, title = {FFLV-type monomial bases for type $B$}, journal = {Algebraic Combinatorics}, pages = {305--322}, publisher = {MathOA foundation}, volume = {2}, number = {2}, year = {2019}, doi = {10.5802/alco.41}, zbl = {07049527}, mrnumber = {3934832}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.41/} }
Makhlin, Igor. 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/articles/10.5802/alco.41/
[1] GelfandâTsetlin polytopes and FeiginâFourierâLittelmannâVinberg polytopes as marked poset polytopes, J. Comb. Theory, Ser. A, Volume 118 (2011) no. 8, pp. 2454-2462 | DOI | MR | Zbl
[2] The PBW filtration and convex polytopes in type , J. Pure Appl. Algebra, Volume 223 (2019) no. 1, pp. 245-276 | DOI | MR | Zbl
[3] Lie Algebras of Finite and Affine Type, Cambridge Studies in Advanced Mathematics, 96, Cambridge University Press, 2005, xvii+632 pages | MR | Zbl
[4] Quiver Grassmannians and degenerate flag varieties, Algebra Number Theory, Volume 6 (2012) no. 1, pp. 165-194 | DOI | MR | Zbl
[5] Extremal part of the PBW-filtration and nonsymmetric Macdonald polynomials, Adv. Math., Volume 282 (2015), pp. 220-264 | DOI | MR | Zbl
[6] The PBW filtration, Represent. Theory, Volume 13 (2009), pp. 165-181 | DOI | MR | Zbl
[7] degeneration of flag varieties, Sel. Math., New Ser., Volume 18 (2012) no. 3, pp. 513-537 | DOI | MR | Zbl
[8] PBW filtration and bases for irreducible modules in type , Transform. Groups, Volume 16 (2011) no. 1, pp. 71-89 | DOI | MR | Zbl
[9] PBW-filtration and bases for symplectic Lie algebras, Int. Math. Res. Not., Volume 2011 (2011) no. 24, pp. 5760-5784 | DOI | MR | Zbl
[10] Vertices of FFLV polytopes, J. Algebr. Comb., Volume 45 (2017) no. 4, pp. 1083-1110 | DOI | MR | Zbl
[11] Marked poset polytopes: Minkowski sums, indecomposables, and unimodular equivalence, J. Pure Appl. Algebra, Volume 220 (2016) no. 2, pp. 606-620 | DOI | MR | Zbl
[12] Degenerate coordinate rings of flag varieties and Frobenius splitting, Sel. Math., New Ser., Volume 20 (2014) no. 3, pp. 823-838 | DOI | MR | Zbl
[13] NewtonâOkounkov polytopes of flag varieties, Transform. Groups, Volume 22 (2017) no. 2, pp. 387-402 | DOI | MR | Zbl
[14] Realization of affine type A KirillovâReshetikhin crystals via polytopes, J. Comb. Theory, Ser. A, Volume 120 (2013) no. 8, pp. 2093-2117 | MR | Zbl
[15] Weight bases of GelfandâTsetlin type for representations of classical Lie algebras, J. Phys. A, Math. Gen., Volume 33 (1999) no. 22, pp. 4143-4168 | DOI | MR | Zbl
[16] Two poset polytopes, Discrete Comput. Geom., Volume 1 (1986), pp. 9-23 | DOI | MR | Zbl
[17] On some canonical bases of representation spaces of simple Lie algebras (2005) (conference talk, Bielefeld)
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