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On truncated Weyl modules. (English) Zbl 07052405
Summary: We study structural properties of truncated Weyl modules. A truncated Weyl module \(W_N(\lambda)\) is a local Weyl module for \(\mathfrak{g}[t]_N=\mathfrak{g}\otimes\frac{\mathbb{C}[t]}{t^N\mathbb{C}[t]}\), where \(\mathfrak{g}\) is a finite-dimensional simple Lie algebra. It has been conjectured that, if \(N\) is sufficiently small with respect to \(\lambda\), the truncated Weyl module is isomorphic to a fusion product of certain irreducible modules. Our main result proves this conjecture when \(\lambda\) is a multiple of certain fundamental weights, including all minuscule ones for simply laced \(\mathfrak{g}\). We also take a further step towards proving the conjecture for all multiples of fundamental weights by proving that the corresponding truncated Weyl module is isomorphic to a natural quotient of a fusion product of Kirillov-Reshetikhin modules. One important part of the proof of the main result shows that any truncated Weyl module is isomorphic to a Chari-Venkatesh module and explicitly describes the corresponding family of partitions. This leads to further results in the case that \(\mathfrak{g}=\mathfrak{sl}_2\) related to Demazure flags and chains of inclusions of truncated Weyl modules.
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B70 Graded Lie (super)algebras
05E10 Combinatorial aspects of representation theory
06A07 Combinatorics of partially ordered sets
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[1] Biswal, R.; Chari, V.; Schneider, L.; Viswanath, S., Demazure flags, Chebyshev polynomials, partial and mock theta functions, J. Combin. Theory Series A, 140, 38-75, (2016) · Zbl 1418.17052
[2] Calixto, L.; Lemay, J.; Savage, A., Weyl modules for Lie superalgebras, Proc. Am. Math. Soc
[3] Chari, V., Minimal affinizations of representations of quantum groups: the rank-2 case, Publ. Res. Inst. Math. Sci, 31, 873-911, (1995) · Zbl 0855.17010
[4] Chari, V., On the fermionic formula and the Kirillov–Reshetikhin conjecture, Int. Math. Res. Notices, 2001, 12, 629-654, (2001) · Zbl 0982.17004
[5] Chari, V.; Fourier, G.; Khandai, T., A categorical approach to Weyl modules, Transf. Groups, 15, 3, 517-549, (2010) · Zbl 1245.17004
[6] Chari, V.; Fourier, G.; Sagaki, D., Posets, tensor products and Schur positivity, Algebra Number Theory, 8, 4, 933-961, (2014) · Zbl 1320.17004
[7] Chari, V.; Loktev, S., Weyl, Demazure and fusion modules for the current algebra of, Adv. Math, 207, 928-960, (2006) · Zbl 1161.17318
[8] Chari, V.; Moura, A., The restricted Kirillov–Reshetikhin modules for the current and twisted current algebras, Commun. Math. Phys, 266, 2, 431-454, (2006) · Zbl 1118.17007
[9] Chari, V.; Pressley, A., Weyl modules for classical and quantum affine algebras, Represent. Theory, 005, 9, 191-223, (2001) · Zbl 0989.17019
[10] Chari, V.; Schneider, L.; Shereen, P.; Wand, J., Modules with Demazure flags and character formulae, SIGMA, 10, 032, (2014) · Zbl 1286.05178
[11] Chari, V.; Venkatesh, R., Demazure modules, fusion products and Q-systems, Commun. Math. Phys, 333, 2, 799-830, (2015) · Zbl 1361.17024
[12] Feigin, B.; Feigin, E., Q-characters of the tensor products in \(##?##\)-case, Mosc. Math. J, 2, 3, 567-588, (2002) · Zbl 1027.05007
[13] Feigin, B.; Loktev, S., On generalized Kostka polynomials and the quantum Verlinde rule differential topology, infinite-dimensional Lie algebras, and applications, Am. Math. Soc. Transl. Ser. 2, 194, 61-79, (1999) · Zbl 0974.17008
[14] Feigin, B.; Loktev, S., Multi-dimensional Weyl modules and symmetric functions, Commun. Math. Phys, 251, 3, 427-445, (2004) · Zbl 1100.17005
[15] Fourier, G., Extended partial order and applications to tensor product, Austral. J. Combin, 58, 178-196, (2014) · Zbl 1296.05198
[16] Fourier, G., New homogeneous ideals for current algebras: filtrations, fusion products and Pieri rules, Mosc. Math. J, 15, 1, 49-72, (2015) · Zbl 1383.17006
[17] Fourier, G.; Littelmann, P., Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions, Adv. Math, 211, 2, 566-593, (2007) · Zbl 1114.22010
[18] Futorny, V.; Rao, S. E.; Sharma, S. S., Weyl modules associated to kac-Moody Lie algebras, Commun. Algebra, 44, 12, 5045-5057, (2016) · Zbl 1403.17025
[19] Hernandez, D., Kirillov–Reshetikhin conjecture: the general case, Int. Math. Res. Notes, 2010, 149-193, (2010) · Zbl 1242.17017
[20] Jakelic, D.; Moura, A., On Weyl modules for quantum and hyper loop algebras, Contemp. Math, 623, 99-134, (2014) · Zbl 1358.17028
[21] Kirillov, A.; Reshetikhin, N., Representations of Yangians and multiplicities of occurrence of the irreducible components of the tensor product of simplie Lie algebras, J. Math. Sci, 52, 3, 3156-3164, (1990)
[22] Kodera, R.; Naoi, K., Loewy series of Weyl modules and the PoincarĂ© polynomials of quiver varieties, Publ. Res. Inst. Math. Sci, 48, 3, 477-500, (2012) · Zbl 1308.17018
[23] Kus, D.; Littelmann, P., Fusion products and toroidal algebras, Pac. J. Math, 278, 2, 427-445, (2015) · Zbl 1381.17014
[24] Martins, V., Truncated Weyl modules as Chari-Venkatesh modules and fusion products, (2017)
[25] Moura, A., Restricted limits of minimal affinizations, Pac. J. Math, 244, 2, 359-397, (2009) · Zbl 1246.17019
[26] Naoi, K., Tensor products of Kirillov–Reshetikhin modules and fusion products, Int. Math. Res. Notices, 2017, 5667-5709, (2017) · Zbl 1405.17050
[27] Neher, E.; Savage, A., A survey of equivariant map algebras with open problems, Contemp. Math, 602, 165-182, (2013) · Zbl 1350.17001
[28] Ravinder, B., Demazure modules, Chari-Venkatesh Modules and fusion products, SIGMA, 10, 110, 10, (2014) · Zbl 1331.17022
[29] Shereen, P., A Steinberg type decomposition theorem for higher level Demazure modules, (2015)
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