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Limits of multiplicities in excellent filtrations and tensor product decompositions for affine Kac-Moody algebras. (English) Zbl 1426.17007
Summary: We express the multiplicities of the irreducible summands of certain tensor products of irreducible integrable modules for an affine Kac-Moody algebra over a simply laced Lie algebra as sums of multiplicities in appropriate excellent filtrations (Demazure flags). As an application, we obtain expressions for the outer multiplicities of tensor products of two fundamental modules for $$\widehat {\mathfrak {sl}}_{2}$$ in terms of partitions with bounded parts, which subsequently lead to certain partition identities.
##### MSC:
 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 05E10 Combinatorial aspects of representation theory 05A17 Combinatorial aspects of partitions of integers
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##### References:
 [1] Andrews, G.: The Theory of Partitions. Cambridge University Press (1998) · Zbl 0996.11002 [2] Bianchi, A; Macedo, T; Moura, A, On Demazure and local Weyl modules for affine hyperalgebras, Pacific J. Math., 274, 257-303, (2015) · Zbl 1394.17050 [3] Biswal, R; Chari, V; Schneider, L; Viswanatha, S, Demazure flags, Chebyshev polynomials, partial and mock theta functions, J. Combinatorial Theory Series A, 140, 38-75, (2016) · Zbl 1418.17052 [4] Chari, V; Loktev, S, Weyl, Demazure and fusion modules for the current algebra of $$\mathfrak{sl}_{r+1}$$𝔰𝔩r+1, Adv. Math., 207, 928-960, (2006) · Zbl 1161.17318 [5] Chari, V; Pressley, A, Weyl modules for classical and quantum affine algebras, Represent. Theory, 5, 191-223, (2001) · Zbl 0989.17019 [6] Chari, V; Schneider, L; Shereen, P; Wand, J, Modules with Demazure flags and character formulae, SIGMA, 10, 16, (2014) · Zbl 1286.05178 [7] Chari, V; Venkatesh, R, Demazure modules, fusion products, and q-systems, Comm. Math. Phys., 333, 799-830, (2015) · Zbl 1361.17024 [8] Feingold, AJ, Tensor products of certain modules for the generalized Cartan matrix Lie algebra $$a_{1}^{(1)}$$a1(1), Comm. Algebra, 9, 1323-1341, (1981) · Zbl 0464.17001 [9] Fourier, G; Littelmann, P, Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions, Adv. Math., 211, 566-593, (2007) · Zbl 1114.22010 [10] Jakelić, D; Moura, A, Finite-dimensional representations of hyper loop algebras, Pacific J. Math., 233, 371-402, (2007) · Zbl 1211.17008 [11] Joseph, A.: Quantum Groups and Their Primitive Ideals. Springer (1995) · Zbl 0808.17004 [12] Joseph, A, A decomposition theorem for Demazure crystals, J. Algebra, 265, 562-578, (2003) · Zbl 1100.17009 [13] Joseph, A.: Modules with a Demazure flag. In: Studies in Lie Theory, Progr. Math., vol. 243, pp. 131-169. Birkhäuser (2006) · Zbl 1195.17010 [14] Kashiwara, M, On crystal bases of the Q-analogue of universal enveloping algebras, Duke Math. J., 63, 465-516, (1991) · Zbl 0739.17005 [15] Kashiwara, M, Crystal base and littelmann’s refined Demazure character formula, Duke Math. J., 71, 839-858, (1993) · Zbl 0794.17008 [16] King, R; Welsh, T, Tensor products for affine Kac-Moody algebras, group theoretical methods in physics (Moscow, 1990), Lect. Notes Phys., 382, 508-511, (1991) [17] Kumar, S.: Kac-Moody Groups, Their Flag Varieties and Representation Theory. Birkhäuser (2002) · Zbl 1026.17030 [18] Lakshmibai, V; Littelmann, P; Magyar, P, Standard monomial theory for Bott-Samelson varieties, Compos. Math., 130, 293-318, (2002) · Zbl 1061.14051 [19] Lepowsky, J; Wilson, R, A Lie theoretic interpretation and proof of the Rogers-Ramanujan identities, Adv. Math., 45, 21-72, (1982) · Zbl 0488.17006 [20] Littelmann, P, A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, Invent. Math., 116, 329-346, (1994) · Zbl 0805.17019 [21] Littelmann, P; Carter, R (ed.); Saxl, J (ed.), The path model, the quantum Frobenius map and standard monomial theory, 175-212, (1998), Dordrecht · Zbl 0938.14031 [22] Mathieu, O, Positivity of some intersections in k0(G/B), J. Pure Appl. Algebra, 152, 231-243, (2000) · Zbl 0978.22016 [23] Misra, K., Wilson, E.: On tensor product decomposition of $$\hat{\mathfrak{sl}}(n)$$-modules. Journal of Algebra and its Applications 12. doi:10.1142/S0219498813500540 (2013) · Zbl 1286.05178 [24] Misra, K., Wilson, E.: Tensor product decomposition of $$\hat{\mathfrak{sl}}(n)$$-modules and identities. Contemp. Math. 627, 131-144 (2014). doi:10.1090/conm/627/12538 · Zbl 0794.17008 [25] Naoi, K, Weyl modules, Demazure modules and finite crystals for non-simply laced type, Adv. Math., 229, 875-934, (2012) · Zbl 1305.17009 [26] Okado, M; Schilling, A; Shimozono, M, A tensor product theorem related to perfect crystals, J. Algebra, 267, 212-245, (2003) · Zbl 1039.17017 [27] Slater, L, Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc., 54, 147-167, (1952) · Zbl 0046.27204 [28] Wand, J.: Demazure flags for local weyl modules. Ph.D thesis, UC Riverside (2015) · Zbl 0046.27204
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