Abstract
A special family of partitions occurs in two apparently unrelated contexts: the evaluation of one-dimensional configuration sums of certain RSOS models, and the modular representation theory of symmetric groups or their Hecke algebras Hm. We provide an explanation of this coincidence by showing how the irreducible Hm-modules which remain irreducible under restriction to Hm_1 (Jantzen–Seitz modules) can be determined from the decomposition of a tensor product of representations sln.
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Andrews, G. E., Baxter, R. J. and Forrester, P. J.: Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities, J. Statist. Phys. 35 (1984), 193–266.
Ariki, S.: On the decomposition numbers of the Hecke algebra of G(m, 1, n), Preprint, 1996.
Baxter, R. J.: Exactly Solved Models in Statistical Mechanics, Academic Press, New York, 1984.
Dipper, R. and James, G. D.: Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. 52 (1986), 20–52.
Date, E., Jimbo, M., Kuniba, A., Miwa, T. and Okado, M.: Paths, Maya diagrams, and representations of \(\widehat{\mathfrak{s}\mathfrak{l}}\)(r, C), Adv. Stud. Pure Math. 19 (1989), 149–191.
Date. E., Jimbo, M., Miwa, T. and Okado, M.:Automorphic properties of local height probabilities for integrable solid-on-solid models, Phys. Rev. B 35 (1987), 2105–2107.
Foda, O., Leclerc, B., Okado, M., Thibon, J.-Y. and Welsh, T. A.: Modular representations of Hecke algebras and solvable lattice models, submitted to Geometric Analysis and Lie Theory in Mathematics and Physics, Lecture Notes Series of the Australian Mathematical Society.
Foda, O., Leclerc, B., Okado, M., Thibon, J.-Y. and Welsh, T. A.: in preparation.
Foda, O., Okado, M. and Warnaar, S. O.: A proof of polynomial identities of type \(\widehat{sl}\)(n)1 ⊗ \(\widehat{sl}\)(n)1/\(\widehat{sl}\)(n)2, J. Math. Phys. 37 (1996), 965–986.
Grojnowski, I.: Affine Hecke algebras (and affine quantum GLn) at roots of unity, Internat.Math. Res. Notices (1994), 215–217.
Hayashi, T.: q-analogues of Clifford andWeyl algebras – spinor and oscillator representations of quantum enveloping algebras, Comm. Math. Phys. 127 (1990), 129–144.
James, G.D.: The decomposition matrices of GLn(q) for n ⩽ 10, Proc. London Math. Soc. 60 (1990), 225–265.
James, G. D. and Kerber, A.: The Representation Theory of the Symmetric Group, Addison-Wesley, New York, 1981.
Jantzen, J. C. and Seitz, G. M.: On the representation theory of the symmetric groups, Proc. London Math. Soc. 65 (1992), 475–504.
Jimbo, M., Misra, K., Miwa, T. and Okado, M.: Combinatorics of representations of Uq(\(\mathop \mathfrak{s}\limits^ \cdot \mathop \mathfrak{l}\limits^ \cdot \)(n)) at q = 0, Comm. Math. Phys. 136 (1991), 543–566.
Jimbo, M., Miwa, T. and Okado, M.: Solvable lattice models whose states are dominant integral weights of A n-1 , Lett. Math. Phys. 14} (1987}), 123–131
Kashiwara, M.: On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), 465–516.
Kashiwara, M.: Global crystal bases of quantum groups, Duke Math. J. 69 (1993), 455–485.
Kashiwara, M., Miwa, T. and Stern, E.: Decomposition of q-deformed Fock spaces, SelectaMath. 1996.
Kleshchev, A. S.: On restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to natural subgroups I, Proc. London Math. Soc. 69 (1994), 515–540.
Kleshchev, A. S.: Branching rules for the modular representations of symmetric groups III; some corollaries and a problem of Mullineux, J. London Math. Soc. 2 (1995).
Lascoux, A., Leclerc, B. and Thibon, J.-Y.: Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), 205–263.
Misra, K. C. and Miwa, T.: Crystal base of the basic representation of Uq(\(\widehat{\mathfrak{s}\mathfrak{l}}\) n), Comm. Math. Phys. 134 (1990), 79–88.
Stern, E.: Semi-infinite wedges and vertex operators, Internat. Math. Res. Notices (1995), 201–220.
Bessenrodt, Ch. and Olsson, J.: Residue symbols and Jantzen–Seitz partitions, to appear.
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Foda, O., Leclerc, B., Okado, M. et al. RSOS Models and Jantzen–Seitz Representations of Hecke Algebras at Roots of Unity. Letters in Mathematical Physics 43, 31–42 (1998). https://doi.org/10.1023/A:1007316705535
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DOI: https://doi.org/10.1023/A:1007316705535