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Fusion Formulas and Fusion Procedure for the Yang-Baxter Equation

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Abstract

We use the fusion formulas of the symmetric group and of the Hecke algebra to construct solutions of the Yang–Baxter equation on irreducible representations of \(\mathfrak {gl}_{N}\), \(\mathfrak {gl}_{N|M}\), \(U_{q}(\mathfrak {gl}_{N})\) and \(U_{q}(\mathfrak {gl}_{N|M})\). The solutions are obtained via the fusion procedure for the Yang–Baxter equation, which is reviewed in a general setting. Distinguished invariant subspaces on which the fused solutions act are also studied in the general setting, and expressed, in general, with the help of a fusion function. Only then, the general construction is specialised to the four situations mentioned above. In each of these four cases, we show how the distinguished invariant subspaces are identified as irreducible representations, using the relevant fusion formula combined with the relevant Schur–Weyl duality.

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References

  1. Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  2. Gomez, C., Ruiz-Altaba, M., Sierra, G.: Quantum Groups in Two-Dimensional Physics. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  3. Isaev, A.: Quantum groups and Yang-Baxter equation. Preprint MPIM 04-132 (2004)

  4. Jimbo, M. (ed.): Yang Baxter Equation in Integrable Systems. Adv. Series in Math. Phys., vol. 10. World Scientific, Singapore (1990)

  5. Kulish, P., Reshetikhin, N., Sklyanin, E.: Yang–Baxter equation and representation theory I. Lett. Math. Phys. 5, 393–403 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  6. Babichenko, A., Regelskis, V.: On boundary fusion and functional relations in the Baxterized affine Hecke algebra. J. Math. Phys. 55, 043503 (2014). arXiv:1305.1941

    Article  MathSciNet  MATH  Google Scholar 

  7. Cherednik, I.: Some finite dimensional representations of generalized Sklyanin algebras. Funct. Anal. Appl. 19, 77–79 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cherednik, I.: On “quantum” deformations of irreducible finite-dimensional representations of \(\mathfrak {gl}_{N}\). Sov. Math. Dokl. 33, 507–510 (1986)

    MATH  Google Scholar 

  9. Date, E., Jimbo, M., Kuniba, A., Miwa, T., Okado, M.: Exactly solvable SOS models II: proof of the star-triangle relation and combinatorial identities. Adv. Stud. Pure Math. 16, 17–122 (1988)

    MathSciNet  MATH  Google Scholar 

  10. Date, E., Jimbo, M., Miwa, T., Okado, M.: Fusion of the eight vertex SOS model. Lett. Math. Phys. 12, 209–215 (1986)

    Article  MathSciNet  Google Scholar 

  11. Hou, B-Y., Zhou, Y.-K.: On the fusion of face and vertex models. J. Phys. A 22, 5089–5096 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jimbo, M: Introduction to the Yang–Baxter equation. Int. J. Mod. Phys. A 04, 3759–3777 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  13. MacKay, N.: The fusion of R-matrices using the Birman–Wenzl–Murakami algebra. J. Math. Phys. 33, 1529–1538 (1992)

    Article  MathSciNet  Google Scholar 

  14. Pearce, P., Zhou, Y.-K.: Fusion of ADE lattice models. Int. J. Mod. Phys. B 8, 3531–3577 (1994). arXiv:hep-th/9405019

    Article  MATH  Google Scholar 

  15. Yue, R-H: Integrable high-spin chain related to the elliptic solution of the Yang-Baxter equation. J. Phys. A 27, 1633–1644 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zabrodin, A: Discrete Hirota’s equation in quantum integrable models. Int. J. Mod. Phys. B 11, 3125–3158 (1997). arXiv:hep-th/9610039

    Article  MathSciNet  MATH  Google Scholar 

  17. Molev, A: On the fusion procedure for the symmetric group. Reports Math. Phys. 61, 181–188 (2008). arXiv:math/0612207

    Article  MathSciNet  MATH  Google Scholar 

  18. Goodman, R., Wallach, R: Representations and Invariants of the Classical Groups. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  19. Weyl, H.: The Classical Groups, Their Invariants and Representations. Princeton University Press, Princeton (1946)

    MATH  Google Scholar 

  20. Jucys, A: On the Young operators of the symmetric group. Liet. Fiz. Rinkinys 6, 163–180 (1966)

    MathSciNet  Google Scholar 

  21. Cherednik, I: On special bases of irreducible finite-dimensional representations of the degenerate affine Hecke algebra. Funct. Anal. Appl. 20, 87–89 (1986)

    Article  Google Scholar 

  22. Nazarov, M.: Yangians and Capelli identities. In: Olshanski, G. I. (ed.) Kirillov’s Seminar on Representation Theory. Amer. Math. Soc. Transl. 181. arXiv:q-alg/9601027, pp 139–163. Amer. Math. Soc., Providence (1998)

    Google Scholar 

  23. Nazarov, M: Mixed hook-length formula for degenerate affine Hecke algebras. Lect. Notes Math. 1815, 223–236 (2003). arXiv:math/9906148

    Article  MathSciNet  MATH  Google Scholar 

  24. Nazarov, M: A mixed hook-length formula for affine Hecke algebras. Eur. J. Comb. 25, 1345–1376 (2004). arXiv:math/0307091

    Article  MathSciNet  MATH  Google Scholar 

  25. Berele, A., Regev, A.: Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras. Adv. Math. 64, 118–175 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sergeev, A.: Representations of the Lie superalgebras \(\mathfrak {gl}(n, m)\) and Q(n) on the space of tensors. Funct. Anal. Appl. 18, 80–81 (1984)

    Article  MathSciNet  Google Scholar 

  27. Isaev, A., Molev, A., Os’kin, A.: On the idempotents of Hecke algebras. Lett. Math. Phys. 85, 79–90 (2008). arXiv:0804.4214

    Article  MathSciNet  MATH  Google Scholar 

  28. Jimbo, M.: A q-Analogue of \(U(\mathfrak {gl}(N+1))\), Hecke algebra, and the Yang–Baxter equation. Lett. Math. Phys. 11, 247–252 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  29. Mitsuhashi, H: Schur–Weyl reciprocity between the quantum superalgebra and the Iwahori–Hecke algebra. Algebr. Represent. Theory 9, 309–322 (2006). arXiv:math/0506156

    Article  MathSciNet  MATH  Google Scholar 

  30. Moon, D.: Highest weight vectors of irreducible representations of the quantum superalgebra U q (g l(m,n)). J. Korean Math. Soc. 40(1), 1–28 (2003). arXiv:math/0105204

    Article  MathSciNet  MATH  Google Scholar 

  31. Isaev, A., Molev, A.: Fusion procedure for the Brauer algebra. Algebra i Analiz 22(3), 142–154 (2010). arXiv:0812.4113

    MathSciNet  MATH  Google Scholar 

  32. Isaev, A., Molev, A., Ogievetsky, O.: A new fusion procedure for the Brauer algebra and evaluation homomorphisms. Int. Math. Res. Not. (11) (2012), 2571–2606. arXiv:1101.1336

  33. Isaev, A., Molev, A., Ogievetsky, O.: Idempotents for Birman-Murakami-Wenzl algebras and reflection equation. Adv. Theor. Math. Phys. 18(1), 1–25 (2011). arXiv:2502.1111

    Article  MathSciNet  MATH  Google Scholar 

  34. Ogievetsky, O., Poulain ’Andecy, L.: Fusion procedure for Coxeter groups of type B and complex reflection groups G(m, 1,n). Proc. Amer. Math. Soc. 142, 2929–2941 (2014). arXiv:1111.6293

    Article  MathSciNet  MATH  Google Scholar 

  35. Ogievetsky, O., Poulain d’Andecy, L.: Fusion procedure for cyclotomic Hecke algebras. SIGMA 10, 039 (2014). 13 pages. arXiv:1301.4237

    MathSciNet  MATH  Google Scholar 

  36. Poulain d’Andecy, L: Fusion procedure for wreath products of finite groups by the symmetric group. Algebr. Represent. Theory 17 (13), 809–830 (2014). arXiv:1301.4399v2

    Article  MathSciNet  MATH  Google Scholar 

  37. Nazarov, M., Tarasov, V.: On irreducibility of tensor products of Yangian modules associated with skew Young diagrams. Duke Math. J. 112, 342–378 (2002). arXiv:math/0012039

    MathSciNet  MATH  Google Scholar 

  38. Nazarov, M: Rational representations of Yangians associated with skew Young diagrams. Math. Z. 247, 21–63 (2004). arXiv:math/0303014

    Article  MathSciNet  MATH  Google Scholar 

  39. El Turkey, H., Kujawa, J.: Presenting Schur superalgebra. Pac. J. Math. 262 (2), 285–316 (2013). arXiv:1209.6327

    Article  MathSciNet  MATH  Google Scholar 

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  1. Université de Reims Champagne-Ardenne, UFR Sciences exactes et naturelles, Laboratoire de Mathématiques EA 4535, Moulin de la Housse BP 1039, 51100, Reims, France

    L. Poulain d’Andecy

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  1. L. Poulain d’Andecy
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d’Andecy, L.P. Fusion Formulas and Fusion Procedure for the Yang-Baxter Equation. Algebr Represent Theor 20, 1379–1414 (2017). https://doi.org/10.1007/s10468-017-9692-1

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