On the Eigenfunctions for the Multi-species q-Boson System
In a previous paper a multi-species version of the q-Boson stochastic particle system is introduced and the eigenfunctions of its backward generator are constructed by using a representation of the Hecke algebra. In this article we prove a formula which expresses the eigenfunctions by means of the q...
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| Published in | Funkcialaj Ekvacioj Vol. 61; no. 3; pp. 349 - 376 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Tokyo
Division of Functional Equations, The Mathematical Society of Japan
2018
Japan Science and Technology Agency |
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| Online Access | Get full text |
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| Abstract | In a previous paper a multi-species version of the q-Boson stochastic particle system is introduced and the eigenfunctions of its backward generator are constructed by using a representation of the Hecke algebra. In this article we prove a formula which expresses the eigenfunctions by means of the q-deformed bosonic operators, which are constructed from the L-operator of higher rank found in the recent work by Garbali, de Gier and Wheeler. The L-operator is obtained from the universal R-matrix of the quantum affine algebra of type Ar(1) by the use of the q-oscillator representation. Thus our formula may be regarded as a bridge between two approaches to studying integrable stochastic systems by means of the quantum affine algebra and the affine Hecke algebra. |
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| AbstractList | In a previous paper a multi-species version of the q-Boson stochastic particle system is introduced and the eigenfunctions of its backward generator are constructed by using a representation of the Hecke algebra. In this article we prove a formula which expresses the eigenfunctions by means of the q-deformed bosonic operators, which are constructed from the L-operator of higher rank found in the recent work by Garbali, de Gier and Wheeler. The L-operator is obtained from the universal R-matrix of the quantum affine algebra of type Ar(1) by the use of the q-oscillator representation. Thus our formula may be regarded as a bridge between two approaches to studying integrable stochastic systems by means of the quantum affine algebra and the affine Hecke algebra. |
| Author | Takeyama, Yoshihiro |
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| Copyright | 2018 by the Division of Functional Equations, The Mathematical Society of Japan Copyright Japan Science and Technology Agency 2018 |
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| References | [4] Garbali, A., de Gier, A. and Wheeler, M., A new generalisation of Macdonald polynomials, preprint, arXiv:1605.07200. [5] Jimbo, M., A q-analogue of U(gl(N + 1)), Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys., 11 (1986), no. 3, 247-252. [9] Sasamoto, T. and Wadati, M., Exact results for one-dimensional totally asymmetric diffusion models, J. Phys. A, 31 (1998), no. 28, 6057-6071. [6] Kuan, J., A multi-species ASEP(q, j) and q-TAZRP with stochastic duality, preprint, arXiv:1605.00691. [1] Borodin, A., On a family of symmetric rational functions, preprint, arXiv:1410.0976. [2] Borodin, A., Corwin, I., Petrov, L. and Sasamoto, T., Spectral theory for the q-Boson particle system, Compos. Math., 151 (2015), no. 1, 1-67. [7] Kuniba, A., Mangazeev, V. V., Maruyama, S. and Okado, M., Stochastic R matrix for Uq(An(1)), preprint, arXiv:1604.08304. [8] Motegi, K. and Sakai, K., K-theoretic boson-fermion correspondence and melting crystals, J. Phys. A, 47 (2014), no. 44, 445202. [10] Takeyama, Y., Algebraic construction of multi-species q-Boson system, preprint, arXiv:1507.02033. [3] Fomin, S. and Kirillov, A., Grothendieck polynomials and the Yang-Baxter equation, Formal power series and algebraic combinatorics (DIMACS), 183-189. [11] Tracy, C. and Widom, H., On the asymmetric simple exclusion process with multiple species, J. Stat. Phys., 150 (2013), no. 3, 457-470. 11 1 2 3 4 5 6 7 8 9 10 |
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| SubjectTerms | Affine Hecke algebra Algebra Deformation Eigenvectors Integrable system Operators (mathematics) Quantum group Representations Stochastic process Stochastic systems |
| Title | On the Eigenfunctions for the Multi-species q-Boson System |
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