Quantum inverse scattering method and generalizations of symplectic Schur functions and Whittaker functions. (English) Zbl 1487.17031
Summary: We introduce generalizations of type \(C\) and \(B\) ice models which were recently introduced by Ivanov and Brubaker-Bump-Chinta-Gunnells, and study in detail the partition functions of the models by using the quantum inverse scattering method. We compute the explicit forms of the wavefunctions and their duals by using the Izergin-Korepin technique, which can be applied to both models. For type \(C\) ice, we show the wavefunctions are expressed using generalizations of the symplectic Schur functions. This gives a generalization of the correspondence by Ivanov. For type \(B\) ice, we prove that the exact expressions of the wavefunctions are given by generalizations of the Whittaker functions introduced by Bump-Friedberg-Hoffstein. The special case is the correspondence conjectured by Brubaker-Bump-Chinta-Gunnells. We also show the factorized forms for the domain wall boundary partition functions for both models. As a consequence of the studies of the partition functions, we obtain dual Cauchy formulas for the generalized symplectic Schur functions and the generalized Whittaker functions.
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 17B80 | Applications of Lie algebras and superalgebras to integrable systems |
| 05E05 | Symmetric functions and generalizations |
| 33C80 | Connections of hypergeometric functions with groups and algebras, and related topics |
| 33E05 | Elliptic functions and integrals |
| 81U40 | Inverse scattering problems in quantum theory |
Keywords:
quantum integrable models; Whittaker functions; symmetric functions; representation theory; quantum dynamical and integrable systems; quantum groupsReferences:
| [1] | Baxter, R. J., Exactly Solved Models in Statistical Mechanics (1982), Academic Press: Academic Press London · Zbl 0538.60093 |
| [2] | Betea, D.; Wheeler, M., Refined Cauchy and littlewood identities, plane partitions and symmetry classes of alternating sign matrices, J. Comb. Theory Ser. A, 137, 126 (2016) · Zbl 1325.05179 |
| [3] | Betea, D.; Wheeler, M.; Zinn-Justin, P., Refined Cauchy/Littlewood identities and six-vertex model partition functions: II. Proofs and new conjectures, J. Algebraic Combin., 42, 555 (2015) · Zbl 1319.05136 |
| [4] | Bogoliubov, N. M., Boxed plane partitions as an exactly solvable boson model, J. Phys. A: Math. Gen., 38, 9415 (2005) · Zbl 1087.82007 |
| [5] | Borodin, A., On a family of symmetric rational functions, Adv. Math., 306, 973 (2017) · Zbl 1355.05250 |
| [6] | Borodin, A.; Petrov, L., Higher spin six vertex model and symmetric rational functions, Sel. Math. New Ser., 24, 1 (2018) · Zbl 1405.60141 |
| [7] | Bressoud, D., Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, MAA Spectrum (1999), Mathematical Association of America: Mathematical Association of America Washington, DC · Zbl 0944.05001 |
| [8] | Brubaker, B.; Buciumas, V.; Bump, D., A Yang-Baxter equation for metaplectic ice, Commun. Number Theory Phys., 13, 101 (2019) · Zbl 1454.11096 |
| [9] | Brubaker, B.; Buciumas, V.; Bump, D.; Friedberg, S., Hecke modules from metaplectic ice, Sel. Math. New Ser., 24, 2523 (2018) · Zbl 1466.20002 |
| [10] | B. Brubaker, V. Buciumas, D. Bump, N. Gray, Duality for metaplectic ice, Appendix to (48). · Zbl 1454.11096 |
| [11] | B. Brubaker, V. Buciumas, D. Bump, H. Gustafsson, Vertex operators, solvable lattice models and metaplectic Whittaker functions, arXiv:1806.07776. · Zbl 1456.82097 |
| [12] | Brubaker, B.; Bump, D.; Chinta, G.; Gunnells, P. E.; Bump, D.; Friedberg, S.; Goldfield, D., Metaplectic functions and crystals of type b, Multiple Dirichlet Series, L-Functions, and Automorphic Forms, vol. 300, 93-118 (2012) · Zbl 1370.11061 |
| [13] | Brubaker, B.; Bump, D.; Friedberg, S., Schur polynomials and the Yang-Baxter equation, Comm. Math. Phys., 308, 281 (2011) · Zbl 1232.05234 |
| [14] | Brubaker, B.; Schultz, A., The 6-vertex model and deformations of the Weyl character formula, J. Algebraic Combin., 42, 917 (2015) · Zbl 1339.82002 |
| [15] | Bump, D.; Friedberg, S.; Hoffstein, J., \(p\)-Adic Whittaker functions on the metaplectic group, Duke Math. J., 63, 379 (1991) · Zbl 0758.22009 |
| [16] | Bump, D.; Gamburd, A., On the averages of characteristic polynomials from classical groups, Comm. Math. Phys., 265, 227 (2006) · Zbl 1107.60004 |
| [17] | Bump, D.; McNamara, P.; Nakasuji, M., Factorial Schur functions and the Yang-Baxter equation, Comm. Math. Univ. Sancti Pauli., 63, 23 (2014) · Zbl 1312.05140 |
| [18] | Caradoc, A. D.; Foda, O.; Wheeler, M.; Zuparic, M., On the trigonometric felderhof model with domain wall boundary conditions, J. Stat. Mech., 2007, P03010 (2007) |
| [19] | Crampe, N.; Mallick, K.; Ragoucy, E.; Vanicat, M., Inhomogeneous discrete-time exclusion processes, J. Phys. A: Math. Theor., 48, Article 484002 pp. (2015) · Zbl 1335.82019 |
| [20] | Deguchi, T.; Akutsu, Y., Colored vertex models, colored IRF models and invariants of trivalent colored graphs, J. Phys. Soc. Japan., 62, 19 (1993) · Zbl 0972.82522 |
| [21] | Deguchi, T.; Fujii, A., IRF models associated with representations of the Lie superalgebras \(g l ( m | n )\) and \(s l ( m | n )\), Mod. Phys. Lett. A., 6, 3413 (1991) · Zbl 1020.82547 |
| [22] | Deguchi, T.; Martin, P., An algebraic approach to vertex models and transfer matrix spectra, Int. J. Mod. Phys. A., 7, suupp01a, 165 (1992) · Zbl 0925.17006 |
| [23] | van Diejen, J. F.; Emsiz, E., Orthogonality of Bethe Ansatz eigenfunctions for the Laplacian on a hyperoctahedral Weyl alcove, Comm. Math. Phys., 350, 1017 (2017) · Zbl 1360.82029 |
| [24] | Drinfeld, V., Hopf algebras and the quantum Yang-Baxter equation, Sov. Math. Dokl., 32, 254 (1985) · Zbl 0588.17015 |
| [25] | Faddeev, L. D.; Sklyanin, E. K.; Takhtajan, L. A., The quantum inverse problem method, Theoret. Math. Phys., 40, 194 (1979), 1 · Zbl 0449.35096 |
| [26] | Felderhof, B., Direct diagonalization of the transfer matrix of the zero-field free-fermion model, Physica, 65, 421 (1973) |
| [27] | Filali, G., Elliptic dynamical reflection algebra and partition function of SOS model with reflecting end, J. Geom. Phys., 61, 1789 (2011) · Zbl 1226.82010 |
| [28] | Filali, G.; Kitanine, N., The partition function of the trigonometric SOS model with a reflecting end, J. Stat. Mech., 2010, L06001 (2010) · Zbl 1456.82256 |
| [29] | Foda, O.; Wheeler, M.; Zuparic, M., Two elliptic height models with factorized domain wall partition functions, J. Stat. Mech., 2008, P02001 (2008) · Zbl 1432.82010 |
| [30] | Galleas, W., Multiple integral representation for the trigonometric SOS model with domain wall boundaries, Nuclear Phys. B, 858, 117 (2012) · Zbl 1246.82059 |
| [31] | Galleas, W.; Lamers, J., Reflection algebra and functional equations, Nuclear Phys. B, 886, 1003 (2014) · Zbl 1325.82003 |
| [32] | Gorbounov, V.; Korff, J., Quantum integrability and generalised quantum schubert calculus, Adv. Math., 313, 282 (2017) · Zbl 1386.14181 |
| [33] | N. Gray, Metaplectic ice for Cartan type \(C\), arXiv:1709.04971. |
| [34] | Hamel, A.; King, R. C., Symplectic shifted tableaux and deformations of Weyl’s denominator formula for \(s p ( 2 n )\), J. Algebraic Combin., 16, 269 (2002) · Zbl 1037.05048 |
| [35] | Hamel, A.; King, R. C., U-turn alternating sign matrices, symplectic shifted tableaux and their weighted enumeration, J. Algebraic Combin., 21, 395 (2005) · Zbl 1066.05012 |
| [36] | Hamel, A.; King, R. C., Bijective proof of a symplectic dual pair identity, SIAM J. Discrete Math., 25, 539 (2011) · Zbl 1290.05154 |
| [37] | Hamel, A. M.; King, R. C., Tokuyamafs identity for factorial Schur \(P\) and \(Q\) functions, Electron. J. Comb., 22 (2015), Paper P2.42 · Zbl 1319.05137 |
| [38] | Hasegawa, K., Spin module versions of Weyl’s reciprocity theorem for classical Kac-Moody Lie algebras - An application to branching rule duality, Publ. RIMS, 25, 741 (1989) · Zbl 0699.17020 |
| [39] | Ivanov, D., Part I, Symplectic Ice, Part II, Global and Local Kubota Symbols (2010), Stanford University: Stanford University USA, (PhD. Thesis) |
| [40] | Ivanov, D., Symplectic ice, (Bump, D.; Friedberg, S.; Goldfield, D., Multiple Dirichlet Series, L-Functions, and Automorphic Forms. Multiple Dirichlet Series, L-Functions, and Automorphic Forms, Progress in Mathematics, vol. 300 (2012), Birkhauser Boston), 205-222 · Zbl 1264.11042 |
| [41] | Izergin, A., Partition function of the 6-vertex model in a finite volume, Sov. Phys. Dokl., 32, 878 (1987) · Zbl 0875.82015 |
| [42] | Jimbo, M., A \(q\)-difference analogue of \(U ( g )\) and the Yang-Baxter equation, Lett. Math. Phys., 10, 63 (1985) · Zbl 0587.17004 |
| [43] | Jimbo, M.; Miwa, T., On a duality of branching rules for affine Lie algebras, Adv. Stud. Pure Math., 6, 17 (1985) · Zbl 0579.17009 |
| [44] | King, R. C., Branching rules for classical Lie groups using tensor and spinor methods, J. Phys. A: Math. Gen., 8, 429 (1975) · Zbl 0301.20029 |
| [45] | Korepin, V. E., Calculation of norms of bethe wavefunctions, Comm. Math. Phys., 86, 391 (1982) · Zbl 0531.60096 |
| [46] | Korepin, V. E.; Bogoliubov, N. M.; Izergin, A. G., Quantum Inverse Scattering Method and Correlation Functions (1993), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0787.47006 |
| [47] | Korff, C., Quantum cohomology via vicious and osculating walkers, Lett. Math. Phys., 104, 771 (2014) · Zbl 1302.14045 |
| [48] | Korff, C.; Stroppel, C., The \(s l ( n )\)-WZNW fusion ring: a combinatorial construction and a realisation as quotient of quantum cohomology, Adv. Math., 225, 200 (2010) · Zbl 1230.17010 |
| [49] | Kuperberg, G., Another proof of the alternating sign matrix conjecture, Int. Math. Res. Not. IMRN, 3, 139 (1996) · Zbl 0859.05027 |
| [50] | Lamers, J., Integral formula for elliptic SOS models with domain walls and a reflecting end, Nuclear Phys. B, 901, 556 (2015) · Zbl 1332.82016 |
| [51] | Lascoux, A., The 6 vertex model and schubert polynomials, SIGMA, 3, 029 (2007) · Zbl 1138.05073 |
| [52] | P.J. McNamara, Factorial Schur functions via the six-vertex model, arXiv:0910.5288. |
| [53] | Morris, A., Spin representation of a direct sum and a direct product, J. Lond. Math. Soc., 33, 326 (1958) · Zbl 0081.25901 |
| [54] | Motegi, K., Dual wavefunction of the Felderhof model, Lett. Math. Phys., 107, 1235 (2017) · Zbl 1370.05215 |
| [55] | Motegi, K., Dual wavefunction of the symplectic ice, Rep. Math. Phys., 80, 414 (2017) · Zbl 1387.81230 |
| [56] | Motegi, K., Elliptic supersymmetric integrable model and multivariable elliptic functions, Prog. Theor. Exp. Phys., 2017, 123A01 (2017) · Zbl 1523.81152 |
| [57] | Motegi, K., Izergin-korepin analysis on the projected wavefunctions of the generalized free-fermion model, Adv. Math. Phys., 2017, Article 7563781 pp. (2017) · Zbl 1403.82006 |
| [58] | Motegi, K., Symmetric functions and wavefunctions of XXZ-type six-vertex models and elliptic Felderhof models by Izergin-Korepin analysis, J. Math. Phys., 59, Article 053505 pp. (2018) · Zbl 1391.82013 |
| [59] | Motegi, K., Izergin-Korepin analysis on the wavefunctions of the \(U_q ( s l_2 )\) six-vertex model with reflecting end, Ann. l’Inst. Henri Poincaré D (2020), in press · Zbl 1444.05144 |
| [60] | Motegi, K.; Sakai, K., Vertex model, TASEP and grothendieck polynomials, J. Phys. A: Math. Theor., 46, Article 355201 pp. (2013) · Zbl 1278.82042 |
| [61] | Motegi, K.; Sakai, K., \(K\)-theoretic boson-fermion correspondence and melting crystals, J. Phys. A: Math. Theor., 47, Article 445202 pp. (2014) · Zbl 1310.82049 |
| [62] | K. Motegi, K. Sakai, S. Watanabe, Partition functions of integrable lattice models and combinatorics of symmetric polynomials, arXiv:1512.07955. |
| [63] | Murakami, J., The free-fermion model in presence of field related to the quantum group \(U_q ( s l_2 )\) of affine type and the multi-variable Alexander polynomial of links, infinite analysis, Adv. Ser. Math. Phys., 16B, 765 (1991) |
| [64] | Okada, S., Alternating sign matrices and some deformations of Weyl’s denominator formula, J. Algebraic Combin., 2, 155 (1993) · Zbl 0781.15008 |
| [65] | Okado, M., Solvable face models related to the Lie superalgebra \(s l ( m | n )\), Lett. Math. Phys., 22, 39 (1991) · Zbl 0726.17007 |
| [66] | Pakuliak, S.; Rubtsov, V.; Silantyev, A., The SOS model partition function and the elliptic weight functions, J. Phys. A: Math. Theor., 41, Article 295204 pp. (2008) · Zbl 1143.81016 |
| [67] | Perk, J.-H.-H.; Schultz, C. L., New families of commuting transfer matrices in \(q\)-state vertex models, Phys. Lett. A, 84, 407 (1981) |
| [68] | Ribeiro, G. A.P.; Korepin, V. E., Thermodynamic limit of the six-vertex model with reflecting end, J. Phys. A: Math. Theor., 48, Article 045205 pp. (2015) · Zbl 1320.81065 |
| [69] | Rosengren, H., An Izergin-Korepin-type identity for the 8VSOS model, with applications to alternating sign matrices, Adv. Appl. Math., 43, 137 (2009) · Zbl 1173.05300 |
| [70] | Sklyanin, E., Boundary conditions for integrable quantum systems, J. Phys. A: Math. Gen., 21, 2375 (1988) · Zbl 0685.58058 |
| [71] | Tabony, S. J., Deformations of Characters, Metaplectic Whittaker Functions and the Yang-Baxter Equation (2011), Massachusetts Institute of Technology: Massachusetts Institute of Technology USA, (PhD. Thesis) |
| [72] | Takeyama, Y., A deformation of affine Hecke algebra and integrable stochastic particle system, J. Phys. A: Math. Theor., 47, Article 465203 pp. (2014) · Zbl 1310.81083 |
| [73] | Takeyama, Y., On the eigenfunctions for the multi-species \(q\)-Boson system, Funkc. Ekvacioj, 61, 349 (2018) · Zbl 1446.17042 |
| [74] | Terada, I., A robinson-schensted-type correspondence for a dual pair on spinors, J. Comb. Theory Ser. A, 63, 90 (1993) · Zbl 0772.05099 |
| [75] | Tokuyama, T., A generating function of strict Gelfand patterns and some formulas on characters of general linear groups, J. Math. Soc. Japan., 40, 671 (1988) · Zbl 0639.20022 |
| [76] | Tsuchiya, O., Determinant formula for the six-vertex model with reflecting end, J. Math. Phys., 39, 5946 (1998) · Zbl 0938.82007 |
| [77] | Wheeler, M., An Izergin-Korepin procedure for calculating scalar products in the six-vertex model, Nuclear Phys. B, 852, 468 (2011) · Zbl 1229.82056 |
| [78] | Wheeler, M.; Zinn-Justin, P., Refined Cauchy/Littlewood identities and six-vertex model partition functions: III. Deformed bosons, Adv. Math., 299, 543 (2016) · Zbl 1341.05249 |
| [79] | Yamane, H., On defining relations of affine Lie superalgebras and affine quantized universal enveloping superalgebras, Publ. RIMS, 35, 321 (1999) · Zbl 0987.17007 |
| [80] | Yang, W.-L.; Chen, X.; Feng, J.; Hao, K.; Shi, K.-J.; Sun, C.-Y.; Yang, Z.-Y.; Zhang, Y.-Z., Domain wall partition function of the eight-vertex model with a non-diagonal reflecting end, Nuclear Phys. B, 847, 367 (2011) · Zbl 1208.82014 |
| [81] | Yang, W.-L.; Chen, X.; Feng, J.; Hao, K.; Wu, K.; Yang, Z.-Y.; Zhang, Y.-Z., Scalar products of the open XYZ chain with non-diagonal boundary terms, Nuclear Phys. B, 848, 523 (2011) · Zbl 1215.82013 |
| [82] | Yang, W.-L.; Zhang, Y.-Z., Partition function of the eight-vertex model with domain wall boundary condition, J. Math. Phys., 50, Article 083518 pp. (2009) · Zbl 1298.82019 |
| [83] | Zhao, S.-Y.; Yang, W.-L.; Zhang, Y.-Z., Determinant representation of correlation functions for the \(U_q ( g l ( 1 | 1 ) )\) free fermion model, J. Math. Phys., 47, Article 013302 pp. (2006) · Zbl 1111.82016 |
| [84] | Zhao, S.-Y.; Zhang, Y.-Z., Supersymmetric vertex models with domain wall boundary conditions, J. Math. Phys., 48, Article 023504 pp. (2007) · Zbl 1121.81115 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
