Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 078, 20 pages      math.QA/0604158      https://doi.org/10.3842/SIGMA.2007.078

Paths and Tableaux Descriptions of Jacobi-Trudi Determinant Associated with Quantum Affine Algebra of Type Cn

Wakako Nakai and Tomoki Nakanishi
Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan

Received May 03, 2007, in final form July 04, 2007; Published online July 18, 2007

Abstract
We study the Jacobi-Trudi-type determinant which is conjectured to be the q-character of a certain, in many cases irreducible, finite-dimensional representation of the quantum affine algebra of type Cn. Like the Dn case studied by the authors recently, applying the Gessel-Viennot path method with an additional involution and a deformation of paths, we obtain an expression by a positive sum over a set of tuples of paths, which is naturally translated into the one over a set of tableaux on a skew diagram.

Key words: quantum group; q-character; lattice path; Young tableau.

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References

  1. Bazhanov V.V., Reshetikhin N., Restricted solid-on-solid models connected with simply laced algebras and conformal field theory, J. Phys. A: Math. Gen. 23 (1990), 1477-1492.
  2. Chari V., Moura A., Characters of fundamental representations of quantum affine algebras, Acta Appl. Math. 90 (2006), 43-63, math.RT/0503020.
  3. Chari V., Pressley A., Quantum affine algebras, Comm. Math. Phys. 142 (1991), 261-283.
  4. Drinfel'd V., Hopf algebras and the quantum Yang-Baxter equation, Soviet. Math. Dokl. 32 (1985), 254-258.
  5. Drinfel'd V., A new realization of Yangians and quantized affine algebras, Soviet. Math. Dokl. 36 (1988), 212-216.
  6. Frenkel E., Mukhin E., Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras, Comm. Math. Phys. 216 (2001), 23-57, math.QA/9911112.
  7. Frenkel E., Reshetikhin N., The q-characters of representations of quantum affine algebras and deformations of W-algebras, Contemp. Math. 248 (1999), 163-205, math.QA/9810055.
  8. Gessel I., Viennot G., Binomial determinants, paths, and hook length formulae, Adv. in Math. 58 (1985), 300-321.
  9. Hernandez D., The Kirillov-Reshetikhin conjecture and solutions of T-systems, J. Reine Angew. Math. 596 (2006), 63-87, math.QA/0501202.
  10. Hernandez D., On minimal affinizations of representations of quantum groups, Comm. Math. Phys., to appear, math.QA/0607527.
  11. Jimbo M., A q-difference analogue of U(g) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63-69.
  12. Kashiwara M., Nakashima T., Crystal graphs for representations of the q-analogue of classical Lie algebras, J. Algebra 165 (1994), 295-345.
  13. Kirillov A. N., Reshetikhin N., Representations of Yangians and multiplicities of the inclusion of the irreducible components of the tensor' product of representations of simple Lie algebras, J. Sov. Math. 52 (1990), 3156-3164.
  14. Kuniba A., Nakanishi T., The Bethe equation at q = 0, the Möbius inversion formula, and weight multiplicities. II. The Xn case, J. Algebra 251 (2002), 577-618, math.QA/0008047.
  15. Kuniba A., Nakanishi T., Suzuki J., Functional relations in solvable lattice models: I. Functional relations and representation theory, Internat. J. Modern Phys. A 9 (1994), 5215-5266, hep-th/9309137.
  16. Kuniba A., Nakamura S., Hirota R., Pfaffian and determinant solutions to a discretized Toda equation for Br, Cr, and Dr, J. Phys. A: Math. Gen. 29 (1996), 1759-1766, hep-th/9509039.
  17. Kuniba A., Ohta Y., Suzuki J., Quantum Jacobi-Trudi and Giambelli formulae for Uq(B(1)r) from the analytic Bethe ansatz, J. Phys. A: Math. Gen. 28 (1995), 6211-6226, hep-th/9506167.
  18. Kuniba A., Okado M., Suzuki J., Yamada Y., Difference L operators related to q-characters, J. Phys. A: Math. Gen. 35 (2002), 1415-1435, math.QA/0109140.
  19. Kuniba A., Suzuki J., Analytic Bethe ansatz for fundamental representations of Yangians, Comm. Math. Phys. 173 (1995), 225-264, hep-th/9406180.
  20. Macdonald I. G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Univ. Press, Oxford, 1995.
  21. Nakajima H., t-analogs of q-characters of quantum affine algebras of type An and Dn, Contemp. Math. 325 (2003), 141-160, math.QA/0204184.
  22. Nakajima H., t-analogs of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras, Represent. Theory 7 (2003), 259-274, math.QA/0204185.
  23. Nakai W., Nakanishi T., Paths, tableaux and q-characters of quantum affine algebras: the Cn case, J. Phys. A: Math. Phys. 39 (2006), 2083-2115, math.QA/0502041.
  24. Nakai W., Nakanishi T., Paths and tableaux descriptions of Jacobi-Trudi determinant associated with quantum affine algebra of type Dn, J. Algebraic Combin., to appear, math.QA/0603160.
  25. Okado M., Schilling A., Shimozono M., Virtual crystals and fermionic formulas of type Dn+1(2), A2n(2), and Cn(1), Represent. Theory 7 (2003), 101-163, math.QA/0105017.
  26. Sagan B., The symmetric group, 2nd ed., Springer, 2000.
  27. Tsuboi Z., Difference L operators and a Casorati determinant solution to the T-system for twisted quantum affine algebras, J. Phys. A: Math. Gen. 35 (2002), 4363-4373.
  28. Tsuboi Z., Kuniba A., Solutions of a discretized Toda field equation for Dr from analytic Bethe ansatz, J. Phys. A: Math. Phys. 29 (1996), 7785-7796, hep-th/9608002.

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