Abstract
We consider a quantum integrable inhomogeneous model based on the Brauer algebra B(1) and discuss the properties of its ground state eigenvector. In particular we derive various sum rules, and show how some of its entries are related to multidegrees of algebraic varieties.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Pearce P.A., Rittenberg V., de Gier J., Nienhuis B.: Temperley–Lieb Stochastic Processes. J. Phys. A35, L661–L668 (2002)
Razumov A.V., Stroganov Yu.G.: Spin chains and combinatorics. J. Phys. A34, 3185 (2001)
Batchelor M.T., de Gier J., Nienhuis B.: The quantum symmetric XXZ chain at Δ=-1/2, alternating sign matrices and plane partitions. J. Phys. A34, L265–L270 (2001)
Razumov A.V., Stroganov Yu.G.: Combinatorial nature of ground state vector of O(1) loop model. Theor. Math. Phys. 138, 333-337 (2004); Teor. Mat. Fiz. 138, 395–400 (2004)
Di Francesco P., Zinn-Justin P.: Around the Razumov–Stroganov conjecture: proof of a multi-parameter sum rule. E. J. Combi. 12(1), R6 (2005)
De Gier J., Nienhuis B.: Brauer loops and the commuting variety. JSTAT P01006 (2005)
Martins M.J., Ramos P.B.: The Algebraic Bethe Ansatz for rational braid-monoid lattice models. Nucl. Phys. B500, 579–620 (1997)
Martins M.J., Nienhuis B., Rietman R.: An Intersecting Loop Model as a Solvable Super Spin Chain. Phys. Rev. Lett. 81 504–507 (1998)
Knutson A: Some schemes related to the commuting variety. http://arxiv.org/list/math.AG/0306275, 2003
Gaudin M.: La fonction d'onde de Bethe. Paris: Masson 1997
Lascoux A., Leclerc B., Thibon J.-Y.: Twisted action of the symmetric group on the cohomology of the flag manifold. Banach Center Publications, Vol. 36, 1996, pp. 111–124
Knutson A., Miller E.: Gröbner geometry of Schubert polynomials. Ann. Math. 161, no. 3, 1245–1318 (2003)
Lindström B.: On the vector representations of induced matroids. Bull. London Math. Soc. 5 85–90 (1973); Gessel I. M., Viennot X.: Binomial determinants, paths and hook formulae. Adv. Math. 58 300–321 (1985)
Cherednik I.: Double affine Hecke algebras, Knizhnik–Zamolodchikov equations, and McDonald's operators. IMRN (Duke Math. Jour.) 9, 171–180 (1992)
Fomin S., Kirillov A.N.: The Yang–Baxter equation, symmetric functions and Schubert polynomials. Discrete Math. 153, 123–143 (1996); The Yang–Baxter equation, symmetric functions and Grothendieck polynomials. http://arxiv.org/list/hep-th/9306005, 1993
Di Francesco P.: A refined Razumov–Stroganov conjecture. JSTAT, P08009 (2004)
Di Francesco P.: A refined Razumov–Stroganov conjecture II. JSTAT, P11004 (2004)
Author information
Authors and Affiliations
Additional information
Communicated by L. Takhtajan
Rights and permissions
About this article
Cite this article
Francesco, P., Zinn-Justin, P. Inhomogenous Model of Crossing Loops and Multidegrees of Some Algebraic Varieties. Commun. Math. Phys. 262, 459–487 (2006). https://doi.org/10.1007/s00220-005-1476-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-005-1476-5