The Brauer loop scheme and orbital varieties. (English) Zbl 1326.81094
Summary: A. Joseph invented multidegrees in his article [J. Algebra 88, 238–278 (1984; Zbl 0539.17006)] to study orbital varieties, which are the components of an orbital scheme, itself constructed by intersecting a nilpotent orbit with a Borel subalgebra. Their multidegrees are known as Joseph polynomials, and these polynomials give a basis of a (Springer) representation of the Weyl group. In the case of the nilpotent orbit \(\{M^2 = 0\}\), the orbital varieties can be indexed by noncrossing chord diagrams in the disk. {
}
In this paper we study the normal cone to the orbital scheme inside this nilpotent orbit \(\{M^2 = 0\}\). This gives a better-motivated construction of the Brauer loop scheme we introduced in [Adv. Math. 214, No. 1, 40–77 (2007; Zbl 1193.14068)], whose components are indexed by all chord diagrams (now possibly with crossings) in the disk.
The multidegrees of its components, the Brauer loop varieties, were shown to reproduce the ground state of the Brauer loop model in statistical mechanics [P. Di Francesco and the second author, Commun. Math. Phys. 262, No. 2, 459–487 (2006; Zbl 1113.82026)]. Here, we reformulate and slightly generalize these multidegrees in order to express them as solutions of the rational quantum Knizhnik-Zamolodchikov equation associated to the Brauer algebra. In particular, the vector of the multidegrees satisfies two sets of equations, corresponding to the \(e_i\) and \(f_i\) generators of the Brauer algebra. The proof of the analogous statement in Knutson and Zinn-Justin (loc. cit.) was slightly roundabout; we verified the \(f_i\) equation using the geometry of multidegrees, and used algebraic results of Di Francesco and Zinn-Justin (loc. cit.) to show that it implied the \(e_i\) equation. We describe here the geometric meaning of both \(e_i\) and \(f_i\) equations in our slightly extended setting. { } We also describe the corresponding actions at the level of orbital varieties: while only the \(e_i\) equations make sense directly on the Joseph polynomials, the \(f_i\) equations also appear if one introduces a broader class of varieties. We explain the connection of the latter with matrix Schubert varieties.
The multidegrees of its components, the Brauer loop varieties, were shown to reproduce the ground state of the Brauer loop model in statistical mechanics [P. Di Francesco and the second author, Commun. Math. Phys. 262, No. 2, 459–487 (2006; Zbl 1113.82026)]. Here, we reformulate and slightly generalize these multidegrees in order to express them as solutions of the rational quantum Knizhnik-Zamolodchikov equation associated to the Brauer algebra. In particular, the vector of the multidegrees satisfies two sets of equations, corresponding to the \(e_i\) and \(f_i\) generators of the Brauer algebra. The proof of the analogous statement in Knutson and Zinn-Justin (loc. cit.) was slightly roundabout; we verified the \(f_i\) equation using the geometry of multidegrees, and used algebraic results of Di Francesco and Zinn-Justin (loc. cit.) to show that it implied the \(e_i\) equation. We describe here the geometric meaning of both \(e_i\) and \(f_i\) equations in our slightly extended setting. { } We also describe the corresponding actions at the level of orbital varieties: while only the \(e_i\) equations make sense directly on the Joseph polynomials, the \(f_i\) equations also appear if one introduces a broader class of varieties. We explain the connection of the latter with matrix Schubert varieties.
MSC:
| 81R15 | Operator algebra methods applied to problems in quantum theory |
| 58D19 | Group actions and symmetry properties |
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
| 17B08 | Coadjoint orbits; nilpotent varieties |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
References:
| [1] | Di Francesco, P.; Zinn-Justin, P., Inhomogeneous model of crossing loops and multidegrees of some algebraic varieties, Comm. Math. Phys., 262, 459-487 (2006) · Zbl 1113.82026 |
| [2] | Knutson, A.; Zinn-Justin, P., A scheme related to Brauer loops, Adv. Math., 214, 40-77 (2007) · Zbl 1193.14068 |
| [3] | Di Francesco, P.; Zinn-Justin, P., Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials, J. Phys. A, 38, L815-L822 (2005) · Zbl 1078.81037 |
| [4] | Rothbach, B., Equidimensionality of the Brauer loop scheme, Electron. J. Combin., 17, R75 (2010) · Zbl 1196.14045 |
| [5] | Spaltenstein, N., The fixed point set of a unipotent transformation on the flag manifold, Nederl. Akad. Wetensch. Proc. Ser. A Indag. Math., 38, 5, 452-456 (1976) · Zbl 0343.20029 |
| [6] | Stanley, R., Enumerative Combinatorics 2 (1997), Cambridge University Press, Catalan interpretations in http://www-math.mit.edu/ rstan/ec/catalan.pdf · Zbl 0889.05001 |
| [7] | Melnikov, A., \(B\)-orbits in solutions to the equation \(X^2 = 0\) in triangular matrices, J. Algebra, 223, 101-108 (2000) · Zbl 0963.15013 |
| [8] | de Gier, J.; Nienhuis, B., Brauer loops and the commuting variety, J. Stat. Mech., P01006 (2005) · Zbl 1072.82585 |
| [9] | Brauer, R., On algebras which are connected with the semisimple continuous groups, Ann. of Math., 38, 857-872 (1937) · JFM 63.0873.02 |
| [10] | Jimbo, M., Introduction to the Yang-Baxter equation, Internat. J. Modern Phys. A, 4, 15, 3759-3777 (1989) · Zbl 0697.35131 |
| [11] | Di Francesco, P.; Zinn-Justin, P., Around the Razumov-Stroganov conjecture: proof of a multi-parameter sum rule, Electron. J. Combin., 12, 1, R6 (2005) · Zbl 1108.05013 |
| [12] | Smirnov, F. A., A general formula for soliton form factors in the quantum sine-Gordon model, J. Phys. A, 19, L575-L578 (1986) · Zbl 0617.58048 |
| [13] | Frenkel, I. B.; Reshetikhin, N. Yu., Quantum affine algebras and holonomic difference equations, Comm. Math. Phys., 146, 1-60 (1992) · Zbl 0760.17006 |
| [14] | Miller, E.; Sturmfels, B., (Combinatorial Commutative Algebra. Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 227 (2004), Springer-Verlag: Springer-Verlag New York) · Zbl 1090.13001 |
| [15] | Joseph, A., Orbital varieties, Goldie rank polynomials and unitary highest weight modules. Algebraic and analytic methods in representation theory (Sønderborg), Perspect. Math., 17, 53-98 (1994) · Zbl 0914.17006 |
| [16] | Joseph, A., On the variety of a highest weight module, J. Algebra, 88, 1, 238-278 (1984) · Zbl 0539.17006 |
| [17] | Rimányi, R.; Tarasov, V.; Varchenko, A.; Zinn-Justin, P., Extended Joseph polynomials, quantized conformal blocks, and a \(q\)-Selberg type integral, J. Geom. Phys., 62, 11, 2188-2207 (2012) · Zbl 1254.22013 |
| [19] | Melnikov, A., Description of \(B\)-orbit closure of order 2 in upper triangular matrices, Transform. Groups, 11, 2, 217-247 (2006) · Zbl 1196.14040 |
| [20] | Lascoux, A., Transition on Grothendieck Polynomials, Physics and Combinatorics, 2000 (Nagoya.), 164-179 (2001), World Sci. Publishing: World Sci. Publishing River Edge, NJ · Zbl 1052.14059 |
| [21] | Knutson, A.; Miller, E.; Yong, A., Gröbner geometry of vertex decompositions and of flagged tableaux, J. für die Reine und Angew. Math. (Crelle’s J.), 2009.630, 1-31 (2009) · Zbl 1169.14033 |
| [22] | Knutson, A.; Miller, E., Gröbner geometry of Schubert polynomials, Ann. of Math., 161, 3, 1245-1318 (2005) · Zbl 1089.14007 |
| [23] | Martins, M. J.; Ramos, P. B., The algebraic Bethe Ansatz for rational braid-monoid lattice models, Nuclear Phys., B500, 579-620 (1997) · Zbl 0933.82016 |
| [24] | Martins, M. J.; Nienhuis, B.; Rietman, R., An intersecting loop model as a solvable super spin chain, Phys. Rev. Lett., 81, 504-507 (1998) · Zbl 0944.82006 |
| [25] | Borho, W.; Brylinski, J.-L.; MacPherson, R., (Nilpotent Orbits, Primitive Ideals, and Characteristic Classes, A Geometric Perspective in Ring Theory. Nilpotent Orbits, Primitive Ideals, and Characteristic Classes, A Geometric Perspective in Ring Theory, Progress in Mathematics, vol. 78 (1989), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA) · Zbl 0697.17006 |
| [26] | Chriss, N.; Ginzburg, V., Representation Theory and Complex Geometry (1997), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA · Zbl 0879.22001 |
| [27] | Nakajima, H., (Lectures on Hilbert Schemes of Points on Surfaces. Lectures on Hilbert Schemes of Points on Surfaces, University Lecture Series, vol. 18 (1999), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 0949.14001 |
| [29] | Fomin, S.; Kirillov, A. N., The Yang-Baxter equation, symmetric functions and Schubert polynomials, Discrete Math., 153, 123-143 (1996) · Zbl 0852.05078 |
| [30] | Ehrenborg, R.; Readdy, M., Juggling and applications to \(q\)-analogues, Discrete Math., 157, 1-3, 107-125 (1996) · Zbl 0859.05010 |
| [31] | Hotta, R., On Joseph’s construction of Weyl group representations, Tohoku Math. J., 36, 49-74 (1984) · Zbl 0545.20029 |
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.
