The Brauer loop scheme and orbital varieties

A. Joseph invented multidegrees in Joseph (1984) 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 {M2=0}{M2=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 {M2=0}{M2=0}. This gives a better-motivated construction of the Brauer loop scheme we introduced in Knutson and Zinn-Justin (2007), 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 (Di Francesco and Zinn-Justin, 2006). 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 eiei and fifi generators of the Brauer algebra. The proof of the analogous statement in Knutson and Zinn-Justin (2007) was slightly roundabout; we verified the fifi equation using the geometry of multidegrees, and used algebraic results of Di Francesco and Zinn-Justin (2006) to show that it implied the eiei equation. We describe here the geometric meaning of both eiei and fifi equations in our slightly extended setting.We also describe the corresponding actions at the level of orbital varieties: while only the eiei equations make sense directly on the Joseph polynomials, the fifi equations also appear if one introduces a broader class of varieties. We explain the connection of the latter with matrix Schubert varieties.