We introduce the_Brauer loop scheme_ E := {M in M_N(C) : M\cp M = 0}, where
\cp is a certain degeneration of the ordinary matrix product. Its components of
top dimension, floor(N^2/2), correspond to involutions \pi in S_N having one or
no fixed points. In the case N even, this scheme contains the upper-upper
scheme from [Knutson '04] as a union of (N/2)! of its components. One of those
is a degeneration of the_commuting variety_ of pairs of commuting matrices.
The_Brauer loop model_ is a quantum integrable stochastic process introduced
in [de Gier--Nienhuis '04], and some of the entries of its Perron-Frobenius
eigenvector were observed (conjecturally) to match the degrees of the
components of the upper-upper scheme. We extend this, with proof, to_all_ the
entries: they are the degrees of the components of the Brauer loop scheme.
Our proof of this follows the program outlined in [Di Francesco--Zinn-Justin
'04]. In that paper, the entries of the Perron-Frobenius eigenvector were
generalized from numbers to polynomials, which allowed them to be calculated
inductively using divided difference operators. We relate these polynomials to
the multidegrees of the components of the Brauer loop scheme, defined using an
evident torus action on E. In particular, we obtain a formula for the degree of
the commuting variety, previously calculated up to 4x4 matrices.