A scheme related to the Brauer loop model

Abstract

We introduce the Brauer loop scheme E : = { M ∈ M N ( C ) : M • M = 0 } , where • is a certain degeneration of the ordinary matrix product. Its components of top dimension, ⌊ N 2 / 2 ⌋ , correspond to involutions π ∈ S N having one or no fixed points. In the case N even, this scheme contains the upper–upper scheme from [A. Knutson, Some schemes related to the commuting variety, J. Algebraic Geom., in press, math.AG/0306275] 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 an integrable stochastic process studied in [J. de Gier, B. Nienhuis, Brauer loops and the commuting variety, J. Stat. Mech. (2005) P01006, math.AG/0410392], based on earlier related work in [M.J. Martins, B. Nienhuis, R. Rietman, An intersecting loop model as a solvable super spin chain, Phys. Rev. Lett. 81 (1998) 504–507, cond-mat/9709051], and some of the entries of its Perron–Frobenius eigenvector were observed (conjecturally) to equal the degrees of the components of the upper–upper scheme. Our proof of this equality follows the program outlined in [P. Di Francesco, P. Zinn-Justin, Inhomogeneous model of crossing loops and multidegrees of some algebraic varieties, math-ph/0412031]. 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. As a consequence, we obtain a formula for the degree of the commuting variety, previously calculated up to 4 × 4 matrices.