A scheme related to the Brauer loop model. (English) Zbl 1193.14068
Summary: We introduce the Brauer loop scheme
\[
E:=\{M\in M_ N({\mathbb C}):\;M\bullet M=0\},
\]
where \(\bullet\) is a certain degeneration of the ordinary matrix product. Its components of top dimension, \(\lfloor N^2/2\rfloor\), 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 [A. Knutson, J. Algebr. Geom. 14, No. 2, 283–294 (2005; Zbl 1074.14044), see also arXiv: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 and B. Nienhuis, J. Stat. Mech. Theory Exp. 2005, No. 1, Paper P01006, 10 p., electronic only (2005; Zbl 1072.82585), see also math.AG/0410392], based on earlier related work in [M. J. Martins, B. Nienhuis and R. Rietman, An intersecting loop model as a solvable super spin chain, Phys. Rev. Lett. 81, No. 3, 504–507 (1998; Zbl 0944.82006), see also 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 and P. Zinn-Justin, Commun. Math. Phys. 262, No. 2, 459–487 (2006; Zbl 1113.82026), see also 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 \(4\times 4\) matrices.
The Brauer loop model is an integrable stochastic process studied in [J. de Gier and B. Nienhuis, J. Stat. Mech. Theory Exp. 2005, No. 1, Paper P01006, 10 p., electronic only (2005; Zbl 1072.82585), see also math.AG/0410392], based on earlier related work in [M. J. Martins, B. Nienhuis and R. Rietman, An intersecting loop model as a solvable super spin chain, Phys. Rev. Lett. 81, No. 3, 504–507 (1998; Zbl 0944.82006), see also 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 and P. Zinn-Justin, Commun. Math. Phys. 262, No. 2, 459–487 (2006; Zbl 1113.82026), see also 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 \(4\times 4\) matrices.
MSC:
| 14M99 | Special varieties |
| 14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 82B23 | Exactly solvable models; Bethe ansatz |
Keywords:
commuting variety; Brauer loop model; Brauer algebra; integrable models; chord diagrams; multidegreesOnline Encyclopedia of Integer Sequences:
Degree of the variety of pairs of commuting n X n matrices.Degree of the n X n Brauer loop scheme. Also, the sum of components of the Brauer loop model in size n.
Degree of the scheme of n X n complex matrices that square to zero.
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