We consider a realization of the real Grassmann manifold Gr(k,n) based on a particular flow defined by the corresponding (singular) solution of the KP equation. Then we show that the KP flow can provide an explicit and simple construction of the incidence graph for the integral cohomology of Gr(k,n). It turns out that there are two types of graphs, one for the trivial coefficients and other for the twisted coefficients, and they correspond to the homology groups of the orientable and non-orientable cases of Gr(k,n) via the Poincare-Lefschetz duality. We also derive an explicit formula of the Poincare polynomial for Gr(k,n) and show that the Poincare polynomial is also related to the number of points on a suitable version of Gr(k,n) over a finite field $\F_q$ with q being a power of a prime. In particular, we find that the number of $\F_q$ points on Gr(k,n) can be computed by counting the number of singularities along the KP flow.
This paper attempts to extract the topology of real Grassmannians based on a realization of the manifolds related to the KP hierarchy. As in the case of the real flag manifolds discussed in [8] using the Toda lattice hierarchy, we here show that the KP equation can be used to obtain similar results for the real Grassmannians.
The KP equation is a two-dimensional extension of the well-known KdV equation, and it was introduced by Kadomtsev and Petviashvili in 1970 to study the stability of one KdV soliton under the influence of weak two-dimensional perturbations [18]. The equation provides also a model to describe a two-dimensional shallow water wave phenomena (see for example [1,27,20,11]. The KP equation is a dispersive wave equation for the scaler function u = u(x, y, t) with spatial variables (x, y) and time variable t, and is given by
∂ 2 u ∂y 2 = 0. We write the solution in the form u(x, y, t) = 2 ∂ 2 ∂x 2 ln τ (x, y, t), where the function τ is called the tau function of the KP equation [26,25]. It is also well-known that some of the exact solutions can be written in the Wronskian determinant form (see for example [15]), (1.1) τ = Wr(f 1 , f 2 , . . . , f k ) :=
, where f (l) j := ∂ l f j /∂x l , and each of the functions {f i (x, y, t) : i = 1, 2, . . . , k} satisfies the simple linear equations, 3 . We then take the following finite dimensional solutions, (i.e. finite Fourie series),
with E j := exp λ j x + λ 2 j y + λ 3 j t ,
where A := (a i,j ) is an k × n real matrix of rank k, and λ j ’s are real and distinct numbers, say
Since λ j ’s are all distinct, the set of exponential functions {E j : j = 1, . . . , n} is linearly independent and forms a basis of R n . Then the set of {f i : i = 1, . . . , k} defines a k-dimensional subspace in R n . This implies that each solution defined by the τ -function (1.1) can be characterized by the A-matrix, hence the solution parametrizes a point of the real Grassmannian. Note here that the Grassmannian denoted by Gr(k, n, R) can be described as
where M k×n (R) is the set of all k × n matrices of rank k, and GL k (R) is the general linear group of k × k matrices. We then expect that some particular solutions of the KP equation contain certain information on the cohomology of Gr(k, n, R). This is our main motivation for the present study. In the previous paper [8], we found the similar results for the case of real flag variety using the Toda lattice equation.
Let us explain how some solutions contain information of the cohomology of Gr(k, n, R) by taking simple example with k = 1 and n = 4. That is, we recover the cohomology of real variety RP 3 ∼ = Gr(1, 4, R) from a KP flow. We consider the τ -function in the form, (1.2)
where j are arbitrary non-zero real constants. Since | j | can be absorbed in the exponential term, we only consider their signs, and here we assume j ∈ {±} (which we refer to as KP signs). Let us consider the case with the specific signs ( 1 , . . . , 4 ) = (+, -, -, +). Using simple asymptotic argument, one can easily find that each E j (x, y, t) becomes dominant in certain region of the xyplane for a fixed t. Figure 1 illustrates the contour plots of the solution u(x, y, t) at t > 0 and t < 0. Each region marked by the number (i) indicates the dominant exponential E i , and at the boundary of the region, the τ -function can be expressed by two exponential functions E i and E j (which is dominant in the adjacent region), i.e. writing E i = e θi with θ i = λ i x + λ 2 i y +
Then at the boundary, the solution has the following form depending on the sign i j ,
In Figure 1, the sech-part (regular) of the solution is shown by the dark colored contour, and the csch-part (singular) is shown in the light colored contour. We construct a polytope whose vertices are labeled by the dominant exponentials. The polytope for our example is a tetrahedron as shown in Figure 1. We then consider a KP flow choosing a particular parameter set (x, y, t), so that the flow is passing near the edges of the polytope. The flow might be expressed as a graph,
where → shows the singular flow, and the ⇒ shows the regular one. It is then interesting to note that the graph is the incidence graph of the real projective space RP 3 , if the arrows are identified as the coboundary operators and their incidence numbers are assigned as 0 for → and ±2 for ⇒. This kind of graph, exemplified by graph (1.3), was introduced in [8] in an initial attempt to compute the number of connected components carved up by the zero divisors of the τ -function, i.e. τ i = 0 for some i, within a moment polytope for the Toda flow. In the case of the KP flow Figure 1 shows that this graph does not contain enough information to attempt to do this (one would have to connect (1) and ( 4) with an edge ⇒). Still the main results in [8], which then motivated this paper, were the result of an observation that such a graph, exclusively defined in terms of the flow crossing or not crossing singularit
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