The present study introduces and investigates a new type of equation which is called Grassmann integral equation in analogy to integral equations studied in real analysis. A Grassmann integral equation is an equation which involves Grassmann integrations and which is to be obeyed by an unknown function over a (finite-dimensional) Grassmann algebra G_m. A particular type of Grassmann integral equations is explicitly studied for certain low-dimensional Grassmann algebras. The choice of the equation under investigation is motivated by the effective action formalism of (lattice) quantum field theory. In a very general setting, for the Grassmann algebras G_2n, n = 2,3,4, the finite-dimensional analogues of the generating functionals of the Green functions are worked out explicitly by solving a coupled system of nonlinear matrix equations. Finally, by imposing the condition G[{\barΨ},Ψ] = G_0[{λ\barΨ}, {λΨ}] + const., 0<λ\in R (\barΨ_k, Ψ_k, k=1,...,n, are the generators of the Grassmann algebra G_2n), between the finite-dimensional analogues G_0 and G of the (``classical'') action and effective action functionals, respectively, a special Grassmann integral equation is being established and solved which also is equivalent to a coupled system of nonlinear matrix equations. If λ\not= 1, solutions to this Grassmann integral equation exist for n=2 (and consequently, also for any even value of n, specifically, for n=4) but not for n=3. If λ=1, the considered Grassmann integral equation has always a solution which corresponds to a Gaussian integral, but remarkably in the case n=4 a further solution is found which corresponds to a non-Gaussian integral. The investigation sheds light on the structures to be met for Grassmann algebras G_2n with arbitrarily chosen n.
Triangulations and soliton graphs for totally positive Grassmannian
KP方程是一个非线性色散波动方程，它为浅水波的共振相互作用提供了一个很好的模型。KP方程的正则孤子解可以由完全非负的Grassmannian Gr $(N,M)_{\geq 0}$中的点构造。Kodama和Williams研究了KP孤子的渐近模式（热带极限），称为孤子图，并表明它们对应于Postnikov的Le图。在本文中，我们考虑孤立子图的KP族，家庭的交换流是兼容的KP方程。对于正的Grassmannian Gr $(2,M)_{>0}$，Kodama和Williams证明了孤子图与$M$ -边形的三角剖分是双射的.当$N=3$和$M=6,7$和$8$同时存在时，我们将这个结果推广到Gr $(N,M)_{>0}$.在每一种情况下，我们表明，孤子图是在双射与Postnikov的plabic图，它推广了乐图。
