Symmetric discrete AKP and BKP equations

We show that when KP (Kadomtsev-Petviashvili) tau functions allow special symmetries, the discrete BKP equation can be expressed as a linear combination of the discrete AKP equation and its reflected symmetric forms. Thus the discrete AKP and BKP equations can share the same tau functions with these symmetries. Such a connection is extended to 4 dimensional (i.e. higher order) discrete AKP and BKP equations in the corresponding discrete hierarchies. Various explicit forms of such tau functions, including Hirota's form, Gramian, Casoratian and polynomial, are given. Symmetric tau functions of Cauchy matrix form that are composed of Weierstrass σ functions are investigated. As a result we obtain a discrete BKP equation with elliptic coefficients.