nLab Sato Grassmannian

Mikio Sato introduced an infinite-dimensional Grassmannian in relation to the integrable systems. It gives a standard way to describe the $\tau$-function. Constructed also by Graeme Segal and Wilson so it is often called Sato-Segal-Wilson Grassmannian.

• M. Sato, The KP hierarchy and infinite dimensional Grassmann manifolds, Proc. Sympos. Pure Math. 49 Part 1. (pp. 51–66) Amer. Math. Soc. 1989.
• Graeme Segal, George Wilson, Loop groups and equations of KdV type, Publ. Math. IHÉS 61, 5–65 (1985) numdam MR87b:58039 Zbl0592.35112

Textbooks include

• Andrew Pressley, Graeme Segal, Loop groups, Clarendon Press 1989

• T. Miwa, M. Jimbo, E. Date, Solitons: differential equations, symmetries and infinite dimensional algebras (translated from Japanese by Miles Reid) Cambridge Tracts in Math. 135, 120 pp.

Other references

• Mattia Cafasso, Chao-Zhong Wu, Tau functions and the limit of block Toeplitz determinants arxiv/1404.5149

A classical way to introduce tau functions for integrable hierarchies of solitonic equationsis by means of the Sato–Segal–Wilson infinite-dimensional Grassmannian. Every point in the Grassmannian is naturally related to a Riemann–Hilbert problem on the unit circle, for which Bertola proposed a tau function that generalizes the Jimbo–Miwa–Ueno tau function for isomonodromic deformation problems. In this paper, we prove that the Sato–Segal–Wilson tau function and the (generalized) Jimbo–Miwa–Ueno isomonodromy tau function coincide under a very general setting, by identifying each of them to the large-size limit of a block Toeplitz determinant. As an application, we give a new definition of tau function for Drinfeld–Sokolov hierarchies (and their generalizations) by means of infinite-dimensional Grassmannians, and clarify their relation with other tau functions given in the literature.

• Dinakar Muthiah, Alex Weekes, Oded Yacobi, The equations defining affine Grassmannians in type A and a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman arxiv/1708.06076

The affine Grassmannian of $SL_n$ admits an embedding into the Sato Grassmannian, which further admits a Plücker embedding into the projectivization of Fermion Fock space…

• Maurice J. Dupré, James F. Glazebrook, Emma Previato, A Banach algebra version of the Sato Grassmannian and commutative rings of differential operators Acta Appl Math (2006) 92: 241 doi; On Banach bundles and operator-valued Baker functions, pdf

• Emma Previato, Mauro Spera, Isometric embeddings of infinite-dimensional Grassmannians, Regul. Chaot. Dyn. (2011) 16: 356 doi

• M. J. Dupré, J. F. Glazebrook, E. Previato, A Banach algebra version of the Sato Grassmannian and commutative rings of differential operators Acta Appl. Math. 92, 241–267 (2006) doi