# 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., Providence, RI, 1989.
• G. Segal, G. Wilson, Loop groups and equations of KdV type, Publ. Math. IHES 61, 5–65 (1985)

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

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 functionand 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

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

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