Most modern treatments of the Schubert calculus typically write about the cohomology ring of the Grassmannian. They also write, almost as an afterthought, that this is derived from Schubert's "symbolic calculus." For example:

"Generally, Schubert calculus refers to Schubert’s methods, involving symbolic manipulations, for the solution of enumerative problems." [1], "The Schubert Calculus is a formal calculus of symbols representing geometric conditions used to solve problems in enumerative geometry." [2], etc.

But what, exactly, was Schubert's symbolic calculus? It was published in his "Kalkül Der Abzählenden Geometrie." Has this ever been translated to English?

Does anyone have a reference that explains Schubert's symbolic formalism in basic terms?

  • $\begingroup$ My guess is that Schubert was doing multiplicative calculations in the cohomology ring without having the notion of cohomology ring as where the calculation "lived." This isn't so crazy to me given how often we do calculations in the Chow ring of a variety without knowing the whole ring, even if in our case we do know that the ring exists. $\endgroup$ – Tabes Bridges May 29 '19 at 19:18

I think the best reference is this article : "Schubert's calculus according to Schubert".

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