Square root crystals and Grothendieck positivity

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The classical theory of type A crystals provides a graphical framework for proving Schur positivity results. 

In this talk we will discuss a new category of "square root crystals" (introduced implicitly in the work of Yu) that can be used to establish instances of Grothendieck positivity. 

For example, Buch's combinatorial interpretation of the coefficients expanding products of symmetric Grothendieck functions has a simple description in terms of the natural tensor product for this category. 

We will also discuss some shifted analogues and applications to a conjectural formula of Cho-Ikeda for K-theoretic Schur P-functions.