Integrable models and \(K\)-theoretic pushforward of Grothendieck classes (Q2233912)

Let \(\pi: Gr_k(A)\to X\) be the Grassmannization of the vector bundle \(A\to X\) over a smooth base space \(X\). The push-forward map \(\pi_*\) along \(\pi\) in cohomology, or in \(K\)-theory, plays an important role in equivariant algebraic geometry. One of the many formulas that expresses \(\pi_*\) is the so-called (equivariant) localization formula. Partition functions of integrable models are fundamental objects in statistical physics. A certain partition functions of the five-vertex model may be regarded as an element of \(K(Gr_k(A))\), the K-algebra of \(Gr_k(A)\). Another partition function may be regarded as an element of \(K(X)\). The authors show that -- in these interpretations -- \(\pi_*\) of the first one is the second one. Halfway between the objects in geometry and those in integrable systems is a commutation relation: an identity -- due to Shigechi and Uchiyama -- involving monodromy matrix elements. While this identity is expressed in non-commutative variables (hence cannot directly be related to the commutative geometric objects), it has explicit similarity to the localization formula for \(\pi_*\). The pushforward formula \(\pi_*\) behaves nicely for various versions of Grothedieck polynomials. The authors present such Grothendieck polynomial interpretations of the their partition functions, and derive ``skew generalizations'' of some known important identities for Grothendieck polynomials.