We explore the connections between two types of integrability phenomena arising from quantum groups: solvable lattice models and Hamiltonian operators from Heisenberg algebras. The fundamental question this paper explores is when there exists a Hamiltonian operator whose discrete time evolution matches the partition function of a solvable lattice model.We deal with two types of lattice models: the classical six-vertex model and a modified six-vertex model involving charge. The six-vertex model can be associated with Hamiltonians from classical Fock space, and we show that such a correspondence exists precisely when the Boltzmann weights are free fermionic. This allows us to prove that the free fermionic partition function is always a (skew) supersymmetric Schur function. In this context, the supersymmetric function involution takes us between two lattice models that are generalizations of the vicious walker and osculating walker models. We also observe that the supersymmetric Jacobi-Trudi formula, through Wick's Theorem, can be seen as a six-vertex analogue of the Lindström-Gessel-Viennot Lemma.The six-vertex model with charge has been used to study metaplectic Whittaker functions. It can be associated with Hamiltonian operators acting on Drinfeld twists of q-Fock space [3], and we determine that such a correspondence exists exactly when the weights satisfy what we call a generalized free fermion condition, in which case the partition function is always a (skew) supersymmetric LLT polynomial.

We prove the existence of several different families of solvable lattice models whose partition functions give the double β-Grothendieck polynomials and the dual double β-Grothendieck polynomials for arbitrary permutations. Moreover, we introduce a new family of double "biaxial" β-Grothendieck polynomials depending on a pair of permutations which simultaneously generalize both the double and dual double polynomials. We then use these models and their Yang-Baxter equations to reprove Fomin-Kirillov's Cauchy identity for β-Grothendieck polynomials, generalize it to a new Cauchy identity for biaxial β-Grothendieck polynomials, and prove a new branching rule for double β-Grothendieck polynomials.and Wheeler [4]. Unlike Borodin-Wheeler, who use color to refine certain partition functions of lattice models from symmetric functions into their nonsymmetric pieces, our models use color to move from permutations with one descent to those with arbitrarily many descents. As a nod to this similarity, we refer to our models as "chromatic" rather than "colo(u)red." As a further distinction, the associated quantum group module for our solutions of the Yang-Baxter equation (at least in the specialization β = −1) is a Drinfeld twist of the standard U q (sl n+1 ) module, while those in [4] arise from symmetric powers of U q (sl 2 ) modules (see Remark 4.2).We conclude by outlining the structure of the subsequent sections. In Section 2, we present two different definitions for double β-Grothendieck polynomials which reflect the Hecke algebra point of view and the combinatorial generating function point of view discussed earlier. Then in Section 3, we introduce two similar, but not bijectively equivalent, lattice models whose partition functions give the double β-Grothendieck polynomials. By allowing more general boundary conditions indexed by a pair of partitions, we obtain what we call the "biaxial" model, which will turn out to simultaneously generalize both the double β-Grothendieck polynomials and their duals. In Section 4, we describe the Yang-Baxter equations and R-matrices associated to each model, and in Section 5, we use these solutions to compute the partition functions. We also briefly discuss the correspondence between the states of the models and pipe dreams. In Section 6, we prove Cauchy-type identities by stacking our models appropriately and calculating the partition functions in two ways. Finally, in Section 7, we provide a branching rule that describes how to express a double β-Grothendieck polynomial for any permutation in S n as a sum over double β-Grothendieck polynomials for permutations in S n−1 .

This paper gives a general algorithm for computing the character table of any Renner monoid Hecke algebra, by adapting and generalizing techniques of Solomon used to study the rook monoid. The character table of the Hecke algebra of the rook monoid (i.e. the Cartan type [Formula: see text] Renner monoid) was computed earlier by Dieng et al. [2], using different methods. Our approach uses analogues of so-called A- and B-matrices of Solomon. In addition to the algorithm, we give explicit combinatorial formulas for the A- and B-matrices in Cartan type [Formula: see text] and use them to obtain an explicit description of the character table for the type [Formula: see text] Renner monoid Hecke algebra.

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We introduce families of two-parameter multivariate polynomials indexed by pairs of partitions v, w -biaxial double (β, q)-Grothendieck polynomials -which specialize at q = 0 and v = 1 to double β-Grothendieck polynomials from torus-equivariant connective K-theory. Initially defined recursively via divided difference operators, our main result is that these new polynomials arise as partition functions of solvable lattice models. Moreover, the associated quantum group of the solvable model for polynomials in n pairs of variables is a Drinfeld twist of the Uq( sl n+1 ) R-matrix. By leveraging the resulting Yang-Baxter equations of the lattice model, we show that these polynomials simultaneously generalize double β-Grothendieck polynomials and dual double β-Grothendieck polynomials for arbitrary permutations. We then use properties of the model and Yang-Baxter equations to reprove Fomin-Kirillov's Cauchy identity for β-Grothendieck polynomials, generalize it to a new Cauchy identity for biaxial double β-Grothendieck polynomials, and prove a new branching rule for double β-Grothendieck polynomials.