Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions

Abstract

Abstract We obtain BGG-type formulas for transfer matrices of irreducible finite-dimensional representations of the classical Lie algebras

$${\mathfrak {g}}$$

g

, whose highest weight is a multiple of a fundamental one and which can be lifted to the representations over the Yangian

$$Y({\mathfrak {g}})$$

Y ( g )

. These transfer matrices are expressed in terms of transfer matrices of certain infinite-dimensional highest weight representations (such as parabolic Verma modules and their generalizations) in the auxiliary space. We further factorise the corresponding infinite-dimensional transfer matrices into the products of two Baxter Q-operators, arising from our previous study Frassek et al. (Adv. Math. 401:108283, 2022), Frassek and Tsymbaliuk (Commun. Math. Phys. 392:545–619, 2022) of the degenerate Lax matrices. Our approach is crucially based on the new BGG-type resolutions of the finite-dimensional

$${\mathfrak {g}}$$

g

-modules, which naturally arise geometrically as the restricted duals of the Cousin complexes of relative local cohomology groups of ample line bundles on the partial flag variety G/P stratified by

$$B_{-}$$

B -

-orbits.