Boundary $q$-characters of evaluation modules for split quantum affine symmetric pairs
Jian-Rong Li, Tomasz Przezdziecki
TL;DR
This work constructs and analyzes evaluation modules for affine quantum symmetric pair coideal subalgebras of split type ${ m AI}$, using Lu–Wang’s Drinfeld-type presentation and a Gelfand–Tsetlin basis to compute the spectrum of the Lu–Wang Cartan subalgebra. It proves an evaluation homomorphism via braid-group action and derives explicit boundary $q$-character formulas for evaluation modules, with a tableaux interpretation that mirrors normal $q$-characters but reveals extra symmetry. The authors show that boundary $q$-characters satisfy a Nakajima-like highest-weight property in this setting and identify genuinely new structures not arising from restriction of ordinary $q$-characters. The results bridge GT-pattern combinatorics with boundary eigenvalues, enabling concrete computations of classical and non-classical boundary characters and providing tools for further exploration of boundary phenomena in quantum symmetric pairs.
Abstract
We study evaluation modules for quantum symmetric pair coideal subalgebras of affine type $\mathsf{AI}$. By computing the action of the generators in Lu and Wang's Drinfeld-type presentation on Gelfand-Tsetlin bases, we determine the spectrum of a large commutative subalgebra arising from the Lu-Wang presentation. This leads to an explicit formula for boundary analogues of $q$-characters in the setting of quantum affine symmetric pairs. We interpret this formula combinatorially in terms of semistandard Young tableaux. Our results imply that boundary $q$-characters share familiar features with ordinary $q$-characters - such as a version of the highest weight property - yet they also display new phenomena, including an extra symmetry. In particular, we provide the first examples of boundary $q$-characters for quantum affine symmetric pairs that do not arise from restriction of ordinary $q$-characters, thereby revealing genuinely new structures in this new setting.