A mathematical preprint reports a complete classification of characters for quantum symmetric-pair coideal subalgebras attached to generalized Satake diagrams of finite and affine type. Here, a character is a one-dimensional representation: a way of assigning values to an algebra while preserving its defining rules.
The central result is a bijection. Each character of the coideal subalgebra B is described by two pieces of information: a character of the torus quotient YTheta/2Q_B and a character of the split part Bns. In practical terms, the paper separates the classification into an Abelian, or commutative, torus component and a split component.
The theorem also identifies generators that cannot carry nonzero values under any character. The E_i and F_i attached to black nodes must map to zero, as must B_i attached to white nodes outside Ins. These vanishing rules narrow the problem before the remaining split data are tested.
The split case supplies the paper’s most concrete test. When the pair is split and YTheta is trivial, assigning values b_i to the generators B_i gives a character exactly when the associated beta_i values form what the paper calls a split character description. The condition is therefore both necessary and sufficient within that setting.
That description is combinatorial, but not arbitrary. It requires the relevant square roots to exist in the field k, requires beta to be constant on each connected component of the diagram Gamma, and imposes the stated rank and edge polynomial conditions. These requirements tie the allowed generator values to the structure of the diagram.
Taken together, the classification says that a character for the generalized quantum affine pairs considered here consists of a character of the Abelian torus quotient, alongside suitable square-root choices for split data on Ins. The scope includes both finite and affine type, as stated in the preprint’s summary of the main results.
How the classification is assembled
The algebraic setup assumes that k is a field of characteristic 0 and that q is not a root of 1. The result is built from algebraic relations rather than empirical observations: the analysis uses generator relations and quotient constructions to determine which assignments remain valid.
A related preliminary step classifies the characters of the ambient quantum group U. Those characters correspond bijectively to characters of the Abelian group Y/2Q. The B classification then uses a surjectivity construction: a character is first defined on the free algebra generated by B′ and Bns, and the proof checks whether it survives the defining quotient relations of B.
The decisive relation checks reduce in part to rank-two Serre cases. The analysis reports no extra condition for A1×A1 or A1; type A2 has zero-or-f−1 alternatives; type B2 or A2 has a one-sided zero-or-f−1 condition; and type G2 has zero-or-f−1/f−3 alternatives. These type-dependent polynomial and zero conditions supply the algebraic tests for split characters.
An example and a further action
For the type CIn example, under the paper’s assumptions of an algebraically closed field and n>1, the listed possibilities include classical characters and a non-classical family. In that non-classical family, Bn is mapped to zero while the other Bi receive the displayed square-root values. The example shows how the general criterion can separate different character families.
The paper also describes a right action of U characters on B characters. Under that action, the torus character is multiplied by ξ2, while each split-generator value bi is sent to ξ2(αi)bi. This gives a way to move among the classified characters using data from the ambient quantum group.
A separate discussion records a narrower form for integrable characters under additional assumptions on the parameters ci and si and on specialisability. In that setting, the paper states q-power and q-number values for Kh and Bi, with all other generators mapped to zero. The formula is not presented as a description of every character in the broader classification.
What the result covers
The work is a preprint, labeled arXiv:2608.25613v1 [math.QA] and dated 26 Aug 2026. It reports support from grant OCENW.M20.108 of the Dutch Research Council NWO and says the project was inspired by a question asked during the MMRT 2026 conference in Ottawa.
The classification is a proof-based algebraic statement within its stated assumptions, not an empirical or causal finding. Its setup excludes roots of unity and fields outside characteristic 0, while the stated classification is limited to finite and affine type. The integrable-character form likewise depends on its extra parameter and specialisability assumptions.
Paper data and sources
Original title: Characters of Quantum Symmetric Pairs
Authors: Philip Schlösser
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text