Presentation of higher-spin \(\mathfrak{sl}_2\) commutant algebras

Develop a presentation for the higher-spin algebra \(\mathcal K^{(j)}:=\operatorname{End}_{U_q(\mathfrak{sl}_2)}(V_j^{\otimes L})\) for every \(j>2\), and, as a first step, construct a web presentation using valenced Temperley–Lieb diagrams and determine the required local reduction rules.

Background

The paper discusses higher-spin rank-one commutant algebras K(j)\mathcal K^{(j)}, where VjV_j is the irreducible (j+1)(j+1)-dimensional sl2\mathfrak{sl}_2-module. These algebras are known as quotients of fused Hecke algebras, but explicit presentations are described only for j=1j=1 and j=2j=2.

For j>2j>2, the authors identify the absence of a presentation as an open problem. They propose constructing a web presentation based on valenced Temperley–Lieb diagrams; this requires working out the local reduction rules for those diagrams.

References

For j>2, finding a presentation remains an open problem, and we propose to find a web presentation as a first step, based on the valenced Temperley-Lieb diagrams in . Indeed, for $j=1$ these diagrams reduce to the usual $A_1$ webs. While for $j=2$, the local reduction rules for the valenced Temperley-Lieb diagrams are known, because $\mathcal K{(2)}$ is isomorphic Thm.\,7.2.1\,\text{content}\,Cor.\,8.2.4 to the chromatic algebra . This algebra has a basis of trivalent graphs, and local reduction rules that preserve the chromatic polynomial. To complete the web presentation for $\mathcal K{(2)}$, it suffices to find the web generators of the chromatic algebra using the results of . For $j>2$, the local reduction rules for the valenced Temperley-Lieb diagrams will have to be worked out.

— Cellular Algebra of $A_2$ webs  (2610.18801 - Gainutdinov et al., 13 Oct 2026) in Section 1, subsection “Outlook,” paragraph “Web presentation for other algebras”