Representation-theoretic interpretation of monotone triangles

Determine whether monotone triangles admit a connection to representation theory.

Background

The paper explains that Gelfand–Tsetlin patterns have a representation-theoretic origin, whereas monotone triangles are obtained by imposing strict increase along the rows. Although monotone triangles are closely connected to alternating sign matrices and related combinatorial structures, the authors state that their relationship with representation theory is unresolved. They mention partial indications from work on generalizations of Littlewood and Cauchy identities, but do not settle the question.

References

Whether there is also a connection of monotone triangles to representation theory is not clear at this point, however, some indications are given in as well as in forthcoming works of the authors and Gangl and the first author on generalizations of the Littlewood identity as well as the Cauchy identity that involve alternating sign matrices.

— The number of monotone trapezoids with prescribed bottom row  (2502.09343 - Fischer et al., 13 Feb 2025) in Section 1, Introduction