Pentad and triangular structures behind the Racah matrices

Abstract

Somewhat unexpectedly, the study of the family of twisted knots revealed a hidden structure behind exclusive Racah matrices $\bar{S}$, which control non-associativity of the representation product in a peculiar channel $R\otimes \bar{R}\otimes R⟶R$. These $\bar{S}$ are simultaneously symmetric and orthogonal, and therefore admit two decompositions: as quadratic forms, $\bar{S}\sim {\mathcal{E}}^{tr}\mathcal{E}$, and as operators: $\bar{T}\bar{S}\bar{T}=S{T}^{-1}{S}^{-1}$. Here $\bar{T}$ and $T$ consist of the eigenvalues of the quantum $\mathcal{R}$-matrices in channels $R\otimes \bar{R}$ and $R\otimes R$ respectively, $S$ is the second exclusive Racah matrix for $\bar{R}\otimes R\otimes R⟶R$ (still orthogonal, but no longer symmetric), and $\mathcal{E}$ is a {\it triangular} matrix. It can be further used to construct the KNTZ evolution matrix $\mathcal{B}=\mathcal{E}{\bar{T}}^2{\mathcal{E}}^{-1}$, which is also triangular and explicitly expressible through the skew Schur and Macdonald functions -- what makes Racah matrices calculable. Moreover, $\mathcal{B}$ is somewhat similar to Ruijsenaars Hamiltonian, which is used to define Macdonald functions, and gets triangular in the Schur basis. Discovery of this pentad structure $(\bar{T},\bar{S},S,\mathcal{E},\mathcal{B})$, associated with the universal $\mathcal{R}$-matrix, can lead to further insights about representation theory, knot invariants and Macdonald-Kerov functions.

8 pages

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Cited by in corpus (7)

• A slow review of the AGT correspondence
• Multiplicity-free $U_q(s{l}_N)$ 6-j symbols: relations, asymptotics, symmetries
• Perspectives of differential expansion
• KNTZ trick from arborescent calculus and the structure of differential expansion
• Differential Expansion for antiparallel triple pretzels: the way the factorization is deformed
• Hook variables: cut-and-join operators and $\tau$-functions
• Evolution properties of the knot's defect