Pentad and triangular structures behind the Racah matrices
Somewhat unexpectedly, the study of the family of twisted knots revealed a hidden structure behind exclusive Racah matrices S, which control non associativity of the representation product in a peculiar channel R RR R. These S are simultaneously symmetric and orthogonal, and therefore admit two decompositions: as quadratic forms, S EtrE, and as operators: T ST ST 1S 1. Here T and T consist of the eigenvalues of the quantum R matrices in channels R R and R R respectively, S is the second exclusive Racah matrix for RRR R (still orthogonal, but no longer symmetric) and E is a triangular matrix. It can be further used to construct the KNTZ evolution matrix B ET 2E 1, which is also triangular and explicitly expressible through the skew Schur and Macdonald functions what makes Racah matrices calculable. Moreover, 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 (T S, S, E,B) associated with the universal R matrix, can lead to further insights about representation theory, knot invariants and Macdonald Kerov functions. Triangular structures are long believed to be intimately related to integrability. For example, triangular are the pseudo differential "dressing operators" in the description of KdV/KP hierarchies [1] In a complementary development, triangularity in the form of Gauss decomposition proved to be crucial for the free field description of generic conformal theories [2] Since then triangular transforms appeared in many places, in particular, in the description of Macdonald Kerov functions, of their "generalized" multi time deformations and, somewhat surprisingly, in the study of "evolution" along the family of twist knots which had direct implication to the theory of Racah matrices (6j symbols) [3] and allowed to explicitly calculate some of them, what remained an unsolvable problem for quite a long period of time. A purpose of this short note is a brief review of this newly emerging and promisingly diverse field. 1. Another example of the fundamental importance is description of Macdonald [4] and, more generally, Kerov functions [5,6] as triangular transformation of Schur functions, see [7] for a recent reminder. In this case one just applies a triangular orthogonalization procedure for the scalar product p|p (g) z d, l i=1 gdi (1) on the space of time variables pk. Here p is the monomial basis, p l i=1 pdi labeled by Young diagrams [d1 d2 dl 0] 2 m2 11 with z k k mk *mk! It is well known, that Schur functions khR{p} form another orthonormal basis when all gi 1, khR|khR' (I) dR,R' (2) and Kerov functions are defined as their triangular transformation, which diagonalizes the product for arbitrary {g} Ker (g) R {p} khR{p} R' 0] R R if r1 r 1 or if r1 r 1, but r2 r 2, or if r1 r 1 and r2 r 2, but r3 r 3, and so on (4)