The irreducible representations of two intermediate Casimir elements associated to the recoupling of three identical irreducible representations of $U_q(\mathfrak{sl}_2)$ are considered. It is shown that these intermediate Casimirs are related by a conjugation involving braid group representations. Consequently, the entries of the braid group matrices are explicitly given in terms of the $q$-Racah polynomials which appear as $6j$-symbols in the Racah problem for $U_q(\mathfrak{sl}_2)$. Formulas for these polynomials are derived from the algebraic relations satisfied by the braid group representations.
This paper is a next step in the project of systematic description of colored knot polynomials started in arXiv:1506.00339. In this paper, we managed to explicitly find the $\textit{inclusive}$ Racah matrices, i.e. the whole set of mixing matrices in channels $R^{\otimes 3}\longrightarrow Q$ with all possible $Q$, for $R=[3,3]$. The case $R=[3,3]$ is a multiplicity free case as well as $R=[2,2]$ obtained in arXiv:1605.03098. The calculation is made possible by the use of highest weight method with the help of Gelfand-Tseitlin tables. The result allows one to evaluate and investigate $[3,3]$-co…
The connection between the recoupling scheme of four copies of $\mathfrak{su}(1,1)$, the generic superintegrable system on the 3 sphere, and bivariate Racah polynomials is identified. The Racah polynomials are presented as connection coefficients between eigenfunctions separated in different spherical coordinate systems and equivalently as different irreducible decompositions of the tensor product representations. As a consequence of the model, an extension of the quadratic algebra ${\rm QR}(3)$ is given. It is shown that this algebra closes only with the inclusion of an additional shift opera…
The oscillator Racah algebra $\mathcal{R}_n(\mathfrak{h})$ is realized by the intermediate Casimir operators arising in the multifold tensor product of the oscillator algebra $\mathfrak{h}$. An embedding of the Lie algebra $\mathfrak{sl}_{n-1}$ into $\mathcal{R}_n(\mathfrak{h})$ is presented. It relates the representation theory of the two algebras. We establish the connection between recoupling coefficients for $\mathfrak{h}$ and matrix elements of $\mathfrak{sl}_n$-representations which are both expressed in terms of multivariate Krawtchouk polynomials of Griffiths type.
By now it is well established that the quantum dimensions of descendants of the adjoint representation can be described in a universal form, independent of a particular family of simple Lie algebras. The Rosso-Jones formula then implies a universal description of the adjoint knot polynomials for torus knots, which in particular unifies the HOMFLY (SU_N) and Kauffman (SO_N) polynomials. For E_8 the adjoint representation is also fundamental. We suggest to extend the universality from the dimensions to the Racah matrices and this immediately produces a unified description of the adjoint knot pol…
One of spectacular results in mathematical physics is the expression of Racah matrices for symmetric representations of the quantum group $SU_q(2)$ through the Askey-Wilson polynomials, associated with the $q$-hypergeometric functions ${_4ϕ_3}$. Recently it was shown that this is in fact the general property of symmetric representations, valid for arbitrary $SU_q(N)$, at least for exclusive Racah matrices $\bar S$. The natural question then is what substitutes the conventional $q$-hypergeometric polynomials when representations are more general? New advances in the theory of matrices $\bar S$,…
Racah matrices and higher $j$-symbols are used in description of braiding properties of conformal blocks and in construction of knot polynomials. However, in complicated cases the logic is actually inverted: they are much better deduced from these applications than from the basic representation theory. Following the recent proposal of arXiv:1612.00422, we obtain the exclusive Racah matrix $\bar S$ for the currently-front-line case of representation $R=[3,1]$ with non-trivial multiplicities, where it is actually operator valued, i.e. depends on the choice of basises in the intertwiner spaces. E…
Recent results on the Racah algebra $\mathcal{R}_n$ of rank $n - 2$ are reviewed. $\mathcal{R}_n$ is defined in terms of generators and relations and sits in the centralizer of the diagonal action of $\mathfrak{su}(1,1)$ in $\mathcal{U}(\mathfrak{su}(1,1))^{\otimes n}$. Its connections with multivariate Racah polynomials are discussed. It is shown to be the symmetry algebra of the generic superintegrable model on the $ (n-1)$ - sphere and a number of interesting realizations are provided.
Université de Montréal（蒙特利尔大学）；Centre de Recherches Mathématiques, Université de Montréal（蒙特利尔大学数学研究中心）；CNRS（法国国家科学研究中心）；University of Miami（迈阿密大学）；Department of Physics, University of Miami（迈阿密大学物理系）；Laboratoire d’Annecy de Physique Théorique（阿讷西理论物理实验室）
There is a well-known zeta function of the $\mathbb{Z}$-dynamical system generated by an element of the symmetric group. By considering this zeta function as a model, we can construct a new zeta function of an element of the braid group.In this paper,we show that the Alexander polynomial which is the most classical polynomial invariant of knots can be expressed in terms of this braid zeta function.Furthermore we define the function $Z_q$ associated with some braids. We show that this function can be expressed by some braid zeta function for the case of special braids whose closures are isotopi…
Assume that ${\mathbb F}$ is a field with $\operatorname{char}{\mathbb F}\not=2$. The Racah algebra $\Re$ is a unital associative ${\mathbb F}$-algebra defined by generators and relations. The generators are $A$, $B$, $C$, $D$ and the relations assert that $[A,B]=[B,C]=[C,A]=2D$ and each of $[A,D]+AC-BA$, $[B,D]+BA-CB$, $[C,D]+CB-AC$ is central in $\Re$. The Bannai-Ito algebra $\mathfrak{BI}$ is a unital associative ${\mathbb F}$-algebra generated by $X$, $Y$, $Z$ and the relations assert that each of $\{X,Y\}-Z$, $\{Y,Z\}-X$, $\{Z,X\}-Y$ is central in $\mathfrak{BI}$. It was discovered that t…
We present and study two families of polynomials with coefficients in the center of the universal enveloping algebra. These polynomials are analogues of a determinant and a characteristic polynomial of a certain non-commutative matrix, labeled by irreducible representations of gl(n,C). The matrix is an image of the universal R-matrix of a Yangian of gl(n,C) under certain representation. We compute the polynomials explicitly for gl(2,C) and establish connections between the first family of polynomials and higher Capelli identities through some sort of plethysm.
We point out that the Homfly polynomial (that is to say, Ocneanu's trace functional) contains two polynomial-valued inner products on the Hecke algebra representation of Artin's braid group. These bear a close connection to the Morton-Franks-Williams inequality. In these structures, the sets of positive, respectively negative permutation braids become orthonormal bases. In the second case, many inner products can be geometrically interpreted through Legendrian fronts and rulings.
The eigenvalue hypothesis claims that any quantum Racah matrix for finite-dimensional representations of $U_q(sl_N)$ is uniquely determined by eigenvalues of the corresponding quantum $\cal{R}$-matrices. If this hypothesis turns out to be true, then it will significantly simplify the computation of Racah matrices. Also due to this hypothesis various interesting properties of colored HOMFLY-PT polynomials will be proved. In addition, it allows one to discover new symmetries of the quantum 6-j symbols, about which almost nothing is known for $N>2$, with the exception of the tetrahedral symmetrie…
The equivariant Kazhdan-Lusztig polynomial of a braid matroid may be interpreted as the intersection cohomology of a certain partial compactification of the configuration space of n distinct labeled points in the plane, regarded as a graded representation of the symmetric group. We show that, in fixed cohomological degree, this sequence of representations of symmetric groups naturally admits the structure of an FS-module, and that the dual FS^op-module is finitely generated. Using the work of Sam and Snowden, we give an asymptotic formula for the dimensions of these representations and obtain …
We study for each $n$ a one-parameter family of complex-valued measures on the symmetric group $S_n$, which interpolate the probability of a monic, degree $n$, square-free polynomial in $\mathbb{F}_q[x]$ having a given factorization type. For a fixed factorization type, indexed by a partition $λ$ of $n$, the measure is known to be a Laurent polynomial. We express the coefficients of this polynomial in terms of characters associated to $S_n$-subrepresentations of the cohomology of the pure braid group $H^{\bullet}(P_n, \mathbb{Q})$. We deduce that the splitting measures for all parameter values…
