arXiv subjects

Loic Poulain d'Andecy

Publications and source records attributed to Loic Poulain d'Andecy.

4 recordsLinked to original sources

arXiv2026

Kazhdan-Lusztig bases of parabolic Hecke algebras and applications to Schur-Weyl duality

With an eye to applications to type A and Schur-Weyl duality, we study Kazhdan-Lusztig bases for a general parabolic Hecke algebra. Parabolic Hecke algebras are idempotent subalgebras of Hecke algebras corresponding to parabolic subgroups, and for type A they coincide with the fused Hecke algebras appearing in a generalisation of the Schur-Weyl duality with the quantum group of GL(N). In this paper we investigate two different Kazhdan-Lusztig bases for parabolic Hecke algebras, together with the associated cells and the corresponding representations. We quickly specialise to type A, for which we describe the cells in terms of the RSK correspondence generalising thus the well-known description for the symmetric group. As a first application we recover the classification of irreducible representations of parabolic Hecke algebras of type A and provide a new construction of these representations. Next we turn to the Schur-Weyl duality and describe the kernel in terms of one the basis studied precedently. Moreover, we formulate some conjectures about a generator of these kernels in terms of Kazhdan-Lusztig basis elements, give some evidence and prove these conjectures in some special cases.

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arXiv2023

Identities for Rankin-Cohen brackets, Racah coefficients and associativity

We prove an infinite family of identities satisfied by the Rankin-Cohen brackets involving the Racah polynomials. A natural interpretation in the representation theory of sl(2) is provided. From these identities and known properties of the Racah polynomials follows a short new proof of the associativity of the Eholzer product. Finally, we discuss, in the context of Rankin-Cohen algebras introduced by D.Zagier, how any algebraic identity satisfied by the Rankin-Cohen brackets can be seen as a consequence of the set of identities presented in this paper.

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arXiv2022

Askey-Wilson braid algebra and centralizer of $U_q(\mathfrak{sl}_2)$

A presentation of the centralizer of the three-fold tensor product of the spin $s$ representation of the quantum group $U_q(\mathfrak{sl}_2)$ is provided. It is expressed as a quotient of the Askey-Wilson braid algebra. This newly defined algebra combines the Askey-Wilson relations with the braid group relations, on three strands, together with a characteristic equation of degree $2s+1$ for the braid generators. Explicit bases are given for the centralizer.

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arXiv2021

The missing label of $\mathfrak{su}_3$ and its symmetry

We present explicit formulas for the operators providing missing labels for the tensor product of two irreducible representations of $\mathfrak{su}_3$. The result is seen as a particular representation of the diagonal centraliser of $\mathfrak{su}_3$ through a pair of tridiagonal matrices. Using these explicit formulas, we investigate the symmetry of this missing label problem and we find a symmetry group of order 144 larger than what can be expected from the natural symmetries. Several realisations of this symmetry group are given, including an interpretation as a subgroup of the Weyl group of type $E_6$, which appeared in an earlier work as the symmetry group of the diagonal centraliser. Using the combinatorics of the root system of type $E_6$, we provide a family of representations of the diagonal centraliser by infinite tridiagonal matrices, from which all the finite-dimensional representations affording the missing label can be extracted. Besides, some connections with the Hahn algebra, Heun--Hahn operators and Bethe ansatz are discussed along with some similarities with the well-known symmetries of the Clebsch--Gordan coefficients.

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