Decomposition results for Gram matrix determinants

  title={Decomposition results for Gram matrix determinants},
  author={Teodor Banica and Stephen J. Curran},
  journal={Journal of Mathematical Physics},
We study the Gram matrix determinants for the groups Sn, On, Bn, Hn, for their free versions Sn+,On+,Bn+,Hn+, and for the half-liberated versions On*,Hn*. We first collect all the known computations of such determinants, along with complete and simplified proofs, and with generalizations where needed. We conjecture that all these determinants decompose as D = ∏πφ(π), with product over all associated partitions. 

Free Quantum Groups and Related Topics

The unitary group U_N has a free analogue U_N^+, and the study of the closed subgroups G\subset U_N^+ is a problem of general interest. We review here the general theory of U_N^+ and its subgroups,

Linear independences of maps associated to partitions

Given a suitable collection of partitions of sets, there exists a connection to easy quantum groups via intertwiner maps. A sufficient condition for this correspondence to be one-to-one are

Quantum permutations, Hadamard matrices, and the search for matrix models

This is a presentation of recent work on quantum permutation groups, complex Hadamard matrices, and the connections between them. A long list of problems is included. We include as well some

Dual bases in Temperley-Lieb algebras

This note is an announcement of the paper [BC16]. We derive a Laurent series expansion in d for the structure coefficients appearing in the dual basis corresponding to the Kauffman diagram basis of

Quantum Permutations and Quantum Reflections

The permutation group $S_N$ has a free analogue $S_N^+$, which is non-classical and infinite at $N\geq4$. We review here the known basic facts on $S_N^+$, with emphasis on algebraic and probabilistic

Quantum groups, from a functional analysis perspective

  • T. Banica
  • Mathematics
    Advances in Operator Theory
  • 2019
It is well-known that any compact Lie group appears as closed subgroup of a unitary group, $G\subset U_N$. The unitary group $U_N$ has a free analogue $U_N^+$, and the study of the closed quantum

Dual bases in Temperley–Lieb algebras, quantum groups, and a question of Jones

We derive a Laurent series expansion for the structure coefficients appearing in the dual basis corresponding to the Kauffman diagram basis of the Temperley-Lieb algebra $\text{TL}_k(d)$, converging

Super-easy quantum groups: definition and examples

We investigate the "two-parameter" quantum symmetry groups that we previously constructed with Skalski, with the conclusion that some of these quantum groups, namely those without singletons, are

Super-easy quantum groups with arbitrary parameters

We discuss an extended easy quantum group formalism, with the Schur-Weyl theoretic Kronecker symbols being as general as possible, and allowed to take values in $\{-1,0,1\}$, and more generally in

Easy quantum groups : linear independencies, models and partition quantum spaces

This work presents results in the context of (unitary) easy quantum groups. These are compact matrix quantum groups featuring a rich combinatorial structure given by partitions (of sets). This thesis



Cumulants, lattice paths, and orthogonal polynomials

Meander Determinants

We prove a determinantal formula for quantities related to the problem of enumeration of (semi-) meanders, namely the topologically inequivalent planar configurations of non-self-intersecting loops

The Matrix of Chromatic Joins

The formula obtained in this paper verifies the conjecture that the determinant of M ( n) is a product of a power of the colour-variable λ and powers of certain polynomials in λ, those called "Beraha polynmials" by combinatorialists.

The Lattice of Non-crossing Partitions and the Birkhoff-Lewis Equations

Abstract A matrix associated with the chromatic join of non-crossing partitions has been introduced by Tutte to generalise the Birkhoff-Lewis equations. A conjecture is given for its determinant in

Liberation of orthogonal Lie groups

Determinants on semilattices

This corollary can be applied to the construction of some (? 1)determinants with large values. For the background on generalized M\4obius functions we refer to the paper [2 ] by Gian-Carlo Rota. 2.

The Gram matrix of a Temperley-Lieb algebra is similar to the matrix of chromatic joins

In this paper we show that the matrix of chromatic joins and the Gram matrix of the Temperley-Lieb algebra are similar (after rescaling), with the change of basis given by diagonal matrices.

Meanders and the Temperley-Lieb algebra

The statistics of meanders is studied in connection with the Temperley-Lieb algebra. Each (multi-component) meander corresponds to a pair of reduced elements of the algebra. The assignment of a

Classification results for easy quantum groups

We study the orthogonal quantum groups satisfying the "easiness" assumption axiomatized in our previous paper, with the construction of some new examples and with some partial classification results.