# Decomposition results for Gram matrix determinants

@article{Banica2010DecompositionRF,
title={Decomposition results for Gram matrix determinants},
author={Teodor Banica and Stephen J. Curran},
journal={Journal of Mathematical Physics},
year={2010},
volume={51},
pages={113503}
}
• Published 21 September 2010
• Mathematics
• 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.
18 Citations

### 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

• Mathematics
• 2019
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

• Mathematics
Quantum Topology
• 2018
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

## References

SHOWING 1-10 OF 39 REFERENCES

### 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

### 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

• Mathematics
• 2008
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

• Mathematics
• 1997
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

• Mathematics
• 2009
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.