The Missing Label of $$\mathfrak {su}_3$$ and Its Symmetry

Crampé, Nicolas; d'Andecy, L. Poulain; Vinet, Luc

doi:10.1007/s00220-022-04596-3

Cited by 2 publications

(5 citation statements)

References 20 publications

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“…We believe that the knowledge of the SU(3) tensor symmetries contributes to the understanding of finer analysis of SU(3) tensor products and the unveiling of unexpected connections with other objects. For instance, the paper [19] solves the missing label problem for the tensor products of SU(3) representations, by introducing operators decomposing univocally each isotypic component. There, the relevance of the choice of these operators is underlined by the fact that they fulfill the same 144 symmetries as the corresponding tensor multiplicities.…”

Section: Discussionmentioning

confidence: 99%

“…Simple, clear proofs are used instead. The recent paper [19] derives again, by other means, the existence of 144 symmetries, and provides alternative descriptions of them, that are of interest.…”

Section: Relation With Other Workmentioning

confidence: 99%

“…Under the isomorphism (19) we have also that: the cone TM × (R + ) 2 corresponds to a cone H + (known as the positive Horn cone), intersection of H with the subset defined by λ 3 ⩾ 0 and µ 3 ⩾ 0; and the set lat(TM) × (Z + ) 2 corresponds to lat(H + ) (lattice points of the positive Horn cone). This is the support of the Littlewood-Richardson coefficients.…”

Section: The Chamber Complex For the Littlewood-richardson Coefficientsmentioning

confidence: 99%

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All linear symmetries of the SU(3) tensor multiplicities

Briand,

Rosas,

Trandafir

2023

J. Phys. A: Math. Theor.

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The SU (3 ) tensor multiplicities are piecewise polynomial of degree 1 in their labels. The pieces are the chambers of a complex of cones. We describe in detail this chamber complex and determine the group of all linear symmetries (of order 144) for these tensor multiplicities. We represent the cells by diagrams showing clearly the inclusions as well as the actions of the group of symmetries and of its remarkable subgroups.

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Section: Discussionmentioning

confidence: 99%

Section: Relation With Other Workmentioning

confidence: 99%

Section: The Chamber Complex For the Littlewood-richardson Coefficientsmentioning

confidence: 99%

See 1 more Smart Citation

All linear symmetries of the SU(3) tensor multiplicities

Briand,

Rosas,

Trandafir

2023

J. Phys. A: Math. Theor.

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“…Following the algorithm in Section 2, we will construct a polynomial algebra under a bilinear form {•, •} defined in (12). Let q h be the set that contains all the monomials of degree h that satisfy {h i , p h } o = 0.…”

Section: Construction Of Cartan Centralizers Of Semisimple Lie Algebrasmentioning

confidence: 99%

“…For instance, this approach was successfully implemented for the Lie algebra su(3) [11] and gl(3) [8], by applying an algorithm on centralizers of subalgebras. More recent examples, which connect with different context of mathematical physics can be found in [7,9,12,21]. In particular, in a recent paper [6], a similar algorithm was also applied to the Cartan centralizer of A n and certain non-semisimple algebras [9].…”

Section: Introductionmentioning

confidence: 99%

Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras

Campoamor-Stursberg

Latini

Marquette

et al. 2023

J. Phys. A: Math. Theor.

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Starting from a purely algebraic procedure based on the commutant of a subalgebra in the universal enveloping algebra of a given Lie algebra, the notion of algebraic Hamiltonians and the constants of the motion generating a polynomial symmetry algebra is proposed. The case of the special linear Lie algebra sl(n) is discussed in detail, where an explicit basis for the commutant with respect to the Cartan subalgebra is obtained, and the order of the polynomial algebra is computed. It is further shown that, with an appropriate realization of sl(n), this provides an explicit connection with the generic superintegrable model on the (n − 1)-dimensional sphere Sn-1 and the related Racah algebra R(n). In particular, we show explicitly how the models on the 2-sphere and 3-sphere and the associated symmetry algebras can be obtained from the quadratic and cubic polynomial algebras generated by the commutants defined in the enveloping algebra of sl(3) and sl(4), respectively. The construction is performed in the classical (or Poisson-Lie) context, where the Berezin bracket replaces the commutator.

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The Missing Label of $$\mathfrak {su}_3$$ and Its Symmetry

Cited by 2 publications

References 20 publications

All linear symmetries of the SU(3) tensor multiplicities

All linear symmetries of the SU(3) tensor multiplicities

Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras

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