The Hopf algebra structure of the character rings of classical groups

Fauser, Bertfried; Jarvis, Peter; King, Ronald C.

doi:10.1088/1751-8113/46/3/035205

Cited by 8 publications

(26 citation statements)

References 30 publications

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“…while the antipode S(s λ ) = (−1) |λ| s λ of the Hopf algebra corresponds to the well-known involution ω of the ring of symmetric functions [28], see e.g. [97,98]; that the antipode S here is an involution is a consequence of bicommutativity of the Hopf algebra structure on Sym.…”

Section: Combinatorial Hopf Algebra Structurementioning

confidence: 99%

“…Of course, by virtue of the identification of Schur functions as characters of irreducible representations of U (∞), this structure makes the Grothendieck group K 0 (Rep(U (∞))) into a graded self-dual, bicommutative Hopf algebra (see e.g. [98]), and ultimately also the representation category Rep(U q (g)) itself in the N → ∞ limit by replacing s λ → U λ in the structure maps above. This Hopf algebra structure is exploited in §5.6 below.…”

Section: Combinatorial Hopf Algebra Structurementioning

confidence: 99%

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q-deformations of two-dimensional Yang–Mills theory: Classification, categorification and refinement

Szabo

Tierz

2013

Nuclear Physics B

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Abstract. We characterise the quantum group gauge symmetries underlying q-deformations of two-dimensional Yang-Mills theory by studying their relationships with the matrix models that appear in Chern-Simons theory and six-dimensional N = 2 gauge theories, together with their refinements and supersymmetric extensions. We develop uniqueness results for quantum deformations and refinements of gauge theories in two dimensions, and describe several potential analytic and geometric realisations of them. We reconstruct standard q-deformed Yang-Mills amplitudes via gluing rules in the representation category of the quantum group associated to the gauge group, whose numerical invariants are the usual characters in the Grothendieck group of the category. We apply this formalism to compute refinements of q-deformed amplitudes in terms of generalised characters, and relate them to refined Chern-Simons matrix models and generalized unitary matrix integrals in the quantum β-ensemble which compute refined topological string amplitudes. We also describe applications of our results to gauge theories in five and seven dimensions, and to the dual superconformal field theories in four dimensions which descend from the N = (2, 0) six-dimensional superconformal theory.

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Section: Combinatorial Hopf Algebra Structurementioning

confidence: 99%

Section: Combinatorial Hopf Algebra Structurementioning

confidence: 99%

q-deformations of two-dimensional Yang–Mills theory: Classification, categorification and refinement

Szabo

Tierz

2013

Nuclear Physics B

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“…Consider the variation under this measurement operation of a homogeneous polynomial f (ρ) which is invariant under local SL(3, C) × SL(3, C) transformations on the density operator, with degree of homogeneity h . We note 11…”

Section: Appendix B Homogeneous Polynomial Entanglement Monotones Anmentioning

confidence: 96%

The mixed two qutrit system: local unitary invariants, entanglement monotones, and the SLOCC group $SL(3,{\mathbb {C}})$

Jarvis

2014

J. Phys. A: Math. Theor.

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We consider local unitary invariants and entanglement monotones for the mixed two qutrit system. Character methods for the local SU (3) × SU (3) transformation group are used to establish the count of algebraically independent polynomial invariants up to degree 5 in the components of the density operator. These are identified up to quartic degree in the standard basis of Gell-Mann matrices, with the help of the calculus of f and d coefficients. Next, investigating local measurement operations, we study a SLOCC qutrit group, which plays the role of a 'relativistic' transformation group analogous to that of the Lorentz group SL(2, C) R ≃ SO(3, 1) for the qubit case. This is the group SL(3, C) R , presented as a group of real 9 × 9 matrices acting linearly on the 9-dimensional space of projective coordinates for the qutrit density matrix. The counterpart, for qutrits, of the invariant 4 × 4 Minkowski metric of the qubit case, proves to be a certain 9 × 9 × 9 totally symmetric three-fold tensor generalizing the Gell-Mann d coefficient. Using this structure, we provide a count of the corresponding LSL polynomial invariants using group character methods. Finally, we give an explicit construction of the lowest degree quantity (the cubic invariant) and its expansion in terms of SU (3) × SU (3) invariants, and we indicate how to construct higher degree analogues. These quantities are proven to yield entanglement monotones. This work generalizes and partly extends the paper of King, Welsh and Jarvis (Journal of Physics A, 40(33):10083, 2007) on the mixed two qubit system, which is reviewed in an appendix.

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“…As said above, we are interested in k -cochains, which are invertible k -1 -maps. Using unitality and a normalization condition the Milnor-Moore recursive inverse is still available via cut coproducts as shown in (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18). Lemma 2.5: Let f be unital, f(1, .…”

Section: Convolution Algebras Over a Hopf Algebramentioning

confidence: 99%

Hopf algebras, distributive (Laplace) pairings and hash products: a unified approach to tensor product decompositions of group characters

Fauser¹,

Jarvis²,

King³

2014

J. Phys. A: Math. Theor.

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We show for bicommutative graded connected Hopf algebras that a certain distributive (Laplace) subgroup of the convolution monoid of 2-cochains parameterizes certain well behaved Hopf algebra deformations. Using the Laplace group, or its Frobenius subgroup, we define higher derived hash products, and develop a general theory to study their main properties. Applying our results to the (universal) bicommutative graded connected Hopf algebra of symmetric functions, we show that classical tensor product and character decompositions, such as those for the general linear group, mixed co-and contravariant or rational characters, orthogonal and symplectic group characters, Thibon and reduced symmetric group characters, are special cases of higher derived hash products. In the Appendix we discuss a relation to formal group laws.

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The Hopf algebra structure of the character rings of classical groups

Cited by 8 publications

References 30 publications

q-deformations of two-dimensional Yang–Mills theory: Classification, categorification and refinement

q-deformations of two-dimensional Yang–Mills theory: Classification, categorification and refinement

The mixed two qutrit system: local unitary invariants, entanglement monotones, and the SLOCC group $SL(3,{\mathbb {C}})$

Hopf algebras, distributive (Laplace) pairings and hash products: a unified approach to tensor product decompositions of group characters

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