Highest Weight Representations Research Articles

Representations of the super-Yangian of type B(n,m)

We are concerned with finite-dimensional irreducible representations of the Yangians associated with the orthosymplectic Lie superalgebras osp2n+1|2m. Every such representation is highest weight and we use embedding theorems and odd reflections of Yangian type to derive necessary conditions for an irreducible highest weight representation to be finite-dimensional. We conjecture that these conditions are also sufficient. We prove the conjecture in the case n=1 and arbitrary m⩾1.

Prime representations in the Hernandez–Leclerc category: classical decompositions

Abstract We use the dual functional realization of loop algebras to study the prime irreducible objects in the Hernandez–Leclerc (HL) category for the quantum affine algebra associated with $\mathfrak {sl}_{n+1}$ . When the HL category is realized as a monoidal categorification of a cluster algebra (Hernandez and Leclerc (2010, Duke Mathematical Journal 154, 265–341); Hernandez and Leclerc (2013, Symmetries, integrable systems and representations, 175–193)), these representations correspond precisely to the cluster variables and the frozen variables are minimal affinizations. For any height function, we determine the classical decomposition of these representations with respect to the Hopf subalgebra $\mathbf {U}_q(\mathfrak {sl}_{n+1})$ and describe the graded multiplicities of their graded limits in terms of lattice points of convex polytopes. Combined with Brito, Chari, and Moura (2018, Journal of the Institute of Mathematics of Jussieu 17, 75–105), we obtain the graded decomposition of stable prime Demazure modules in level two integrable highest weight representations of the corresponding affine Lie algebra.

Highest weight modules over the quantum periplectic superalgebra of type P

In this paper, we begin the study of highest weight representations of the quantized enveloping superalgebra Uqpn of type P. We introduce a Drinfeld-Jimbo representation and establish a triangular-decomposition of Uqpn. We explain how to relate modules over Uqpn to modules over pn, the Lie superalgebra of type P, and we prove that the category of tensor modules over Uqpn is not semisimple.

Asymptotic dynamics of three dimensional supergravity and higher spin gravity revisited

We reconsider the Hamiltonian reduction of the action for three dimensional AdS supergravity and W3 higher spin AdS gravity in the Chern-Simons formulation under asymptotically anti-de Sitter boundary conditions. We show that the reduction gives two copies of chiral bosons on the boundary. In particular, we take into account the holonomy of the Chern-Simons connection which manifests itself as zero mode of the momentum of the boundary chiral boson. We provide an equivalent formulation of the boundary action which we claim to be the geometric action on symplectic leaves of a (super-)Virasoro or a higher spin WN Poisson manifold in the case of supergravity or higher spin gravity respectively, where the intersection of leaves (given in terms of leaves representatives) can be identified as the bulk holonomy. This concludes the extension to non-linear algebras where the notion of coadjoint representation is not well-defined. The boundary Hamiltonian depends on a choice of boundary conditions and is equivalent to the Schwarzian action for corresponding Brown-Henneaux boundary conditions. We make this connection explicit in the extended supersymmetric case. Moreover, we discuss the geometric action in the case of W3 AdS3 gravity in both mathfrak{sl} (3) highest weight representations based on principal and diagonal mathfrak{sl} (2) embeddings.

BMS field theories with u(1) symmetry

We investigate quantum field theories in two dimensions (2D) with an underlying Bondi--van der Burgh--Metzner-Sachs symmetry augmented by $\mathfrak{u}(1)$ currents. These field theories are expected to holographically capture features of charged versions of cosmological solutions in asymptotically flat 3D spacetimes called flat space cosmologies. We conduct a study of the modular properties of these field theories. The characters for the highest weight representations of the symmetry algebra are constructed, and the partition function of the theory is obtained from them. We derive the density of (primary) states and find the entropy and asymptotic values of the structure constants exploiting the modular properties of the partition function and the torus one-point function. The expression for the asymptotic structure constants shows shifts in the weights and one of the central terms and an extra phase compared to the earlier results in the literature for Bondi--van der Burgh--Metzner-Sachs invariant theories without $\mathfrak{u}(1)$ currents present. We reproduce our field results for the structure constants by a bulk computation involving a scalar probe in the background of a charged flat space cosmology.

Affinization of q-oscillator representations of $$U_q(\mathfrak {gl}_n)$$

We introduce a category $\widehat{\mathcal{O}}_{\rm osc}$ of $q$-oscillator representations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_n)$. We show that $\widehat{\mathcal{O}}_{\rm osc}$ has a family of irreducible representations, which naturally corresponds to finite-dimensional irreducible representations of quantum affine algebra of untwisted affine type $A$. It is done by constructing a category of $q$-oscillator representations of the quantum affine superalgebra of type $A$, which interpolates these two family of irreducible representations. The category $\widehat{\mathcal{O}}_{\rm osc}$ can be viewed as a quantum affine analogue of the semisimple tensor category generated by unitarizable highest weight representations of $\mathfrak{gl}_{u+v}$ ($n=u+v$) appearing in the $(\mathfrak{gl}_{u+v},\mathfrak{gl}_\ell)$-duality on a bosonic Fock space.

On emergent conformal symmetry near the photon ring

In this note we revisit the emergent conformal symmetry in the near-ring region of warped spacetime. In particular, we propose a novel construction of the emergent near-ring sl(2, ℝ)QNM symmetry. We show that each eikonal QNM family falls into one highest-weight representation of this algebra, and sl(2, ℝ)QNM can be related to the near-ring isometry group sl(2, ℝ)ISO in a simple way. Furthermore we find that the coherent state space of sl(2, ℝ)QNM can be identified with the phase space of the photon ring.

On higher-dimensional Carrollian and Galilean conformal field theories

In this paper, we study the Carrollian and Galilean conformal field theories (CCFT and GCFT) in d>2d>2 dimensions. We construct the highest weight representations (HWR) of Carrollian and Galilean conformal algebra (CCA and GCA). Even though the two algebras have different structures, their HWRs share similar structure, because their rotation subalgebras are isomorphic. In both cases, we find that the finite dimensional representations are generally reducible but indecomposable, and can be organized into the multiplets. Moreover, it turns out that the multiplet representations in d>2d>2 CCA and GCA carry not only the simple chain structure appeared in logCFT or 2d2d GCFT, but also more generally the net structures. We manage to classify all the allowed chain representations. Furthermore we discuss the two-point and three-point correlators by using the Ward identities. We mainly focus on the two-point correlators of the operators in chain representations. Even in this relative simple case, we find some novel features: multiple-level structure, shortage of the selection rule on the representations, undetermined 2-pt coefficients, etc.. We find that the non-trivial correlators could only appear for the representations of certain structure, and the correlators are generally polynomials of time coordinates for CCFT (spacial coordinates for GCFT), whose orders depend on the levels of the correlators.

Newton–Okounkov polytopes of flag varieties and marked chain-order polytopes

Marked chain-order polytopes are convex polytopes constructed from a marked poset. They give a discrete family relating a marked order polytope with a marked chain polytope. In this paper, we consider the Gelfand–Tsetlin poset of type A A , and realize the associated marked chain-order polytopes as Newton–Okounkov bodies of the flag variety. Our realization connects previous realizations of Gelfand–Tsetlin polytopes and Feigin–Fourier–Littelmann–Vinberg polytopes as Newton–Okounkov bodies in a uniform way. As an application, we prove that the flag variety degenerates into the irreducible normal projective toric variety corresponding to a marked chain-order polytope. We also construct a specific basis of an irreducible highest weight representation. The basis is naturally parametrized by the set of lattice points in a marked chain-order polytope.

Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions

We obtain BGG-type formulas for transfer matrices of irreducible finite-dimensional representations of the classical Lie algebras $\mathfrak{g}$, whose highest weight is a multiple of a fundamental one and which can be lifted to the representations over the Yangian $Y(\mathfrak{g})$. These transfer matrices are expressed in terms of transfer matrices of certain infinite-dimensional highest weight representations (such as parabolic Verma modules and their generalizations) in the auxiliary space. We further factorise the corresponding infinite-dimensional transfer matrices into the products of two Baxter $Q$-operators, arising from our previous study (arXiv:2001.04929, arXiv:2104.14518) of the degenerate Lax matrices. Our approach is crucially based on the new BGG-type resolutions of the finite-dimensional $\mathfrak{g}$-modules, which naturally arise geometrically as the restricted duals of the Cousin complexes of relative local cohomology groups of ample line bundles on the partial flag variety $G/P$ stratified by $B_{-}$-orbits.

Representations of cyclotomic oriented Brauer categories

The cyclotomic Brauer category OB(u,u′) is an upper finite weakly triangular category over an arbitrary field in the sense of [14, Definition 2.2] and the category of locally finite dimensional representations of A is an upper finite fully stratified category in the sense of [10], where A is the locally unital algebra associated to OB(u,u′). The category of locally finite dimensional representations of A is used to give a direct proof of the tensor product categorification (in the general sense of Losev and Webster) for an integrable lowest weight with an integrable highest weight representation of the same level for the Lie algebra g, where g is a direct sum of copies of sl∞ (resp., slˆp) if the characteristic p of the ground field is 0 (resp., positive). Such a result was expected in [4] when k=C and proved previously in [2] when the level is 1.

An extended trace formula for vertex operators

We present an extension of the trace of a vertex operator and explain a representation-theoretic interpretation of the trace. Specifically, we consider a twist of the vertex operator with infinitely many Casimir operators and compute its trace as a character formula. To do this, we define the Fock space of infinite level F∞. Then, we prove a duality between gl∞ and a∞=glˆ∞ of Howe type, which provides a decomposition of F∞ into irreducible representations with joint highest weight vector for gl∞ and a∞. The decomposition of the Fock space F∞ into highest weight representations provides a method to calculate and interpret the extended trace.

Ground state representations of topological groups

Let α:R→Aut(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha : {{\\mathbb {R}}}\\rightarrow \\mathop {\ extrm{Aut}}\ olimits (G)$$\\end{document} define a continuous R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {R}}}$$\\end{document}-action on the topological group G. A unitary representation (π♭,H)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\pi ^\\flat ,\\mathcal {H})$$\\end{document} of the extended group G♭:=G⋊αR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G^\\flat := G \\rtimes _\\alpha {{\\mathbb {R}}}$$\\end{document} is called a ground state representation if the unitary one-parameter group π♭(e,t)=eitH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi ^\\flat (e,t) = e^{itH}$$\\end{document} has a non-negative generator H≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H \\ge 0$$\\end{document} and the subspace H0:=kerH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}^0 := \\ker H$$\\end{document} of ground states generates H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}$$\\end{document} under G. In this paper, we introduce the class of strict ground state representations, where (π♭,H)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\pi ^\\flat ,\\mathcal {H})$$\\end{document} and the representation of the subgroup G0:=Fix(α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G^0 := \\mathop {\ extrm{Fix}}\ olimits (\\alpha )$$\\end{document} on H0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}^0$$\\end{document} have the same commutant. The advantage of this concept is that it permits us to classify strict ground state representations in terms of the corresponding representations of G0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G^0$$\\end{document}. This is particularly effective if the occurring representations of G0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G^0$$\\end{document} can be characterized intrinsically in terms of concrete positivity conditions. To find such conditions, it is natural to restrict to infinite dimensional Lie groups such as (1) Heisenberg groups (which exhibit examples of non-strict ground state representations); (2) Finite dimensional groups, where highest weight representations provide natural examples; (3) Compact groups, for which our approach provides a new perspective on the classification of unitary representations; (4) Direct limits of compact groups, as a class of examples for which strict ground state representations can be used to classify large classes of unitary representations.

Determinantal Ideals and the Canonical Commutation Relations: Classically or Quantized

We construct homomorphic images of $$su(n,n)^{{\mathbb {C}}}$$ in Weyl Algebras $${{\mathcal {H}}}_{2nr}$$ . More precisely, and using the Bernstein filtration of $${{\mathcal {H}}}_{2nr}$$ , $$su(n,n)^{{\mathbb {C}}}$$ is mapped into degree 2 elements with the negative non-compact root spaces being mapped into second order creation operators. Using the Fock representation of $${{\mathcal {H}}}_{2nr}$$ , these homomorphisms give all unitary highest weight representations of $$su(n,n)^{{\mathbb {C}}}$$ thus reconstructing the Kashiwara–Vergne List for the Segal–Shale–Weil representation. Using an idea from the derivation of the their list, we construct a homomorphism of $$u(r)^{{\mathbb {C}}}$$ into $${{\mathcal {H}}}_{2nr}$$ whose image commutes with the image of $$su(n,n)^{{\mathbb {C}}}$$ , and vice versa. This gives the multiplicities. The construction also gives an easy proof that the ideal of $$(r+1)\times (r+1)$$ minors is prime. Here, of course, $$r\le n-1$$ and for a fixed such r, the space of any irreducible representation of $$su(n,n)^{{\mathbb {C}}}$$ is annihilated by this ideal. As a consequence, these representations can be realized in spaces of solutions to Maxwell type equations. We actually go one step further and determine exactly for which representations from our list there is a non-trivial homomorphism between generalized Verma modules, thereby revealing, by duality, exactly which covariant differential operators have unitary null spaces. We prove the analogous results for $${{\mathcal {U}}}_q(su(n,n)^{{\mathbb {C}}})$$ . The Weyl Algebras are replaced by the Hayashi–Weyl Algebras $${{\mathcal {H}}}{{\mathcal {W}}}_{2nr}$$ and the Fock space by a q-Fock space. Further, determinants are replaced by q-determinants, and a homomorphism of $${{\mathcal {U}}}_q(u(r)^{{\mathbb {C}}})$$ into $${{\mathcal {H}}}{{\mathcal {W}}}_{2nr}$$ is constructed with analogous properties. For this purpose a Drinfeld Double is used. We mention one difference: The quantized negative non-compact root spaces, while still of degree 2, are no longer given entirely by second order creation operators.

Prolate-shape dominance and dual-shell mechanism

The dominance of prolate over oblate deformations in the ground states of deformed even-even nuclei has long been known empirically. Although various theoretical approaches have been applied, the underlying mechanism is not yet considered as being adequately understood. One of those theoretical approaches has found an interesting interference effect between the spin-orbit coupling and the surface diffuseness in the single-particle potential. Recently the proxy-SU(3) model claimed that the prolate dominance would emerge naturally from the SU(3) symmetry assuming the highest-weight irreducible representations as ground-state configurations. This article presents a possible interpretation of the interference effect in the former approach in the context of the latter approach, more specifically invoking the dual-shell mechanism that includes both the three-dimensional harmonic-oscillator shells and the spin-orbit-like shells.

Super Hirota bilinear equations for the super modified BKP hierarchy

In this paper, the super modified BKP (SmBKP) hierarchy is constructed from the perspective of the neutral free superfermions by using highest weight representations of the infinite-dimensional Lie superalgebra b∞|∞(g). Based upon this, the corresponding super Hirota bilinear identity of the SmBKP hierarchy is obtained by using the super Boson-Fermion correspondence of type B, and some specific examples of super Hirota bilinear equations are given. The super bilinear identity with respect to super wave and adjoint wave functions is also constructed. At last, we also give a class of solutions other than group orbit by the neutral free superfermions.

Super modified KP hierarchy in Kac–van de Leur construction

In this paper, the modified KP hierarchy in the Kupershmidt–Kiso version is extended to the super case by Kac–van de Leur construction, that is, using highest weight representations of the even part in the tensor product of the infinite-dimensional Lie superalgebra gl∞|∞ with Grassmann algebra G. First, the super modified KP (SmKP) hierarchy is constructed in terms of superfermionic bilinear equations. Then, the superbosonic form of the SmKP hierarchy is given by super boson–fermion correspondence. With the help of super Hirota bilinear operators, the corresponding super Hirota bilinear equations of the SmKP hierarchy are obtained. Next, the Darboux transformations of this new SmKP hierarchy are expressed in the form of free superfermions and various solutions are derived. Finally, the super bilinear equations in the form of super wave functions are also constructed from the superbosonic ones, which is hoped to be helpful to obtain the corresponding Lax structures.

The N = 1 super Heisenberg–Virasoro vertex algebra at level zero

We study the representation theory of the [Formula: see text] super Heisenberg–Virasoro vertex algebra at level zero, which extends the previous work on the Heisenberg–Virasoro vertex algebra [D. Adamović and G. Radobolja, Free field realization of the twisted Heisenberg–Virasoro algebra at level zero and its applications, J. Pure Appl. Algebra 219(10) (2015) 4322–4342; D. Adamović and G. Radobolja, Self-dual and logarithmic representations of the twisted Heisenberg–Virasoro algebra at level zero, Commun. Contemp. Math. 21(2) (2019) 1850008; Y. Billig, Representations of the twisted Heisenberg–Virasoro algebra at level zero, Can. Math. Bull. 46(4) (2003) 529–537] to the super case. We calculated all characters of irreducible highest weight representations by investigating certain Fock space representations. Quite surprisingly, we found that the maximal submodules of certain Verma modules are generated by subsingular vectors. The formulas for singular and subsingular vectors are obtained using screening operators appearing in a study of certain logarithmic vertex algebras [D. Adamović and A. Milas, On W-algebras associated to (2, [Formula: see text]) minimal models and their representations, Int. Math. Res. Notices 2010(20) (2010) 3896–3934].

On the tensor semigroup of affine Kac-Moody lie algebras

The support of the tensor product decomposition of integrable irreducible highest weight representations of a symmetrizable Kac-Moody Lie algebra g \mathfrak {g} defines a semigroup of triples of weights. Namely, given λ \lambda in the set P + P_+ of dominant integral weights, V ( λ ) V(\lambda ) denotes the irreducible representation of g \mathfrak {g} with highest weight λ \lambda . We are interested in the tensor semigroup Γ N ( g ) ≔ { ( λ 1 , λ 2 , μ ) ∈ P + 3 | V ( μ ) ⊂ V ( λ 1 ) ⊗ V ( λ 2 ) } , \begin{equation*} \Gamma _{\mathbb {N}}(\mathfrak {g})≔\{(\lambda _1,\lambda _2,\mu )\in P_{+}^3\,|\, V(\mu )\subset V(\lambda _1)\otimes V(\lambda _2)\}, \end{equation*} and in the tensor cone Γ ( g ) \Gamma (\mathfrak {g}) it generates: Γ ( g ) ≔ { ( λ 1 , λ 2 , μ ) ∈ P + , Q 3 | ∃ N ≥ 1 V ( N μ ) ⊂ V ( N λ 1 ) ⊗ V ( N λ 2 ) } . \begin{equation*} \Gamma (\mathfrak {g})≔\{(\lambda _1,\lambda _2,\mu )\in P_{+,{\mathbb {Q}}}^3\,|\,\exists N\geq 1 \quad V(N\mu )\subset V(N\lambda _1)\otimes V(N\lambda _2)\}. \end{equation*} Here, P + , Q P_{+,{\mathbb {Q}}} denotes the rational convex cone generated by P + P_+ . In the special case when g \mathfrak {g} is a finite-dimensional semisimple Lie algebra, the tensor semigroup is known to be finitely generated and hence the tensor cone to be convex polyhedral. Moreover, the cone Γ ( g ) \Gamma (\mathfrak {g}) is described in Belkale and Kumar [Invent. Math. 166 (2006), pp. 185–228] by an explicit finite list of inequalities. In general, Γ ( g ) \Gamma (\mathfrak {g}) is neither polyhedral, nor closed. In this article we describe the closure of Γ ( g ) \Gamma (\mathfrak {g}) by an explicit countable family of linear inequalities for any untwisted affine Lie algebra, which is the most important class of infinite-dimensional Kac-Moody algebra. This solves a Brown-Kumar’s conjecture in this case (see Brown and Kumar [Math. Ann. 360 (2014), pp. 901–936]). The difference between the tensor cone and the tensor semigroup is measured by the saturation factors. Namely, a positive integer d d is called a saturation factor, if V ( N λ 1 ) ⊗ V ( N λ 2 ) V(N\lambda _1)\otimes V(N\lambda _2) contains V ( N μ ) V(N\mu ) for some positive integer N N then V ( d λ 1 ) ⊗ V ( d λ 2 ) V(d\lambda _1)\otimes V(d\lambda _2) contains V ( d μ ) V(d\mu ) , assuming that μ − λ 1 − λ 2 \mu -\lambda _1-\lambda _2 belongs to the root lattice. For g = s l n \mathfrak {g}={\mathfrak {sl}}_n , the famous Knutson-Tao theorem asserts that d = 1 d=1 is a saturation factor (see Knutson and Tao [J. Amer. Math. Soc. 12 (1999), pp. 1055–1090]). More generally, for any simple Lie algebra, explicit saturation factors are known. In the Kac-Moody case, Γ N ( g ) \Gamma _{\mathbb {N}}(\mathfrak {g}) is not necessarily finitely generated and hence the existence of such a factor is unclear a priori. Here, we obtain explicit saturation factors for any affine Kac-Moody Lie algebra. For example, in type A ~ n \tilde A_n , we prove that any integer d ≥ 2 d\geq 2 is a saturation factor, generalizing the case A ~ 1 \tilde A_1 shown in Brown and Kumar [Math. Ann. 360 (2014), pp. 901–936].

Hidden Symmetry of Vanishing Love Numbers

We show that perturbations of massless fields in a black hole background enjoy a hidden SL(2,R)×U(1) ("Love") symmetry in the properly defined near zone approximation. Love symmetry mixes low- and high-frequency modes. Still, this approximate symmetry allows us to derive exact results about static tidal responses. Generators of the Love symmetry are globally well defined for any value of black hole spin. Generic regular solutions of the near zone equation for linearized perturbations form infinite-dimensional SL(2,R) representations. In some special cases, these are highest weight representations. This situation corresponds to vanishing Love numbers. Other known facts about static Love numbers also acquire an elegant explanation in terms of the SL(2,R) representation theory.