Irreducible Subrepresentations Research Articles

Monodromy of subrepresentations and irreducibility of low degree automorphic Galois representations

AbstractLet be a smooth, separated, geometrically connected scheme defined over a number field and a system of semisimple ‐adic representations of the étale fundamental group of such that for each closed point of , the specialization is a compatible system of Galois representations under mild local conditions. For almost all , we prove that any type A irreducible subrepresentation of is residually irreducible. When is totally real or CM, , and is the compatible system of Galois representations of attached to a regular algebraic, polarized, cuspidal automorphic representation of , for almost all , we prove that is (i) irreducible and (ii) residually irreducible if in addition .

Split Casimir operator for simple Lie algebras in the cube of ad-representation and Vogel parameters

In this paper, we constructed characteristic identities for the 3-split (polarized) Casimir operators of simple Lie algebras in the adjoint representations [Formula: see text] and deduced a certain class of subrepresentations in [Formula: see text]. The projectors onto invariant subspaces for these subrepresentations were directly constructed from the characteristic identities for the 3-split Casimir operators. For all simple Lie algebras, universal expressions for the traces of higher powers of the 3-split Casimir operators were found and dimensions of the subrepresentations in [Formula: see text] were calculated. All our formulas are in agreement with the universal description of (irreducible) subrepresentations in [Formula: see text] for simple Lie algebras in terms of the Vogel parameters.

From harmonic analysis of translation-invariant valuations to geometric inequalities for convex bodies

The Alesker–Bernig–Schuster theorem asserts that each irreducible representation of the special orthogonal group appears with multiplicity at most one as a subrepresentation of the space of continuous translation-invariant valuations with fixed degree of homogeneity. Moreover, the theorem describes in terms of highest weights which irreducible representations appear with multiplicity one. In this paper, we present a refinement of this result, namely the explicit construction of a highest weight vector in each irreducible subrepresentation. We then describe how important natural operations on valuations (pullback, pushforward, Fourier transform, Lefschetz operator, Alesker–Poincaré pairing) act on these highest weight vectors. We use this information to prove the Hodge–Riemann relations for valuations in the case of Euclidean balls as reference bodies. Since special cases of the Hodge–Riemann relations have recently been used to prove new geometric inequalities for convex bodies, our work immediately extends the scope of these inequalities.

Invariant Einstein metrics on real flag manifolds with two or three isotropy summands

We study the existence of invariant Einstein metrics on real flag manifolds associated to simple and non-compact split real forms of complex classical Lie algebras whose isotropy representation decomposes into two or three irreducible subrepresentations. In this situation, one can have equivalent submodules, leading to the existence of non-diagonal homogeneous Riemannian metrics. In particular, we prove the existence of non-diagonal Einstein metrics on real flag manifolds.

Strong representation equivalence for compact symmetric spaces of real rank one

Let $G/K$ be a simply connected compact irreducible symmetric space of real rank one. For each $K$-type $\tau$ we compare the notions of $\tau$-representation equivalence with $\tau$-isospectrality. We exhibit infinitely many $K$-types $\tau$ so that, for arbitrary discrete subgroups $\Gamma$ and $\Gamma'$ of $G$, if the multiplicities of $\lambda$ in the spectra of the Laplace operators acting on sections of the induced $\tau$-vector bundles over $\Gamma\backslash G/K$ and $\Gamma'\backslash G/K$ agree for all but finitely many $\lambda$, then $\Gamma$ and $\Gamma'$ are $\tau$-representation equivalent in $G$ (i.e.\ $\dim \operatorname{Hom}_G(V_\pi, L^2(\Gamma\backslash G))=\dim \operatorname{Hom}_G(V_\pi, L^2(\Gamma'\backslash G))$ for all $\pi\in \widehat G$ satisfying $\operatorname{Hom}_K(V_\tau,V_\pi)\neq0$). In particular $\Gamma\backslash G/K$ and $\Gamma'\backslash G/K$ are $\tau$-isospectral (i.e.\ the multiplicities agree for all $\lambda$). We specially study the case of $p$-form representations, i.e. the irreducible subrepresentations $\tau$ of the representation $\tau_p$ of $K$ on the $p$-exterior power of the complexified cotangent bundle $\bigwedge^p T_{\mathbb C}^*M$. We show that for such $\tau$, in most cases $\tau$-isospectrality implies $\tau$-representation equivalence. We construct an explicit counter-example for $G/K= \operatorname{SO}(4n)/ \operatorname{SO}(4n-1)\simeq S^{4n-1}$.

Split Casimir operator for simple Lie algebras, solutions of Yang–Baxter equations, and Vogel parameters

We construct characteristic identities for the split (polarized) Casimir operators of the simple Lie algebras in defining (minimal fundamental) and adjoint representations. By means of these characteristic identities, for all simple Lie algebras, we derive explicit formulas for invariant projectors onto irreducible subrepresentations in T⊗2 in two cases, when T is the defining and the adjoint representation. In the case when T is the defining representation, these projectors and the split Casimir operator are used to explicitly write down invariant solutions of the Yang–Baxter equations. In the case when T is the adjoint representation, these projectors and characteristic identities are considered from the viewpoint of the universal description of the simple Lie algebras in terms of the Vogel parameters.

Flat manifolds with homogeneous holonomy representation

We show that a rational holonomy representation of any flat manifold except torus must have at least two non-equivalent irreducible subrepresentations. As an application we show that if a K\"ahler flat manifold is not a torus then its holonomy representation is reducible.

Split Casimir Operator and Universal Formulation of the Simple Lie Algebras

We construct characteristic identities for the split (polarized) Casimir operators of the simple Lie algebras in adjoint representation. By means of these characteristic identities, for all simple Lie algebras we derive explicit formulae for invariant projectors onto irreducible subrepresentations in T⊗2 in the case when T is the adjoint representation. These projectors and characteristic identities are considered from the viewpoint of the universal description of the simple Lie algebras in terms of the Vogel parameters.

Rank-deficient representations in the theta correspondence over finite fields arise from quantum codes

Let V V be a symplectic vector space and let μ \mu be the oscillator representation of Sp ⁡ ( V ) \operatorname {Sp}(V) . It is natural to ask how the tensor power representation μ ⊗ t \mu ^{\otimes t} decomposes. If V V is a real vector space, then the theta correspondence asserts that there is a one-one correspondence between the irreducible subrepresentations of Sp ⁡ ( V ) \operatorname {Sp}(V) and the irreps of an orthogonal group O ( t ) O(t) . It is well-known that this duality fails over finite fields. Addressing this situation, Gurevich and Howe have recently assigned a notion of rank to each Sp ⁡ ( V ) \operatorname {Sp}(V) representation. They show that a variant of the Theta correspondence continues to hold over finite fields, if one restricts attention to subrepresentations of maximal rank. The nature of the rank-deficient components was left open. Here, we show that all rank-deficient Sp ⁡ ( V ) \operatorname {Sp}(V) -subrepresentations arise from embeddings of lower-order tensor products of μ \mu and μ ¯ \bar \mu into μ ⊗ t \mu ^{\otimes t} . The embeddings live on spaces that have been studied in quantum information theory as tensor powers of self-orthogonal Calderbank-Shor-Steane (CSS) quantum codes. We then find that the irreducible Sp ⁡ ( V ) \operatorname {Sp}(V) -subrepresentations of μ ⊗ t \mu ^{\otimes t} are labelled by the irreps of orthogonal groups O ( r ) O(r) acting on certain r r -dimensional spaces for r ≤ t r\leq t . The results hold in odd charachteristic and the “stable range” t ≤ 1 2 dim ⁡ V t\leq \frac 12 \dim V . Our work has implications for the representation theory of the Clifford group. It can be thought of as a generalization of the known characterization of the invariants of the Clifford group in terms of self-dual codes.

The Infinitesimal Characters of Discrete Series for Real Spherical Spaces

Let $$Z=G/H$$ be the homogeneous space of a real reductive group and a unimodular real spherical subgroup, and consider the regular representation of G on $$L^2(Z)$$ . It is shown that all representations of the discrete series, that is, the irreducible subrepresentations of $$L^2(Z)$$ , have infinitesimal characters which are real and belong to a lattice. Moreover, let K be a maximal compact subgroup of G. Then each irreducible representation of K occurs in a finite set of such discrete series representations only. Similar results are obtained for the twisted discrete series, that is, the discrete components of the space of square integrable sections of a line bundle, given by a unitary character on an abelian extension of H.

Born's Rule from the Principle of Unitary Equivalence

Complex phase factors are viewed not only as redundancies of the quantum formalism but instead as remnants of unitary transformations under which the probabilistic properties of observables are invariant. It is postulated that a quantum observable corresponds to a unitary representation of an abelian Lie group, the irreducible subrepresentations of which correspond to the observable's outcomes. It is shown that this identification agrees with the conventional identification as self-adjoint operators. The upshot of this formalism is that one may 'second quantize' the representation to which an observable corresponds, thus obtaining the corresponding Fock space representation. This Fock space representation is then also identifiable as an observable in the same sense, the outcomes of which are naturally interpretable as ensembles of outcomes of the corresponding non-second quantized observable. The frequency interpretation of probability is adopted, i.e. probability as the average occurrence, from which Born's rule is deduced by enforcing the notion of 'average' to such that are invariant under the second quantized unitary representation which defines the quantum observable to which the initial state is an outcome. The enforcement of this invariance is an application the principle referred to as the 'principle of unitary equivalence'.

On typical representations for depth-zero components of split classical groups

Let G \mathbf {G} be a split classical group over a non-Archimedean local field F F with the cardinality of the residue field q F > 5 q_F>5 . Let M M be the group of F F -points of a Levi factor of a proper F F -parabolic subgroup of G \mathbf {G} . Let [ M , σ M ] M [M, \sigma _M]_M be an inertial class such that σ M \sigma _M contains a depth-zero Moy–Prasad type of the form ( K M , τ M ) (K_M, \tau _M) , where K M K_M is a hyperspecial maximal compact subgroup of M M . Let K K be a hyperspecial maximal compact subgroup of G ( F ) \mathbf {G}(F) such that K K contains K M K_M . In this article, we classify s \mathfrak {s} -typical representations of K K . In particular, we show that the s \mathfrak {s} -typical representations of K K are precisely the irreducible subrepresentations of ind J K ⁡ λ \operatorname {ind}_J^K\lambda , where ( J , λ ) (J, \lambda ) is a level-zero G G -cover of ( K ∩ M , τ M ) (K\cap M, \tau _M) .

Extremal unipotent representations for the finite Howe correspondence

We study the Howe correspondence for unipotent representations of irreducible dual pairs (G,G′)=(Um(Fq),Un(Fq)) and (G,G′)=(Sp2m(Fq),O2nϵ(Fq)), where Fq denotes the finite field with q elements (q odd) and ϵ=±1. We show how to extract extremal (i.e. minimal and maximal) irreducible subrepresentations from the image Θ(π′) of a unipotent representation π′ of G′.

Permutative representations of the 2-adic ring C∗-algebra

The notion of permutative representation is generalized to the 2-adic ring C∗-algebra Q2. Permutative representations of Q2 are then investigated with a particular focus on the inclusion of the Cuntz algebra O2⊂Q2. Notably, every permutative representation of O2 is shown to extend automatically to a permutative representation of Q2 provided that an extension whatever exists. Moreover, all permutative extensions of a given representation of O2 are proved to be unitarily equivalent to one another. Irreducible permutative representations of Q2 are classified in terms of irreducible permutative representations of the Cuntz algebra. Apart from the canonical representation of Q2, every irreducible representation of Q2 is the unique extension of an irreducible permutative representation of O2. Furthermore, a permutative representation of Q2 will decompose into a direct sum of irreducible permutative subrepresentations if and only if it restricts to O2 as a \textit{regular} representation in the sense of Bratteli--Jorgensen. As a result, a vast class of pure states of O2 is shown to enjoy the unique pure extension property with respect to the inclusion O2⊂Q2.

A continuous frame representation for an abstract shearlet group

Let \({\mathbb {S}}\) be a locally compact abstract shearlet group and \(\sigma :{\mathbb {S}}\rightarrow {\mathcal {U}}({\mathcal {H}})\) be a representation of \({\mathbb {S}}\) on a separable Hilbert space \({\mathcal {H}}\). We give a necessary and sufficient condition for the family of representation coefficients to be a continuous shearlet frame. Furthermore relations between continuous shearlet frames associated to a given representation and its irreducible sub-representations are investigated.

Representations of the multi-qubit Clifford group

The q-qubit Clifford group, that is, the normalizer of the q-qubit Pauli group in U(2q), is a fundamental structure in quantum information with a wide variety of applications. We characterize all irreducible subrepresentations of the two-copy representation φ⊗2 of the Clifford group on the two-fold tensor product of the space of linear operators M2q⊗2. In the companion paper [Helsen et al., e-print arXiv:1701.04299 (2017)], we apply this result to improve the statistics of randomized benchmarking, a method for characterizing quantum systems.

Local diffeomorphisms and smooth embeddings to gravitational field II: spherical symmetry and their breaking in the space-time

Consequences of the diffeomorphisms induced by K− invariant connections of the space of 1-forms of certain endomorphisms defined over a Lie algebra that is isomorphic to the tangent space seated in the identity element, of homogeneous spaces G/K⊂G/H , are analized. The images of these diffeomorphisms in G/H , are 2-form of curvatures that can be induced to each class of the G/K . Then using the K− invariant connection of this homogeneous space, the curvature can be determined as a regular representation that admits a finite discomposing of irreducible sub-representations of finite type, accord with the generalizing in dimensions of the Gauss-Bonnet theorem and the generalized Radon transform to obtain curvature through of co-cycles of the image of the corresponding space. Such irreducible sub-representations will be isotopic components of the certain smoothly embedded image in a manifold modelled this last, by a generalized function space. Likewise, through these realizations we have the curvature integrals as dual case of their field equations. Finally, using the complex Riemannian structure of our model of the space-time, and the K− invariant G− structure of the orbits used to obtain curvature, are obtained as consequences of the diffeomorphisms the field equations to the energy-matter tensor density in each case of the gravitational field. Of this manner, is determined their energy-mass tensor density as an integral which represents the energy spectra of the curvature when this is obtained in duality to the homogeneous field equations to the Riemann tensor Rμν−12gμν R.

Generic Steady State Bifurcations in Monoid Equivariant Dynamics with Applications in Homogeneous Coupled Cell Systems

We prove that steady state bifurcations in finite-dimensional dynamical systems that are symmetric with respect to a monoid representation generically occur along an absolutely indecomposable subrepresentation. This is stated as a conjecture in B. Rink and J. Sanders, "Coupled cell networks and their hidden symmetries", SIAM J. Math. Anal., 46 (2014). It is a generalization of the well-known fact that generic steady state bifurcations in equivariant dynamical systems occur along an absolutely irreducible subrepresentation if the symmetries form a group - finite or compact Lie. Our generalization also includes non-compact symmetry groups. The result has applications in bifurcation theory of homogeneous coupled cell networks as they can be embedded (under mild additional assumptions) into monoid equivariant systems.

Frame vector multipliers for finite group representations

A frame vector (or generator) for a group representation π of a countable or finite group G on a Hilbert space H is a vector ξ∈H such that {π(g)ξ}g∈G is a Parseval frame for H. Frame vector multipliers are the unitary operators on H that map frame vectors to frame vectors. Based on a characterization of frame vectors with respect to the standard decomposition of a group representation as the direct sums of irreducible subrepresentations (with multiplicity), we obtain explicit characterizations of frame generator multipliers for two basic cases for finite group representations. With the help of these characterizations we obtain some necessary conditions of frame vector multipliers for general frame representations, and present several examples to demonstrate how these results can be used to get explicit characterizations for frame vector multipliers.

Local cohomology with support in generic determinantal ideals

For positive integers m >= n >= p, we compute the GL_m x GL_n-equivariant description of the local cohomology modules of the polynomial ring S of functions on the space of m x n matrices, with support in the ideal of p x p minors. Our techniques allow us to explicitly compute all the modules Ext_S(S/I_x,S), for x a partition and I_x the ideal generated by the irreducible sub-representation of S indexed by x. In particular we determine the regularity of the ideals I_x, and we deduce that the only ones admitting a linear free resolution are the powers of the ideal of maximal minors of the generic matrix, as well as the products between such powers and the maximal ideal of S.