Representations Of SL Research Articles

Construction of even-level representations of [formula omitted] with residue field of characteristic two

We construct the finite-dimensional continuous complex representations of SL2 over compact discrete valuation rings of even residual characteristic, assuming the level is large enough compared to the ramification index, in the mixed characteristic case. We also prove that the complex group algebras of SL2 over finite quotient rings of such compact discrete valuation rings depend on the characteristic of the ring. In particular, we prove that the group algebras C[SL2(Z/2rZ)] and C[SL2(F2[t]/(tr))] are not isomorphic for any r≥4.

On the asymptotics of matrix coefficients of representations of [formula omitted]

We study matrix coefficients of the unitary (and also the completely bounded) representations of SL(2,R) and its universal covering group. We describe the asymptotic distribution of column vectors in terms of Whittaker functions, exhibiting also a relationship to the Fourier transform.

SL2 representations and relative property (T)

We investigate relative property (T) of Kazhdan-Margulis for (SL2(R)⋉Rn,Rn), mainly where R is a discrete finitely generated commutative ring. The action of SL2(R) on Rn is defined through admissible lattices of the irreducible representations of sl2(C). Both positive and negative results are obtained. Applications of these results will be given, including the establishment of property (T) for certain Kac-Moody type groups over commutative rings.

Note on the Classification of the Unitary $\rm{SL}_2(\mathbb{R})$ Representations

The aim of this paper is to explain the classification of the unitary SL_2(R) representations done by Gelfand by using the induced representation technique. We induce the SL_2(R) representation from the subgroup N. We get a representation constructed on a space of homogeneous functions in two variables. Then, we move to induce the SL_2(R) representation in stages. Consequently, the representation of SL_2(R) acts on a space of functions of one variable.

Tableau evacuation and webs

Webs are certain planar diagrams embedded in disks. They index and describe bases of tensor products of representations of s l 2 \mathfrak {sl}_2 and s l 3 \mathfrak {sl}_3 . There are explicit bijections between webs and certain rectangular tableaux. Work of Petersen–Pylyavskyy–Rhoades (2009) and Russell (2013) shows that these bijections relate web rotation to tableau promotion. We describe the analogous relation between web reflection and tableau evacuation.

Embedding quantum PSU(2) representations of SL 2(ℤ) in quantum PSU(3) representations

We demonstrate how the quantum PSU(2) representations of S L 2 ( Z ) can be embedded as a direct summand in the quantum PSU(3) representations.

On the structure of principal series representations of SL2(𝔽̄p)

Let [Formula: see text] be a prime number and [Formula: see text] be the algebraic closure of the finite field [Formula: see text] with [Formula: see text] elements. Let [Formula: see text] be an algebraically closed field of characteristic [Formula: see text]. Let [Formula: see text] and [Formula: see text] be the standard Borel subgroup of [Formula: see text]. For any character [Formula: see text] of [Formula: see text], define [Formula: see text], where [Formula: see text] is the group algebra of [Formula: see text]. In this paper, we prove two interesting properties of [Formula: see text]: (a) [Formula: see text] is either of finite length or residually finite; (b) Any nontrivial quotient of [Formula: see text] is finite dimensional.

Counterexamples to the Zassenhaus conjecture on simple modular Lie algebras

We provide an infinite family of counterexamples to the conjecture of Zassenhaus on the solvability of the outer derivation algebra of a simple modular Lie algebra. In fact, we show that the simple modular Lie algebras H(2;(1,n))(2) of dimension 3n+1−2 in characteristic p=3 do not have a solvable outer derivation algebra for all n≥1. For n=1 this recovers the known counterexample of psl3(F). We show that the outer derivation algebra of H(2;(1,n))(2) is isomorphic to (sl2(F)⋉V(2))⊕Fn−1, where V(2) is the natural representation of sl2(F). We also study other known simple Lie algebras in characteristic three, but they do not yield a new counterexample.

On symmetric representations of 𝑆𝐿₂(ℤ)

We introduce the notions of symmetric and symmetrizable representations of SL 2 ⁡ ( Z ) {\operatorname {SL}_2(\mathbb {Z})} . The linear representations of SL 2 ⁡ ( Z ) {\operatorname {SL}_2(\mathbb {Z})} arising from modular tensor categories are symmetric and have congruence kernel. Conversely, one may also reconstruct modular data from finite-dimensional symmetric, congruence representations of SL 2 ⁡ ( Z ) {\operatorname {SL}_2(\mathbb {Z})} . By investigating a Z / 2 Z \mathbb {Z}/2\mathbb {Z} -symmetry of some Weil representations at prime power levels, we prove that all finite-dimensional congruence representations of SL 2 ⁡ ( Z ) {\operatorname {SL}_2(\mathbb {Z})} are symmetrizable. We also provide examples of unsymmetrizable noncongruence representations of SL 2 ⁡ ( Z ) {\operatorname {SL}_2(\mathbb {Z})} that are subrepresentations of a symmetric one.

Representation growth of compact special linear groups of degree two

We study the finite-dimensional continuous complex representations of SL2 over the ring of integers of non-Archimedean local fields of even residual characteristic. We prove that for characteristic two, the abscissa of convergence of the representation zeta function is 1, resolving the last remaining open case of this problem. We additionally prove that, contrary to the expectation, the group algebras of C[SL2(Z/(22rZ))] and C[SL2(F2[t]/(t2r))] are not isomorphic for any r>1. This is the first known class of reductive groups over finite rings wherein the representation theory in the equal and mixed characteristic settings is genuinely different. From our methods, we explicitly obtain the primitive representation zeta polynomials of SL2(F2[t]/(t2r)) and SL2(Z/22rZ) for 1≤r≤3.

Components of eleven-dimensional supergravity with four off-shell supersymmetries

We derive the component structure of 11D, N = 1/8 supergravity linearized around eleven-dimensional Minkowski space. This theory represents 4 local supersymmetries closing onto 4 of the 11 spacetime translations without the use of equations of motion. It may be interpreted as adding 201 auxiliary bosons and 56 auxiliary fermions to the physical supergravity multiplet for a total of 376 + 376 components. These components and their transformations are organized into representations of SL(2; C) × G2.

Plethysms of symmetric functions and representations of SL 2 (C)

Let ∇λ denote the Schur functor labelled by the partition λ and let E be the natural representation of SL2(C). We make a systematic study of when there is an isomorphism ∇λSymlE≅∇μSymmE of representations of SL2(C). Generalizing earlier results of King and Manivel, we classify all such isomorphisms when λ and μ are conjugate partitions and when one of λ or μ is a rectangle. We give a complete classification when λ and μ each have at most two rows or columns or is a hook partition and a partial classification when l=m. As a corollary of a more general result on Schur functors labelled by skew partitions we also determine all cases when ∇λSymlE is irreducible. The methods used are from representation theory and combinatorics; in particular, we make explicit the close connection with MacMahon’s enumeration of plane partitions, and prove a new q-binomial identity in this setting

(𝔤,K)-module of O(p,q) associated with the finite-dimensional representation of 𝔰𝔩2

The main aim of this paper is to show that one can construct [Formula: see text]-modules of [Formula: see text] associated with the finite-dimensional representation of [Formula: see text] by quantizing the moment map on the symplectic vector space [Formula: see text] and using the fact that [Formula: see text] is a dual pair. Then one obtains the [Formula: see text]-type formula, the Gelfand–Kirillov dimension and the Bernstein degree of them for all non-negative integers [Formula: see text] satisfying [Formula: see text] when [Formula: see text] and [Formula: see text] is even. In fact, one finds that the Gelfand–Kirillov dimension is equal to [Formula: see text] and the Bernstein degree is equal to [Formula: see text].

More on Wilson toroidal networks and torus blocks

We consider the Wilson line networks of the Chern-Simons 3d gravity theory with toroidal boundary conditions which calculate global conformal blocks of degenerate quasi-primary operators in torus 2d CFT. After general discussion that summarizes and further extends results known in the literature we explicitly obtain the one-point torus block and two-point torus blocks through particular matrix elements of toroidal Wilson network operators in irreducible finite-dimensional representations of sl(2, ℝ) algebra. The resulting expressions are given in two alternative forms using different ways to treat multiple tensor products of sl(2, ℝ) representations: (1) 3mj Wigner symbols and intertwiners of higher valence, (2) totally symmetric tensor products of the fundamental sl(2, ℝ) representation.

Covariants, Invariant Subsets, and First Integrals

Let $k$ be an algebraically closed field of characteristic 0, and let $V$ be a finite-dimensional vector space. Let $End(V)$ be the semigroup of all polynomial endomorphisms of $V$. Let $E$ be a subset of $End(V)$ which is a linear subspace and also a semi-subgroup. Both $End(V)$ and $E$ are ind-varieties which act on $V$ in the obvious way. In this paper, we study important aspects of such actions. We assign to $E$ a linear subspace $D_{E}$ of the vector fields on $V$. A subvariety $X$ of $V$ is said to $D_{E}$ -invariant if $h(x)$ is in the tangent space of $x$ for all $h$ in $D_{E}$ and $x$ in $X$. We show that $X$ is $D_{E}$ -invariant if and only if it is the union of $E$-orbits. For such $X$, we define first integrals and construct a quotient space for the $E$-action. An important case occurs when $G$ is an algebraic subgroup of $GL(V$) and $E$ consists of the $G$-equivariant polynomial endomorphisms. In this case, the associated $D_{E}$ is the space the $G$-invariant vector fields. A significant question here is whether there are non-constant $G$-invariant first integrals on $X$. As examples, we study the adjoint representation, orbit closures of highest weight vectors, and representations of the additive group. We also look at finite-dimensional irreducible representations of SL2 and its nullcone.

Codimension growth for weak polynomial identities, and non-integrality of the PI exponent

Let K be a field of characteristic zero. In this paper, we study the polynomial identities of representations of Lie algebras, also called weak identities, or identities of pairs. These identities are determined by pairs of the form (A, L) where A is an associative enveloping algebra for the Lie algebra L. Then a weak identity of (A, L) (or an identity for the representation of L associated to A) is an associative polynomial which vanishes when evaluated on elements of L⊆ A. One of the most influential results in the area of PI algebras was the theory developed by Kemer. A crucial role in it was played by the construction of the Grassmann envelope of an associative algebra and the close relation of the identities of the algebra and its Grassmann envelope. Here we consider varieties of pairs. We prove that under some restrictions one can develop a theory similar to that of Kemer's in the study of identities of representations of Lie algebras. As a consequence, we establish that in the case when K is algebraically closed, if a variety of pairs does not contain pairs corresponding to representations of sl2(K), and if the variety is generated by a pair where the associative algebra is PI then it is soluble. As another consequence of the methods used to obtain the above result, and applying ideas from papers by Giambruno and Zaicev, we were able to construct a pair (A, L) such that its PI exponent (if it exists) cannot be an integer. We recall that the PI exponent exists and is an integer whenever A is an associative (a theorem by Giambruno and Zaicev), or a finite-dimensional Lie algebra (Zaicev). Gordienko also proved that the PI exponent exists and is an integer for finite-dimensional representations of Lie algebras.

Harmonic symmetries for Hermitian manifolds

Complex manifolds with compatible metric have a naturally defined subspace of harmonic differential forms that satisfy Serre, Hodge, and conjugation duality, as well as hard Lefschetz duality. This last property follows from a representation of s l ( 2 , C ) \mathfrak {sl}(2,\mathbb {C}) , generalizing the well-known structure on the harmonic forms of compact Kähler manifolds. Some topological implications are deduced.

On the space of 𝐾-finite solutions to intertwining differential operators

In this paper we give Peter–Weyl-type decomposition theorems for the space of K K -finite solutions to intertwining differential operators between parabolically induced representations. Our results generalize a result of Kable for conformally invariant systems. The main idea is based on the duality theorem between intertwining differential operators and homomorphisms between generalized Verma modules. As an application we uniformly realize on the solution spaces of intertwining differential operators all small representations of S L ~ ( 3 , R ) \widetilde {\mathrm {SL}}(3,\mathbb {R}) attached to the minimal nilpotent orbit.

The Loewy series of principal indecomposable modules in the regular nilpotent representations of 𝔰𝔩(3,k)

Let [Formula: see text] be a simple Lie algebra of type [Formula: see text] over an algebraically closed field [Formula: see text] of prime characteristic [Formula: see text] and [Formula: see text] be the reduced enveloping algebra of [Formula: see text]. In this paper, when the [Formula: see text]-character [Formula: see text] is regular nilpotent and has standard Levi form, we precisely determine the Lowey series of principal indecomposable [Formula: see text]-modules and the dimensions for the self-extension of irreducible [Formula: see text]-modules.

A combinatorial 𝔰𝔩₂-action and the Sperner property for the weak order

We construct a simple combinatorially-defined representation of s l 2 \mathfrak {sl}_2 which respects the order structure of the weak order on the symmetric group. This is used to prove that the weak order has the strong Sperner property, and is therefore a Peck poset, solving a problem raised by Björner [Orderings of Coxeter groups, Amer. Math. Soc., Providence, RI, 1984, pp. 175–195]; a positive answer to this question had been conjectured by Stanley [Some Schubert shenanigans, preprint, 2017].