Representation Ring Research Articles

Hitchin's conjecture for simply-laced Lie algebras implies that for any simple Lie algebra

Let g be any simple Lie algebra over C. Recall that there exists an embedding of sl2 into g, called a principal TDS, passing through a principal nilpotent element of g and uniquely determined up to conjugation. Moreover, ∧(g⁎)g is freely generated (in the super-graded sense) by primitive elements ω1,…,ωℓ, where ℓ is the rank of g. N. Hitchin conjectured that for any primitive element ω∈∧d(g⁎)g, there exists an irreducible sl2-submodule Vω⊂g of dimension d such that ω is non-zero on the line ∧d(Vω). We prove that the validity of this conjecture for simple simply-laced Lie algebras implies its validity for any simple Lie algebra.Let G be a connected, simply-connected, simple, simply-laced algebraic group and let σ be a diagram automorphism of G with fixed subgroup K. Then, we show that the restriction map R(G)→R(K) is surjective, where R denotes the representation ring over Z. As a corollary, we show that the restriction map in the singular cohomology H⁎(G)→H⁎(K) is surjective. Our proof of the reduction of Hitchin's conjecture to the simply-laced case relies on this cohomological surjectivity.

On $p$-adic Speh representations

This note contains simple proofs of some known results (unitarity, character formula) on Speh representations of a group GL n (D) where D is a local non Archimedean division algebra of any characteristic. Introduction. In this note I give simple proofs of some known results on Speh representations of groups GL n (D) where D is a central division algebra of finite dimension over a local non archimedean field F of any characteristic. The new idea is to use the Moeglin-Waldspurger algorithm (MWA) for computing the dual of an irreducible representation. For the unitarizability of Speh representations , previous proofs were based either on the trace formula and the close fields theory, or on deep results of type theory. The proof here is combinatoric , independent of D, the characteristic , and Bernstein's (also called U0) theorem. Short general proofs using MWA are also given for other known facts. The proof is always the same: one wants to prove a relation (R) involving some representations (for example an induced representation π is irreducible). One starts by writing the naive relation (R') between these representations, known from standard theory, but not as strong as (R) (for example the semi simplification of π is a sum with non negative coefficients of some irreducible representations). Usually (R') has more terms that (R), because it is weaker, and one wants to prove that some terms which are supposed to appear in (R') are actually not there (for example all the subquotients of π except of the expected one have coefficient zero). The method then is to consider also the dual relation (R ) to (R'), and to play with the MWA, in order to show that the mild constraints one has on (R') and (R ) are enough to show that the extra terms are not there and (R') reduces actually to (R). All the results in this paper are already known, and here I only give new short proofs. So I kindly ask the reader, when using one of these facts, to quote, at least in first place, the original reference (see the historical notice at the end of the paper). Beside the Zelevinsky and Tadi´cTadi´c classification of the admissible dual ([Ze], [Ta2]), the proofs here rely on [Au] (dual of an irreducible representation is irreducible), [MW] and [BR2] (algorithm for computing the dual) and some easy tricks from [Ta1], [Ba3] and [CR], and do not involve any complicated technique. The idea of searching for simple proof for classification of unitary representations, together with a list of basic tricks to use, are due to Marko Tadi´cTadi´c ([Ta3] for example). He also was the first to formulate some properties of Speh representations (formula for ends of complimentary series, character formula, Speh representations are prime elements in the ring of representations, dual of a Speh representation is Speh) and to prove them when D = F. The starting point of my proof here of the assertion Speh representations are unitary is also due to Tadi´cTadi´c who found the simple but brilliant trick reducing the problem of unitarity to a problem of irreducibility. I would like to thank Guy Henniart who read the paper and made useful observations.

The Green Ring of Drinfeld Double D(H 4)

In this paper, we study the Green ring (or the representation ring) of Drinfeld quantum double $D(H_4)$ of Sweedler's 4-dimensional Hopf algebra $H_4$. We first give the decompositions of the tensor products of finite dimensional indecomposable modules into the direct sum of indecomposable modules over $D(H_4)$. Then we describe the structure of the Green ring $r(D(H_4))$ of $D(H_4)$ and show that $r(D(H_4))$ is generated, as a ring, by infinitely many elements subject to a family of relations.

Brauer relations in finite groups II: Quasi-elementary groups of order paq

Abstract. This is the second in a series of papers investigating the space of Brauer relations of a finite group, the kernel of the natural map from its Burnside ring to the rational representation ring. The first paper classified all primitive Brauer relations, that is those that do not come from proper subquotients. In the case of quasi-elementary groups the description is intricate, and it does not specify groups that have primitive relations in terms of generators and relations. In this paper we provide such a classification in terms of generators and relations for quasi-elementary groups of order paq.

Functors from association schemes

We construct a wide subcategory of the category of finite association schemes with a collection of desirable properties. Our subcategory has a first isomorphism theorem analogous to that of groups. Also, standard constructions taking schemes to groups (thin radicals and quotients by thin residues) or algebras (adjacency algebras) become functorial when restricted to our category. We use our category to give a more conceptual account for a result of Hanaki concerning products of characters of association schemes; i.e. we show that the virtual representations of an association scheme form a module over the representation ring of the thin quotient of the association scheme.

Equivariant K-theory of flag varieties revisited and related results

In this article we obtain many results on the multiplicative structure constants of $T$-equivariant Grothendieck ring of the flag variety $G/B$. We do this by lifting the classes of the structure sheaves of Schubert varieties in $K_{T}(G/B)$ to $R(T)\otimes R(T)$, where $R(T)$ denotes the representation ring of the torus $T$. We further apply our results to describe the multiplicative structure constants of $K(X)_{\mathbb Q}$ where $X$ is the wonderful compactification of the adjoint group of $G$, in terms of the structure constants of Schubert varieties in the Grothendieck ring of $G/B$.

The Hopf algebra structure of the character rings of classical groups

The character ring of covariant irreducible tensor representations of the general linear group admits a Hopf algebra structure isomorphic to the Hopf algebra of symmetric functions. Here we study the character rings and of the orthogonal and symplectic subgroups of the general linear group within the same framework of symmetric functions. We show that and also admit natural Hopf algebra structures that are isomorphic to that of , and hence to . The isomorphisms are determined explicitly, along with the specification of standard bases for and analogous to those used for . A major structural change arising from the adoption of these bases is the introduction of new orthogonal and symplectic Schur–Hall scalar products. Significantly, the adjoint with respect to multiplication no longer coincides, as it does in the case, with a Foulkes derivative or skew operation. The adjoint and Foulkes derivative now require separate definitions, and their properties are explored here in the orthogonal and symplectic cases. Moreover, the Hopf algebras and are not self-dual. The dual Hopf algebras and are identified. Finally, the Hopf algebra of the universal rational character ring of mixed irreducible tensor representations of the general linear group is introduced and its structure maps identified.

Idempotents in representation rings of quivers

For an acyclic quiver Q, we solve the Clebsch-Gordan problem for the projective representations by computing the multiplicity of a given indecomposable projective in the tensor product of two indecomposable projectives. Motivated by this problem for arbitrary representations, we study idempotents in the representation ring of Q (the free abelian group on the indecomposable representations, with multiplication given by tensor product). We give a general technique for constructing such idempotents and for decomposing the representation ring into a direct product of ideals, utilizing morphisms between quivers and categorical Moebius inversion.

Equivariant K -theory of compact Lie group actions with maximal rank isotropy

Let G denote a compact connected Lie group with torsion-free fundamental group acting on a compact space X such that all the isotropy subgroups are connected subgroups of maximal rank. Let T⊂G be a maximal torus with Weyl group W. If the fixed-point set XT has the homotopy type of a finite W–CW complex, then we prove that the rationalized complex equivariant K-theory of X is a free module over the representation ring of G. Given additional conditions on the W-action on the fixed-point set XT, we show that the equivariant K-theory of X is free over R(G). We use this to provide computations for a number of examples, including the ordered n-tuples of commuting elements in G with the conjugation action.

Ideal triangles in Euclidean buildings and branching to Levi subgroups

Let G̲ denote a connected reductive group, defined and split over Z, and let M̲⊂G̲ denote a Levi subgroup. In this paper we study varieties of geodesic triangles with fixed vector-valued side-lengths α,β,γ in the Bruhat–Tits buildings associated to G̲, along with varieties of ideal triangles associated to the pair M̲⊂G̲. The ideal triangles have a fixed side containing a fixed base vertex and a fixed infinite vertex ξ such that other infinite side containing ξ has fixed “ideal length” λ and the remaining finite side has fixed length μ. We establish an isomorphism between varieties in the second family and certain varieties in the first family (the pair (μ,λ) and the triple (α,β,γ) satisfy a certain relation). We apply these results to the study of the Hecke ring of G̲ and the restriction homomorphism R(Gˆ̲)→R(Mˆ̲) between representation rings. We deduce some new saturation theorems for constant term coefficients and for the structure constants of the restriction homomorphism.

Hooks generate the representation ring of the symmetric group

We prove that the representation ring of the symmetric group on n letters is generated by the exterior powers of its natural (n−1)-dimensional representation. The proof we give illustrates a strikingly simple formula due to Dvir. We provide an application and investigate a possible generalization of this result to some other reflection groups.

Real representation spheres and the real monomial Burnside ring

We introduce a restriction morphism, called the Boltje morphism, from a given ordinary representation functor to a given monomial Burnside functor. In the case of a sufficiently large fibre group, this is Robert Boltjeʼs splitting of the linearization morphism. By considering a monomial Lefschetz invariant associated with real representation spheres, we show that, in the case of the real representation ring and the fibre group { ± 1 } , the image of a modulo 2 reduction of the Boltje morphism is contained in a group of units associated with the idempotents of the 2-local Burnside ring. We deduce a relation on the dimensions of the subgroup-fixed subspaces of a real representation.

Augmentation quotients for complex representation rings of dihedral groups

Denote by Dm the dihedral group of order 2m. Let ℛ(Dm) be its complex representation ring, and let Δ(Dm) be its augmentation ideal. In this paper, we determine the isomorphism class of the n-th augmentation quotient Δn(Dm)/Δn+1(Dm) for each positive integer n.

A generalization of the sign character

For every q ∈ N we define a class function on finite symmetric groups, which generalizes the sign character. We show that this class function is a generalized character. In case q a prime we determine all its irreducible constituents. As an application we show that a certain inner product on the representation ring of a symmetric group can be evaluated for Young modules by the aid of this class function.

The additivity of theρ–invariant and periodicity in topological surgery

For a closed topological manifold M with dim (M) >= 5 the topological structure set S(M) admits an abelian group structure which may be identified with the algebraic structure group of M as defined by Ranicki. If dim (M) = 2d-1, M is oriented and M is equipped with a map to the classifying space of a finite group G, then the reduced rho-invariant defines a function, \wrho : S(M) \to \QQ R_{hat G}^{(-1)^d}, to a certain sub-quotient of the complex representation ring of G. We show that the function \wrho is a homomorphism when 2d-1 >= 5. Along the way we give a detailed proof that a geometrically defined map due to Cappell and Weinberger realises the 8-fold Siebenmann periodicity map in topological surgery.

Quotients of representation rings

We give a proof using so-called fusion rings andqq-deformations of Brauer algebras that the representation ring of an orthogonal or symplectic group can be obtained as a quotient of a ringGr(O(∞))Gr(O(\infty )). This is obtained here as a limiting case for analogous quotient maps for fusion categories, with the level going to∞\infty. This in turn allows a detailed description of the quotient map in terms of a reflection group. As an application, one obtains a general description of the branching rules for the restriction of representations ofGl(N)Gl(N)toO(N)O(N)andSp(N)Sp(N)as well as detailed information about the structure of theqq-Brauer algebras in the nonsemisimple case for certain specializations.

Cross-sections, quotients, and representation rings of semisimple algebraic groups

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny \( \tau :\hat{G} \to G \) is bijective; this answers Grothendieck’s question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg’s theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck’s questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T ⇢ G/T where T is a maximal torus of G and W the Weyl group.

On the isomorphism problem for the ring of monomial representations of a finite group

In this paper we are concerned with the problem of finding properties of a finite group G in the ring D ( G ) of monomial representations of G. We determine the conductors of the primitive idempotents of Q ( ζ ) ⊗ Z D ( G ) , where ζ ∈ C is a primitive | G | -th root of unity, and prove a structure theorem for the torsion units of D ( G ) . Using these results we show that an abelian group G is uniquely determined by the ring D ( G ) . We state an explicit formula for the primitive idempotents of Z [ ζ ] p ⊗ Z D ( G ) , where Z [ ζ ] p is a localization of Z [ ζ ] . We get further results for nilpotent and p-nilpotent groups and we obtain properties of Sylow subgroups of G from D ( G ) .

Completing Verlinde algebras

We compute the completion of the Verlinde algebra of a simply connected simple compact Lie group G at the augmentation ideal of the representation ring. By results of Freed, Hopkins, Teleman and Dwyer and Lahtinen, this gives a computation of (non-equivariant) twisted K-theory of the free loop space of BG.

The Bernstein–Gelfand–Gelfand complex and Kasparov theory for [formula omitted

For G = SL ( 3 , C ) , we construct an element of G -equivariant analytic K-homology from the Bernstein–Gelfand–Gelfand complex for G . This furnishes an explicit splitting of the restriction map from the Kasparov representation ring R ( G ) to the representation ring R ( K ) of its maximal compact subgroup SU ( 3 ) , and the splitting factors through the equivariant K-homology of the flag variety X of G . In particular, we obtain a new model for the γ-element of G . The construction is made using SU ( 3 ) -harmonic analysis associated to the canonical fibrations of X . On this matter, we prove results which demonstrate the compatibility of both the G -action and the order zero longitudinal pseudodifferential operators with the SU ( 3 ) -harmonic analysis.