Tensor Induction Research Articles

Tensor induction for M-Burnside rings

Let G be a finite group and H a subgroup. The M-Burnside ring ΩM(H) arising from a monoid functor M for G is a generalization of the Burnside ring Ω(H). We present a certain hypothesis on M such that if M satisfies the hypothesis, then there is a multiplicative map θHG:ΩM(H)→ΩM(G) derived from the tensor induction jndHG:Ω(H)→Ω(G). Let A be a finite abelian G-group. For the monoid functor HA assigning each K≤G the 1st cohomology group H1(K,A) of K with coefficients in A, the HA-Burnside ring ΩHA(H) is isomorphic to the ring of monomial representations of H with coefficients in A. We show that HA satisfies the hypothesis and the associated multiplicative map defines the known tensor (or multiplicative) induction for the rings of monomial representations. Let S be a finite commutative G-monoid. The monoid functor CS assigning each K≤G the set of K-invariants in S satisfies the hypothesis. To extend this fact, we introduce the concept of hereditary monoid functors which satisfy the hypothesis. The monoid functor MS∘ assigning each K≤G the set of normal subgroups of K is a hereditary monoid functor for G on the subgroup lattice of G. In short, we present tensor inductions for HA-, CS-, and MS∘-Burnside rings derived from the tensor induction for Burnside rings.

P-adic Asai L-functions of Bianchi modular forms

The Asai (or twisted tensor) $L$-function of a Bianchi modular form $\Psi$ is the $L$-function attached to the tensor induction to $\mathbb{Q}$ of its associated Galois representation. In this paper, when $\Psi$ is ordinary at $p$ we construct a $p$-adic analogue of this $L$-function: that is, a $p$-adic measure on $\mathbb{Z}_p^\times$ that interpolates the critical values of the Asai $L$-function twisted by Dirichlet characters of $p$-power conductor. The construction uses techniques analogous to those used by Lei, Zerbes and the first author in order to construct an Euler system attached to the Asai representation of a quadratic Hilbert modular form.

Asai cube L-functions and the local Langlands correspondence

Let F be a non-archimedean locally compact field. We study a class of Langlands-Shahidi pairs (H,L), consisting of a quasi-split connected reductive group H over F and a Levi subgroup L which is closely related to a product of restriction of scalars of GL1's or GL2's. We prove the compatibility of the resulting local factors with the Langlands correspondence. In particular, let E be a cubic separable extension of F. We consider a simply connected quasi-split semisimple group H over F of type D4, with triality corresponding to E, and let L be its Levi subgroup with derived group ResE/FSL2. In this way we obtain Asai cube local factors attached to irreducible smooth representations of GL2(E); we prove that they are Weil-Deligne factors obtained via the local Langlands correspondence for GL2(E) and tensor induction from E to F. A consequence is that Asai cube γ- and ε-factors become stable under twists by highly ramified characters.

Idempotent characters and equivariantly multiplicative splittings of K‐theory

We classify the primitive idempotents of the $p$-local complex representation ring of a finite group $G$ in terms of the cyclic subgroups of order prime to $p$ and show that they all come from idempotents of the Burnside ring. Our results hold without adjoining roots of unity or inverting the order of $G$, thus extending classical structure theorems. We then derive explicit group-theoretic obstructions for tensor induction to be compatible with the resulting idempotent splitting of the representation ring Mackey functor. Our main motivation is an application in homotopy theory: we conclude that the idempotent summands of $G$-equivariant topological $K$-theory and the corresponding summands of the $G$-equivariant sphere spectrum admit exactly the same flavors of equivariant commutative ring structures, made precise in terms of Hill-Hopkins-Ravenel norm maps. This paper is a sequel to the author's earlier work on multiplicative induction for the Burnside ring and the sphere spectrum, see arXiv:1802.01938.

Deflation and tensor induction on the Frobenius–Wielandt morphism

Abstract We explore conditions for the Frobenius–Wielandt morphism to commute with the operations of deflation and tensor induction on the Burnside ring. In doing this, we review the commutativity with induction. The techniques used for induction and tensor induction no longer work for deflation, so, in this case, we make use of tools coming from the theory of biset functors.

Multiplicative induction and units for the ring of monomial representations

Let G be a finite group, and let A be a finite abelian G-group. For each subgroup H of G, Ω(H,A) denotes the ring of monomial representations of H with coefficients in A, which is a generalization of the Burnside ring Ω(H) of H. We research the multiplicative induction map Ω(H,A)→Ω(G,A) derived from the tensor induction map Ω(H)→Ω(G), and also research the unit group of Ω(G,A). The results are explained in terms of the first cohomology groups H1(K,A) for K≤G. We see that tensor induction for 1-cocycles plays a crucial role in a description of multiplicative induction. The unit group of Ω(G,A) is identified as a finitely generated abelian group. We especially study the group of torsion units of Ω(G,A), and study the unit group of Ω(G) as well.

Monomial G-posets and their Lefschetz invariants

Let G be a finite group, and C be an abelian group. We introduce the notions of C-monomial G-sets and C-monomial G-posets, and state some of their categorical properties. This gives in particular a new description of the C-monomial Burnside ring BC(G). We also introduce Lefschetz invariants of C-monomial G-posets, which are elements of BC(G). These invariants allow for a definition of a generalized tensor induction multiplicative map TU,λ:BC(G)→BC(H) associated to any C-monomial (G,H)-biset (U,λ), which in turn gives a group homomorphism BC(G)×→BC(H)× between the unit groups of C-monomial Burnside rings.

Oddness of residually reducible Galois representations

We show that suitable congruences between polarized automorphic forms over a CM field always produce elements in the Selmer group for exactly the ±-Asai (aka tensor induction) representation that is critical in the sense of Deligne. For this, we relate the oddness of the associated polarized Galois representations (in the sense of the Bellaïche-Chenevier sign being [Formula: see text]) to the parity condition for criticality. Under an assumption similar to Vandiver’s conjecture this also provides evidence for the Fontaine–Mazur conjecture for residually reducible polarized Galois representations.

Modular group actions on algebras and p-local Galois extensions for finite groups

Let k be a field of positive characteristic p and let G be a finite group. In this paper we study the category TsG of finitely generated commutative k-algebras A on which G acts by algebra automorphisms with surjective trace. If A=k[X], the ring of regular functions of a variety X, then trace-surjective group actions on A are characterized geometrically by the fact that all point stabilizers on X are p′-subgroups or, equivalently, that AP≤A is a Galois extension for every Sylow p-group of G. We investigate categorical properties of TsG, using a version of Frobenius-reciprocity for group actions on k-algebras, which is based on tensor induction for modules. We also describe projective generators in TsG, extending and generalizing the investigations started in [7–9] in the case of p-groups. As an application we show that for an abelian or p-elementary group G and k large enough, there is always a faithful (possibly nonlinear) action on a polynomial ring such that the ring of invariants is also a polynomial ring. This would be false for linear group actions by a result of Serre. If A is a normal domain and G≤Autk(A) an arbitrary finite group, we show that AOp(G) is the integral closure of k[Soc(A)], the subalgebra of A generated by the simple kG-submodules in A. For p-solvable groups this leads to a structure theorem on trace-surjective algebras, generalizing the corresponding result for p-groups in [8].

Characterization of γ-Factors: the Asai Case

Let $E$ be a separable quadratic extension of a locally compact field $F$ of positive characteristic. Asai \gamma-factors are defined for smooth irreducible representations \pi of ${\rm GL}_n(E)$. If \sigma is the Weil-Deligne representation of $\mathcal{W}_E$ corresponding to \pi under the local Langlands correspondence, we show that the Asai \gamma-factor is the same as the Deligne-Langlands \gamma-factor of the Weil-Deligne representation of $\mathcal{W}_F$ obtained from \sigma under tensor induction. This is achieved by proving that Asai \gamma-factors are characterized by their local properties together with their role in global functional equations for $L$-functions. As an immediate application, we establish the stability property of \gamma-factors under twists by highly ramified characters.

Diagrammes de Diamond et (φ, Γ)-modules

Let ρ be a 2-dimensional continuous semi-simple generic representation of Gal(ℚp/ℚp) over Fp. The modulo p Langlands correspondence for GL2(ℚp) defined in [5], as realized in [9], can be reformulated as a quite simple recipee giving back the (φ, Γ)-module of the dual of ρ starting from the “Diamond diagram” associated to ρ. Let F be a finite unramified extension of ℚp and ρ a 2-dimensional continuous semi-simple generic representation of Gal(ℚp/F) over Fp. When one formally extends this recipee to the Diamond diagrams associated to ρ in [6], we show that one essentially finds the (φ, Γ)-module of the tensor induction from F to ℚp of the dual of ρ.

On Frobenius and tensor induction

Tensor inducing a Frobenius induced class function leads to a sum of class functions where each summand is given by first tensor induction, then Frobenius induction. This qualitative statement is made precise. As an example, tensor induction of monomial characters is considered.

On tensor induction for representations of finite groups

It is well known that if D is an irreducible complex representation of a finite group G, then every direct summand of the restriction of D to a subgroup H must have degree at least as large as the degree of D divided by the index | G : H | ; moreover, D is induced from H if and only if the restriction does have a direct summand whose dimension is equal to this quotient. This paper explores the possibility of an analogous result for tensor induction, under the additional assumption that D is faithful, quasi-primitive and not a tensor product (of projective representations of degree greater than 1), and that the Fitting subgroup F ( G ) is not in the centre Z ( G ) . The main question is this: if the restriction has a (projective) tensor factor whose degree is the | G : H | th root of the degree of D, does it follow that D is tensor induced from H? Among other results, examples are given to show that the answer can be negative when the index is 2. An affirmative answer is proved for normal subgroups of odd index, and also for arbitrary subgroups of odd prime index. As might be expected, the key lies in the study of F ( G ) / Z ( G ) as a symplectic module over a finite prime field; in particular, in exploring the connection between (ordinary) induction and form-induction of such modules.

A remark on the Dade group and the Burnside group

The object of this note is to show that the formula for tensor induction of relative syzygies in the Dade group, stated in [S. Bouc, J. Reine Angew. Math. 523 (2000) 113], can be viewed as a special case of a functorial homomorphism from the dual B ∗ of the Burnside group to the subgroup D Ω of the Dade group generated by relative syzygies. It follows that there exists a short exact sequence of functors 0→R Q ∗→B ∗→D Ω/D Ω tors →0 , where R Q is the functor of rational representations. This may be viewed as an improvement (from Q to Z ) of Theorem D of [Invent. Math. 139 (2000) 275].

Non-additive exact functors and tensor induction for Mackey functors

Introduction Non additive exact functors Permutation Mackey functors Tensor induction for Mackey functors Relations with the functors ${\mathcal L}_U$ Direct product of Mackey functors Tensor induction for Green functors Cohomological tensor induction Tensor induction for $p$-permutation modules Tensor induction for modules Bibliography.

Tensor induction of projective representations

TO HELMUT WIELANDT ON HIS 80~~ BIRTHDAY There are some standard reduction techniques in representation theory of finite groups, mainly based on work by Clifford and Fong. Apparently less known is a method suggested by Berger [l] which refers to tensor induction of (projective) representations. It applies in the following situation: (TI) E is a nonabelian normal subgroup

On tensor induction of group representations

AbstractLetGbe a (not necessarily finite) group andρa finite dimensional faithful irreducible representation ofGover an arbitrary field; writeρ¯forρviewed as a projective representation. Suppose thatρis not induced (from any proper subgroup) and thatρ¯is not a tensor product (of projective representations of dimension greater than 1). LetKbe a noncentral subgroup which centralizes all its conjugates inGexcept perhaps itself, writeHfor the normalizer ofKinG, and suppose that some irreducible constituent, σ say, of the restrictionp↓Kis absolutely irreducible. It is proved that then (ρis absolutely irreducible and) ρ¯ is tensor induced from a projective representation ofH, namely from a tensor factor π ofρ¯↓Hsuch that π↓K= σ¯ and ker π is the centralizer ofKinG.

Clifford theory and tensor induction

Clifford theory and tensor induction

A remark on Brauer's k(B)-conjecture

Forgetting about context, one could say that this paper does just one thing, namely, it constructs a square root of a generalized character. To be sure, it is a very special generalized character for a very special group (namely, S,). All the same, this seems a rather unusual thing to do in character theory. As far as I am aware, there is no a priori argument known of why a root should exist, let alone that explains the role played by the Jacobi symbol in its construction. Brauer’s k(B)-conjecture-in a special case, namely, the “k(GV)problem,” see [4, 6]-provides the motivation; in fact, the generalized character 6 (see below) whose square root is constructed came up when studying this problem. The corollary clarifies the connection. The reader should be familiar with the elements of character theory, in particular those of the symmetric group. Also, a few facts about tensor induction are needed. Let G be a finite group acting on a finite set Q and let F be a finite field with char F= p. Let R be the permutation character of G on FQ; i.e., 7r( g) = IC,(g)l is the order of the space of fixed points of g E G. Let 6(g) = n(g))’ rc( l), so if 10) = n and g has m orbits on Q, then 6(g) = IF-Jn-m. Let us assume from now on that p 1 IGI. Then 6 is a generalized character by [6, Prop. 1.11. Denote by 9 the ring of algebraic integers in @ and let .r$ = m, SO ~2 is an ideal of 9; see [2, Sects. 1, 41 for more details. Finally, let a=:(1 +i&)oc and Y=Z++a, so .Z is a Z-lattice in C.

TENSOR INDUCTION AND TRANSFER

TENSOR INDUCTION AND TRANSFER