Representation theory of 2-groups on Kapranov and Voevodsky's 2-vector spaces

In this paper the representation theory of 2-groups in 2-categories is considered, focusing the attention on the 2-category Rep2MatK(G) of representations of a 2-group G in (a version of) Kapranov and Voevodsky's 2-category of 2-vector spaces over a field K. The set of equivalence classes of such representations is computed in terms of the invariants π₀(G), π₁(G) and [α]∈H³(π₀(G),π₁(G)) classifying G, and the categories of intertwiners are described in terms of categories of vector bundles endowed with a projective action. In particular, it is shown that the monoidal category of finite dimensional linear representations (more generally, the category of [z]-projective representations, for any given cohomology class [z]∈H²(π₀(G),K^∗)) of the first homotopy group π₀(G) as well as its category of representations on finite sets both live in Rep2MatK(G), the first as the monoidal category of endomorphisms of the trivial representation (more generally, as the category of intertwiners between suitable 1-dimensional representations) and the second as a non-full subcategory of the homotopy category of Rep2MatK(G).     

Primitives and central detection numbers in group cohomology

Fix a prime p. Given a finite group G, let H^∗(G) denote its mod p cohomology. In the early 1990s, Henn, Lannes, and Schwartz introduced two invariants d₀(G) and d₁(G) of H^∗(G) viewed as a module over the mod p Steenrod algebra. They showed that, in a precise sense, H^∗(G) is respectively detected and determined by Hd(CG(V)) for d?d₀(G) and d?d₁(G), with V running through the elementary abelian p-subgroups of G.The main goal of this paper is to study how to calculate these invariants. We find that a critical role is played by the image of the restriction of H^∗(G) to H^∗(C), where C is the maximal central elementary abelian p-subgroup of G. A measure of this is the top degree e(G) of the finite dimensional Hopf algebra H^∗(C)_⊗H^∗(G)Fp, a number that tends to be quite easy to calculate.Our results are complete when G has a p-Sylow subgroup P in which every element of order p is central. Using the Benson-Carlson duality, we show that in this case, d₀(G)=d₀(P)=e(P), and a similar exact formula holds for d₁. As a bonus, we learn that He(G)(P) contains nontrivial essential cohomology, reproving and sharpening a theorem of Adem and Karagueuzian.In general, we are able to show that d₀(G)?max{e(CG(V))|V<G} if certain cases of Benson's Regularity Conjecture hold. In particular, this inequality holds for all groups such that the difference between the p-rank of G and the depth of H^∗(G) is at most 2. When we look at examples with p=2, we learn that d₀(G)?14 for all groups with 2-Sylow subgroup of order up to 64, with equality realized when G=SU(3,4).En route we study two objects of independent interest. If C is any central elementary abelian p-subgroup of G, then H^∗(G) is an H^∗(C)-comodule, and we prove that the subalgebra of H^∗(C)-primitives is always Noetherian of Krull dimension equal to the p-rank of G minus the p-rank of C. If the depth of H^∗(G) equals the rank of Z(G), we show that the depth essential cohomology of G is nonzero (reproving and extending a theorem of Green), and Cohen-Macauley in a certain sense, and prove related structural results.     

The Multiplicity Free Permutation Representations of the Ree Groups 2 G 2(q), the Suzuki Groups 2 B 2(q), and Their Automorphism Groups

Abstract

Let G a simple group of type ² B ₂(q) or ² G ₂(q), where q is an odd power of 2 or 3, respectively. The main goal of this paper is to determine the multiplicity free permutation representations of G and A ≤ Aut(G) where A is a subgroup containing a copy of G. Let B be a Borel subgroup of G. If G = ² B ₂(q) we show that there is only one non-trivial multiplicity free permutation representation, namely the representation of G associated to the action on G/B. If G = ² G ₂(q) we show that there are exactly two such non-trivial representations, namely the representations of G associated to the action on G/B and the action on G/M, where M = UC with U the maximal unipotent subgroup of B and C the unique subgroup of index 2 in the maximal split torus of B. The multiplicity free permutation representations of A correspond to the actions on A/H where H is isomorphic to a subgroup containing B if G = ² B ₂(q), and containing M if G = ² G ₂(q). The problem of determining the multiplicity free representations of the finite simple groups is important, for example, in the classification of distance-transitive graphs.     

Bimeasure algebras on locally compact groups

For locally compact groups G and H, let BM(G, H) denote the Banach space of bounded bilinear forms on C₀(G) × C₀(H). Using a consequence of the fundamental inequality of A. Grothendieck. a multiplication and an adjoint operation are introduced on BM(G, H) which generalize the convolution structure of M(G × H) and which make BM(G, H) into a K_G²-Banach ${}^1$-algebra, where K_G is Grothendieck's universal constant. Various topics relating to the ideal structure of BM(G, H) and the lifting of unitary representations of G × H to ${}^1$-representations of BM(G, H) are investigated.     

Equivalence between the Morita categories of étale Lie groupoids and of locally grouplike Hopf algebroids

Any étale Lie groupoid G is completely determined by its associated convolution algebra C_c^∞(G) equipped with the natural Hopfalgebroid structure. We extend this result to the generalized morphisms between étale Lie groupoids: we show that any principal H-bundle P over G is uniquely determined by the associated C_c^∞(G)-C_c^∞(H)-bimodule C_c^∞(P) equipped with the natural coalgebra structure. Furthermore, we prove that the functor C_c^∞gives an equivalence between the Morita category of étale Lie groupoids and the Morita category of locally grouplike Hopf algebroids.     

Colorings at minimum cost

We define by min_c∑_{u,v}∈E(G)|c(u)−c(v)| the min-costMC(G) of a graph G, where the minimum is taken over all proper colorings c. The min-cost-chromatic numberχ_M(G) is then defined to be the (smallest) number of colors k for which there exists a proper k-coloring c attaining MC(G). We give constructions of graphs G where χ(G) is arbitrarily smaller than χ_M(G). On the other hand, we prove that for every 3-regular graph G^′, χ_M(G^′)≤4 and for every 4-regular line graph G^″, χ_M(G^″)≤5. Moreover, we show that the decision problem whether χ_M(G)=k is -hard for k≥3.     

On automorphisms of some finite <Emphasis Type="Italic">p</Emphasis>-groups

We give a sufficient condition on a finite p-group G of nilpotency class 2 so that Aut _c (G) = Inn(G), where Aut _c (G) and Inn(G) denote the group of all class preserving automorphisms and inner automorphisms of G respectively. Next we prove that if G and H are two isoclinic finite groups (in the sense of P. Hall), then Aut _c (G) ≃ Aut _c (H). Finally we study class preserving automorphisms of groups of order p ⁵, p an odd prime and prove that Aut _c (G) = Inn(G) for all the groups G of order p ⁵ except two isoclinism families.     

On algebraic connectivity augmentation

Suppose that G is an undirected graph, and that H is a spanning subgraph of G^c whose edges induce a subgraph on p vertices. We consider the expression α(G∪H)-α(G), where α denotes the algebraic connectivity. Specifically, we provide upper and lower bounds on α(G∪H)-α(G) in terms of p, and characterise the corresponding equality cases. We also discuss the density of the expression α(G∪H)-α(G) in the interval [0,p]. A bound on α(G∪H)-α(G) is provided in a special case, and several examples are considered.     

Embeddings of minimal non-abelian p-groups

By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. For a given finite group G, let p(G) denote the minimal degree of a faithful representation of G by permutation matrices, and let c(G) denote the minimal degree of a faithful representation of G by quasi-permutation matrices. See [4]. It is easy to see that c(G) is a lower bound for p(G). Behravesh [H. Behravesh, The minimal degree of a faithful quasi-permutation representation of an abelian group, Glasg. Math. J. 39 (1) (1997) 51-57] determined c(G) for every finite abelian group G and also [H. Behravesh, Quasi-permutation representations of p-groups of class 2, J. Lond. Math. Soc. (2) 55 (2) (1997) 251-260] gave the algorithm of c(G) for each finite group G. In this paper, we first improve this algorithm and then determine c(G) and p(G) for an arbitrary minimal non-abelian p-group G.     

Étale slices for representation varieties in characteristic p

Let R(F, G) be the variety of representations of a finitely generated group F into a connected reductive algebraic group G, and let C(F, G) be the variety of closed conjugacy classes of representations. We examine the question of whether an étale slice for the conjugation action of G exists through a representation ρR(F, G) when the ground field k has characteristic p > 0. We show that an étale slice through ρ may exist for the action of an enlarged group Ĝ, even when there is no étale slice for the G-action. As an application, we generalise a result known to hold in characteristic zero, which expresses the tangent space to C(F, G) at the conjugacy class of a suitable representation ρ as a subspace of the 1-cohomology H¹ (F, %plane1D;524;(ρ)) of an F-module %plane1D;524;(ρ). A similar result holds in characteristic p, but with H¹ (F%plane1D;524;(ρ)) replaced by a quotient of H¹ (F%plane1D;524;(ρ)).     

Correspondance de Jacquet-Langlands pour les corps locaux de caractéristique non nulle

In this article we prove the Jacquet-Langlands local correspondence in non-zero characteristic. Let F be a local field of non-zero charactersitic and G′ an inner form of GL_n(F); then, following [17], we prove relations between the representation theory of G′ and the representation theory of an inner form of GL_n(L), where L is a local field of zero characteristic close to F. The proof of the Jacquet-Langlands correspondence between G′ and GL_n(F) is done using the above results and ideas from the proof by Deligne, Kazhdan and Vignéras [10] of the zero characteristic case. We also get the following, already known in zero characteristic: orthogonality relations for G′, inequality involving conductor and level for representations of G′ and finiteness for automorphic cuspidal representations with fixed component at almost every place for an inner form of GL_n over a global field of non-zero characteristic.     

The similarity problem for Fourier algebras and corepresentations of group von Neumann algebras

Let G be a locally compact group, and let A(G) and VN(G) be its Fourier algebra and group von Neumann algebra, respectively. In this paper we consider the similarity problem for A(G): Is every bounded representation of A(G) on a Hilbert space H similar to a *-representation? We show that the similarity problem for A(G) has a negative answer if and only if there is a bounded representation of A(G) which is not completely bounded. For groups with small invariant neighborhoods (i.e. SIN groups) we show that a representation π:A(G)→B(H) is similar to a *-representation if and only if it is completely bounded. This, in particular, implies that corepresentations of VN(G) associated to non-degenerate completely bounded representations of A(G) are similar to unitary corepresentations. We also show that if G is a SIN, maximally almost periodic, or totally disconnected group, then a representation of A(G) is a *-representation if and only if it is a complete contraction. These results partially answer questions posed in Effros and Ruan (2003) [7] and Spronk (2002) [25].     

On subword decomposition and balanced polynomials

Let H(x) be a monic polynomial over a finite field F=GF(q). Denote by Na(n) the number of coefficients in Hn which are equal to an element a∈F, and by G the set of elements a∈F^× such that Na(n)>0 for some n. We study the relationship between the numbers (Na(n))_a∈G and the patterns in the base q representation of n. This enables us to prove that for “most” n's we have Na(n)≈Nb(n), a,b∈G. Considering the case H=x+1, we provide new results on Pascal's triangle modulo a prime. We also provide analogous results for the triangle of Stirling numbers of the first kind.     

Completing an extension of Tunnell's theorem

We look at a special case of a familiar problem: Given a locally compact group G, a subgroup H and a complex representation π₊ of G how does π₊ decompose on restriction to H. Here G is GL⁺(2,F), where F is a nonarchimedian local field of characteristic not two, K a separable quadratic extension of F, GL⁺(2,F) the subgroup of index 2 in GL(2,F) consisting of those matrices whose determinant is in NK/F(K^∗), π₊ is an irreducible, admissible supercuspidal representation of GL⁺(2,F) and H=K^∗ under an embedding of K^∗ into GL(2,F).     

On conjugate characteristic polynomial of a graph

Let $c=a+b\sqrt{m}$ and $\overline{c}=a-b\sqrt{m}$ , where a and b are two nonzero integers and m is a positive integer such that m is not a perfect square. We say that A ^c =[c _ij ] is the conjugate adjacency matrix of a graph G if c _ij =c for any two adjacent vertices i and j, $c_{ij}=\overline{c}$ for any two nonadjacent vertices i and j, and c _ij =0 if i=j. Let P _G ^c (λ)=|λ I?A ^c | denote the conjugate characteristic polynomial of G. Further, let e=e(G) and Δ=Δ(G) be the number of edges and number of triangles of G, respectively. Let G and H be two graphs of order n and let e(G)=e(H). In this work we prove that c ₃(G)=c ₃(H) if and only if Δ(G)=Δ(H) and $\Delta(\overline{G})=\Delta(\overline{H})$ , where $\overline{G}$ denotes the complement of G and c _k is the coefficient which corresponds to λ ^n?k with respect to P _G ^c (λ). Besides, we here give the conjugate spectrum and conjugate characteristic polynomial of all connected graphs of order n=2,3,4,5, with respect to the constant $c=1+\sqrt{2}$ .     

Induced representations and finite volume homogeneous spaces

We prove that the representation of $\text{C}{}^1\text{(G ×}\text{G}\text{H}\text{)}$ induced from the restriction to H of a unitary representation π of G can be constructed directly from π in the framework of Rieffel's theory of induced representations of C¹-algebras, with the inducing process defined by a generalized conditional expectation. We then show, in the general context of Rieffel's theory, that if the induced representation is CCR, so is the original. In a more special situation, which still generalizes that of a conditional expectation onto a subalgebra, and which includes the operation of inducing from an open subgroup and the above-mentioned process when $\text{G}\text{H}$ is of finite volume, we prove that if the induced representation is type I, so is the original, and obtain a result on intertwining operators. This provides a unified treatment, as well as an extension to the nonseparable case, of certain known results on induction and restriction of representations.     

Partitioning 2-edge-colored graphs by monochromatic paths and cycles

We present results on partitioning the vertices of 2-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of Sárközy: the vertex set of every 2-edge-colored graph can be partitioned into at most 2α(G) monochromatic cycles, where α(G) denotes the independence number of G. Another direction, emerged recently from a conjecture of Schelp, is to consider colorings of graphs with given minimum degree. We prove that apart from o(|V (G)|) vertices, the vertex set of any 2-edge-colored graph G with minimum degree at least \(\tfrac{{(1 + \varepsilon )3|V(G)|}} {4}\) can be covered by the vertices of two vertex disjoint monochromatic cycles of distinct colors. Finally, under the assumption that \(\bar G\) does not contain a fixed bipartite graph H, we show that in every 2-edge-coloring of G, |V (G)| ? c(H) vertices can be covered by two vertex disjoint paths of different colors, where c(H) is a constant depending only on H. In particular, we prove that c(C ₄)=1, which is best possible.     

Representations of algebraic groups over a 2-dimensional local field

We introduce a categorical framework for the study of representations of G(F), where G is a reductive group, and F is a 2-dimensional local field, i.e. F = K((t)), where K is a local field. Our main result says that the space of functions on G(F), which is an object of a suitable category of representations of G(F) with the respect to the action of G on itself by left translations, becomes a representation of a certain central extension of G(F), when we consider the action by right translations.     

On an approximation property for nondiscrete inequivalent representations of finitely generated groups

Let R(Γ, G) be the variety of representations of a finitely generated group Γ in a simple complex algebraic group G. We establish some sufficient conditions for the image of the diagonal representation ϱ = (ϱ₁, …, ϱ_t), ϱ_i ε R(Γ, G), to be dense in G^f in the complex topology (“weak approximation”).     

Grundy number and products of graphs

The Grundy number of a graph G, denoted by Γ(G), is the largest k such that G has a greedyk-colouring, that is a colouring with k colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this paper, we study the Grundy number of the lexicographic and cartesian products of two graphs in terms of the Grundy numbers of these graphs.Regarding the lexicographic product, we show that Γ(G)×Γ(H)≤Γ(G[H])≤2^Γ(G)−1(Γ(H)−1)+Γ(G). In addition, we show that if G is a tree or Γ(G)=Δ(G)+1, then Γ(G[H])=Γ(G)×Γ(H). We then deduce that for every fixed c≥1, given a graph G, it is CoNP-Complete to decide if Γ(G)≤c×χ(G) and it is CoNP-Complete to decide if Γ(G)≤c×ω(G).Regarding the cartesian product, we show that there is no upper bound of Γ(G□H) as a function of Γ(G) and Γ(H). Nevertheless, we prove that Γ(G□H)≤Δ(G)⋅2^Γ(H)−1+Γ(H).