Invariant theory for wreath product groups

This thesis records an attempt to prove the two conjecture: Conjecture A: Every finite non-regular primitive permutation group of degree n contains permutations fixing one point but fixing at most $n^{1/2}$ points. Conjecture C: Every finite irreducible linear group of degree m > 1 contains an element whose fixed-point space has dimension at most m/2. Variants of these conjectures are formulated, and C is reduced to a special case of A. The main results of the investigation are: Theorem 2: Every finite non-regular primitive permutation group of degree n contains permutations which fix one point but fix fewer than (n+3)/4 points. Theorem 3: Every finite non-regular primitive soluble permutation group of degree n contains permutations which fix one point but fix fewer than $n^{7/18}$ points. Theorem 4: If H is a finite group, F is a field whose characteristic is 0 or does not divide the order of H, and M is a non-trivial irreducible H-module of dimension m over F, then there is an element h in H whose fixed-point space in M has dimension less than m/2. Theorem 5: If H is a finite soluble group, F is any field, and M is a non-trivial irreducible H-module of dimension m over F, then there is an element h in H whose fixed-point space in M has dimension less than 7m/18. Proofs of these assertions are to be found in Chapter II; examples which show the limitations on possible strenghtenings of the conjectures and results are marshalled in Chapter III. A detailed formulation of the problems and results is contained in section 1.

We say that a compact Lie group G has the Borsuk-Ulam property in the weak sense if for every orthogonal representation V of G and every G-equivariant map /: S{V) -> S(V), VG = {0}, of the unit sphere we have deg/ Φ 0 . We say that G has the Borsuk-Ulam property in the strong sense if for any two orthogonal representations V, W of G with dim W = dim V and WG = VG = {0} and every G-equivariant map /: S(V) —> S{W) of the unit spheres we have deg/ Φ 0. In this paper a complete classification, up to isomorphism, of group with the weak Borsuk-Ulam property is given. A classification of groups with the strong Borsuk-Ulam property does not cover nonabelian p-groups with all elements of the order p . In fact we deal with a more general definition admitting a nonempty fixed point set of G on the sphere S(V). 1. The main theorems. In order to formulate our main results we introduce the following notation. Let G be a compact Lie group. We denote by Go the component of identity of and by Γ the quotient group G/GQ . We use standard notation of the theory of compact transformation groups (see for instance [4] or [5]). In particular, for every subgroup H c G, the fixed point set of H on a G-space X is denoted by XH. Also, for a G-equivariant map f:X—>Y between two G-spaces, we denote by fH its restriction to the space XH . The symbol (n, m) stands for the greatest common divisor of the integers n, m with the notation (0, 1) = 0 and \G for the rank of the (finite) group G. We will work with the following definition of the BorsukUlam property (cf. [11]). DEFINITION I. (A) We say that G has the Borsuk-Ulam property in the weak sense A if for every orthogonal representation V of G and every G-equivariant map /: S(V) -> S(V) if (deg/ G, |Γ|) = 1 then

We perform a comprehensive analysis of the symmetry-resolved (SR) entanglement entropy (EE) for one single interval in the ground state of a 1 + 1D conformal field theory (CFT), that is invariant under an arbitrary finite or compact Lie group, G. We utilize the boundary CFT approach to study the total EE, which enables us to find the universal leading order behavior of the SREE and its first correction, which explicitly depends on the irreducible representation under consideration and breaks the equipartition of entanglement. We present two distinct schemes to carry out these computations. The first relies on the evaluation of the charged moments of the reduced density matrix. This involves studying the action of the defect-line, that generates the symmetry, on the boundary states of the theory. This perspective also paves the way for discussing the infeasibility of studying symmetry resolution when an anomalous symmetry is present. The second scheme draws a parallel between the SREE and the partition function of an orbifold CFT. This approach allows for the direct computation of the SREE without the need to use charged moments. From this standpoint, the infeasibility of defining the symmetry-resolved EE for an anomalous symmetry arises from the obstruction to gauging. Finally, we derive the symmetry-resolved entanglement spectra for a CFT invariant under a finite symmetry group. We revisit a similar problem for CFT with compact Lie group, explicitly deriving an improved formula for U(1) resolved entanglement spectra. Using the Tauberian formalism, we can estimate the aforementioned EE spectra rigorously by proving an optimal lower and upper bound on the same. In the abelian case, we perform numerical checks on the bound and find perfect agreement.

Weights for symmetric and general linear groups

Quantifying local embeddings into finite groups

Terms like commutativity degree, non-commuting graph and isoclinism are far well-known for much of the group theorists nowadays. There are so many papers about each of these concepts and also about their relationships in finite groups. Also, there are some recent researches about generalizing these notions in finite rings and their connexions.The concepts of commutativity degree and non-commuting graph are also extended to non-associative structures such as Moufang loops and some part of the known results in group theory in these contexts have been expanded to them.In this paper, we are going to generalize the notion of isoclinism in finite Moufang loops and then study the relationships between these three concepts. Among other results, we prove that two isoclinic finite Moufang loops have the same commutativity degree and if they have the same sizes of centers and commutants then they have isomorphic non-commuting graphs. Also, the converses of these results have been investigated.Furthermore, it has been proved that a finite simple group can be characterized by its non-commuting graph. We will prove the same is true for a finite simple Moufang loop by imposing one additional hypothesis, namely, the isoclinism of the regarding loops.

The aim of the workshop was to give a panoramic view of the main lines of research on various aspects of growth and expansion in finite groups. The main topics included: diameter bounds for Cayley graphs of alternating and classical groups, with results for generating sets that can be arbitrary, random, or containing special elements; constructions of expander families of finite simple groups, with the use of property (T); and character bounds for finite simple groups, yielding applications to word map problems and random walks on Cayley graphs. A series of smaller topics explored connections and similarities to different problems in group theory.

A finite group W generated by reflections is called a finite reflection group and have been studied deeply in different important contexts such as geometry and group theory, matrix algebra and Lie Theory. In this paper we study a class of fuzzy subgroups of finite reflection groups which are called parabolic fuzzy subgroups using preferential equality. In the classical case parabolic subgroups are associated with a set of generators, each a simple reflection, indexed by elements of subsets of a fixed root system Δ . We establish a one-to-one correspondence between the class of fuzzy parabolic subgroups of W and the class of fuzzy subsets of S where S is a set of simple reflections associated with Δ .

The wreath product of groups $A\wr B$ is one of basic constructions in group theory. We construct its analogue, a wreath product of Lie algebras. Consider Lie algebras $H$ and $G$ over a field $K$. Let $U(G)$ be the universal enveloping algebra. Then $\bar H=\operatorname {Hom}_K(U(G),H)$ has the natural structure of a Lie algebra, where the multiplication is defined via the comultiplication in $U(G)$. Also, $G$ acts by derivations on $\bar H$ via the (left) coregular action. The semidirect sum $\bar H \leftthreetimes G$ we call the wreath product and denote by $H\wr G$. As a main result, we prove that an arbitrary extension of Lie algebras $0\to H\to L\to G\to 0$ can be embedded into the wreath product $L\hookrightarrow H\wr G$.

We introduce a notion of partition wreath product of a finite group by a partition quantum group, a construction motivated on the one hand by classical wreath products and on the other hand by the free wreath product of J. Bichon.We identify the resulting quantum group in several cases, establish some of its properties and show that when the finite group in question is abelian, the partition wreath product is itself a partition quantum group. This allows us to compute its representation theory, using earlier results of the first named author.

We study the neighborhoods of a typical point $$Z_n$$ visited at n-th step of a random walk, determined by the condition that the transition probabilities stay close to $$\mu ^{*n}(Z_n)$$ . If such neighborhood contains a ball of radius $$C \sqrt{n}$$ , we say that the random walk has almost invariant transition probabilities. We prove that simple random walks on wreath products of $$\mathbb {Z}$$ with finite groups have almost invariant distributions. A weaker version of almost invariance implies a necessary condition of Ozawa’s criterion for the property $$H_\mathrm{FD}$$ . We define and study the radius of almost invariance. We estimate this radius for random walks on iterated wreath products and show this radius can be asymptotically strictly smaller than n / L(n), where L(n) denotes the drift function of the random walk. We show that the radius of individual almost invariance of a simple random walk on the wreath product of $$\mathbb {Z}^2$$ with a finite group is asymptotically strictly larger than n / L(n). Finally, we show the existence of groups such that the radius of almost invariance is smaller than a given function, but remains unbounded. We also discuss possible limiting distribution of ratios of transition probabilities on non almost invariant scales.

We show that a finite permutation group containing a regular abelian self-normalizing subgroup is soluble.

On the tensor product of permutation modules

We show that a compact group G has finite conjugacy classes, i.e., is an FC-group if and only if its center Z(G) is open if and only if its commutator subgroup G′ is finite. Let d(G) denote the Haar measure of the set of all pairs (x,y) in G×G for which [x,y]=1; this, formally, is the probability that two randomly picked elements commute. We prove that d(G) is always rational and that it is positive if and only if G is an extension of an FC-group by a finite group. This entails that G is abelian by finite. The proofs involve measure theory, transformation groups, Lie theory of arbitrary compact groups, and representation theory of compact groups. Examples and references to the history of the discussion are given at the end of the paper.

We consider the distance enumerator Δ G (x) of a finite permutation group G, which is the polynomial ∑ g∈G x n−π(g), where n is the degree of G and π(g) the number of fixed points of g ∈ G. In particular, we introduce a bivariate polynomial which is a special case of the cycle index of G, and from which Δ G (x) can be obtained, and then use this new polynomial to prove some identities relating the distance enumerators of groups G and H with those of their direct and wreath products. In the case of the direct product, this answers a question of Blake et al. (1979). We also use the identity for the wreath product to find an explicit combinatorial expression for the distance enumerators of the generalised hyperoctahedral groups C m ≀ S n .