Ribbon blocks for centraliser algebras of symmetric groups

For each pair (𝔤,𝔞) consisting of a real Lie algebra 𝔤 and a subalgebra a of some Cartan subalgebra 𝔥 of 𝔤 such that [𝔞, 𝔥]∪ [𝔞, 𝔞] we define a Weyl group W(𝔤, 𝔞) and show that it is finite. In particular, W(𝔤, 𝔥,) is finite for any Cartan subalgebra h. The proof involves the embedding of 0 into the Lie algebra of a complex algebraic linear Lie group to which the structure theory of Lie algebras and algebraic groups is applied. If G is a real connected Lie group with Lie algebra 𝔤, the normalizer N(𝔥, G) acts on the finite set Λ of roots of the complexification 𝔤c with respect to hc, giving a representation π : N(𝔥, G)→ S(Λ) into the symmetric group on the set Λ. We call the kernel of this map the Cartan subgroup C(𝔥) of G with respect to h; the image is isomorphic to W(𝔤, 𝔥), and C(𝔥)= {g G : Ad(g)(h)— h ε [h,h] for all h ε h }. All concepts introduced and discussed reduce in special situations to the familiar ones.The information on the finiteness of the Weyl groups is applied to show that under very general circumstance, for b ∪ 𝔥 the set 𝔥⊂ ϕ(b) remains finite as ϕ ranges through the full group of inner automorphisms of 𝔤.

Preface. Introduction. Linear and Nonlinear Group Actions, and the Newton Institute Program L.L. Scott. Tilting Modules for Algebraic Groups H.H. Andersen. Semisimplicity in Positive Characteristic G.J. McNinch. Homology Bases Arising from Reductive Groups over a Finite Field G. Lusztig. Highest Weight Modules Associated to Parabolic Subgroups with Commutative Unipotent Radicals T. Tanisaki. Symmetric Groups and Schur Algebras G. James. Branching Rules for Symmetric Groups and Applications A.S. Kleshchev. Endomorphism Algebras and Representation Theory E. Cline, et al. Representations of Simple Lie Algebras: Modern Variations on a Classical Theme R.W. Carter. The Path Model, the Quantum Frobenius Map and Standard Monomial Theory P. Littelmann. Arithmetical Properties of Blocks G.R. Robinson. The Isomorphism and Isogeny Theorems for Reductive Algebraic Groups R. Steinberg. Double Cosets in Algebraic Groups G.M. Seitz. Dense Orbits and Double Cosets J. Brundan. Subgroups of Exceptional Groups M.W. Liebeck. Overgroups of Special Elements in Simple Algebraic Groups and Finite Groups of Lie Type J. Saxl. Some Applications of Subgroup Structure to Probabilistic Generation and Covers of Curves R.M. Guralnick. Quasithin Groups M. Aschbacher. Tame Groups of Odd and Even Type A.V. Borovik. Index.

Using a new colored analogue of $$P$$P-partitions, we prove the existence of a colored Eulerian descent algebra which is a subalgebra of the Mantaci---Reutenauer algebra. This algebra has a basis consisting of formal sums of colored permutations with the same number of descents (using Steingrimsson's definition of the descent set of a colored permutation). The colored Eulerian descent algebra extends familiar Eulerian descent algebras from the symmetric group algebra and the hyperoctahedral group algebra to colored permutation group algebras. We also describe orthogonal idempotents that span the colored Eulerian descent algebra and include, as a special case, the familiar Eulerian idempotents in the group algebra of the symmetric group.

On Extensions of Simple Modules over Symmetric and Algebraic Groups

The study of finite subgroups of a simple algebraic group $G$ reduces in a sense to those which are almost simple. If an almost simple subgroup of $G$ has a socle which is not isomorphic to a group of Lie type in the underlying characteristic of $G$, then the subgroup is called non-generic. This paper considers non-generic subgroups of simple algebraic groups of exceptional type in arbitrary characteristic. A finite subgroup is called Lie primitive if it lies in no proper subgroup of positive dimension. We prove here that many non-generic subgroup types, including the alternating and symmetric groups $\text{Alt}_{n}$, $\text{Sym}_{n}$ for $n \ge 10$, do not occur as Lie primitive subgroups of an exceptional algebraic group. A subgroup of $G$ is called $G$-completely reducible if, whenever it lies in a parabolic subgroup of $G$, it lies in a conjugate of the corresponding Levi factor. Here, we derive a fairly short list of possible isomorphism types of non-$G$-completely reducible, non-generic simple subgroups. As an intermediate result, for each simply connected $G$ of exceptional type, and each non-generic finite simple group $H$ which embeds into $G/Z(G)$, we derive a set of feasible characters, which restrict the possible composition factors of $V \downarrow S$, whenever $S$ is a subgroup of $G$ with image $H$ in $G/Z(G)$, and $V$ is either the Lie algebra of $G$ or a non-trivial Weyl module for $G$ of least dimension. This has implications for the subgroup structure of the finite groups of exceptional Lie type. For instance, we show that for $n \ge 10$, $\text{Alt}_n$ and $\text{Sym}_n$, as well as numerous other almost simple groups, cannot occur as a maximal subgroup of an almost simple group whose socle is a finite simple group of exceptional Lie type.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Multi-matrix invariants, and in particular the scalar multi-trace operators of N = 4 SYM with U(N) gauge symmetry, can be described using permutation centraliser algebras (PCA), which are generalisations of the symmetric group algebras and independent of N. Free-field two-point functions define an N-dependent inner product on the PCA, and bases of operators have been constructed which are orthogonal at finite N. Two such bases are well-known, the restricted Schur and covariant bases, and both definitions involve representation-theoretic quantities such as Young diagram labels, multiplicity labels, branching and Clebsch-Gordan coefficients for symmetric groups. The explicit computation of these coefficients grows rapidly in complexity as the operator length increases. We develop a new method for explicitly constructing all the operators with specified Young diagram labels, based on an N-independent integer eigensystem formulated in the PCA. The eigensystem construction naturally leads to orthogonal basis elements which are integer linear combinations of the multi-trace operators, and the N-dependence of their norms are simple known dimension factors. We provide examples and give computer codes in SageMath which efficiently implement the construction for operators of classical dimension up to 14. While the restricted Schur basis relies on the Artin-Wedderburn decomposition of symmetric group algebras, the covariant basis relies on a variant which we refer to as the Kronecker decomposition. Analogous decompositions exist for any finite group algebra and the eigenvalue construction of integer orthogonal bases extends to the group algebra of any finite group with rational characters.

We prove that finite groups have the same complex character tables iff the group algebras are twisted forms of each other as Drinfel’d quasi-bialgebras or iff there is non-associative bi-Galois algebra over these groups. The interpretations of class-preserving automorphisms and permutation representations with the same character in terms of Drinfel’d algebras are also given. 1. Introduction. The theory of quasi-Hopf algebras was developed by V.G.Drinfel’d for the description of quantizations of Lie groups and algebras or so-called quantum groups. Althought the deformational quantization approach which is so useful in the theory of quantum groups can’t be applied for the the case of finite groups, the idea of twisting seems to be very suitible for reformulating of various problems from representation theory of finite groups. The key observations of this article is that any bijection between character tables of finite groups corresponds to the quasi-isomorphism of the group algebras considered as quasi-Hopf algebras and any two homomorphisms of the group algebras define the same map of character tables iff they are twisted forms. In particular, we can give the definitions in terms of (quasi-)Hopf algebras of such objects as class-preserving automorphisms, permutation representations with the same character, groups with the same character tables. Namely, any class-preserving automorphism is twisted form of identity maps as homomorphisms of Hopf algebras. Two permutation representations have the same complex character iff the corresponding homomorphisms into symmetric group are twisted forms as homomorphisms of Hopf algebras. Two groups have the same character tables iff their group algebras are twisted forms as quasi-Hopf algebras. This point of view allows to select the subclass of pairs of groups with the same character tables. This subclass consists of pairs of groups whose group algebas are twisted forms as Hopf algebras.

The Donald–Flanigan problem for a finite group H and coefficient ring k asks for a deformation of the group algebra kH to a separable algebra. It is solved here for dihedral groups and Weyl groups of types Bn and Dn (whose rational group algebras are computed), leaving but six finite reflection groups with solutions unknown. We determine the structure of a wreath product of a group with a sum of central separable algebras and show that if there is a solution for H over k which is a sum of central separable algebras and if Sn is the symmetric group then (i) the problem is solvable also for the wreath product H\(\wr\)Sn = H × ··· × H (n times) ⋊ Sn and (ii) given a morphism from a finite Abelian or dihedral group G to Sn it is solvable also for H\(\wr\)G. The theorems suggested by the Donald–Flanigan conjecture and subsequently proven follow, we also show, from a geometric conjecture which although weaker for groups applies to a broader class of algebras than group algebras.

A deformation-theoretic version of Maschke's theorem for modular group algebras: The commutative case

The extended golay codes considered as ideals

This chapter discusses the group $${\mathfrak{S}}_{n}$$ , the symmetric group on $$n$$ elements, which is one of the most important examples of finite groups and is widely used in applications to geometry and physics. The importance of symmetry groups in abstract algebra is due to the fact that for any finite group $$G$$ , there is a symmetric group $${\mathfrak{S}}_{n}$$ that contains a copy of $$G$$ . For each $$n \in {\mathbb{N}}$$ , the group $${\mathfrak{S}}_{n}$$ consists of all the bijective maps of $$\left\{ {1,2, \ldots ,n} \right\}$$ to itself, called permutations of $$\left\{ {1,2, \ldots ,n} \right\}$$ . These permutations are usually denoted by symbols such as $$\phi \;{\text{and}}\;\psi$$ . The identity permutation that corresponds to the identity map of $$\left\{ {1,2, \ldots ,n} \right\}$$ is denoted by $$e$$ . In this chapter, Sect. 6.1 provides a representation of the elements of $${\mathfrak{S}}_{n}$$ as matrices and specifies the order of $${\mathfrak{S}}_{n}$$ in terms of the integer $$n$$ . Additionally, the notion of pairwise disjoint permutations is discussed, and their commutativity is verified. In Sect. 6.2, cycles, a special case of permutations, are defined and studied. The main result of this section is Proposition 6.2.9, which states that any permutation can be written as a finite product of disjoint cycles. The proof of this proposition requires a study of orbits of a permutation which discussed in Sect. 6.3 and followed by the proof of Proposition 6.2.9. The last two sections of this chapter discuss methods for determining the order of permutations and classifying permutations as odd and even.

The non-existence set forth in the title is proved. It is known that for numerical Campedelli surfaces the algebraic fundamental group is of order ⩽9 and that the dihedral group of order 8 cannot occur. Therefore the quaternionic group is the only non-abelian algebraic fundamental group in this range.

In this paper, we give the characterization of the unit groups of semisimple group algebras of some non-metabelian groups of order [Formula: see text]. This study completes the study of unit groups of semisimple group algebras of all groups up to order [Formula: see text], except that of the symmetric group [Formula: see text] and groups of order [Formula: see text].

The purpose of this paper is to describe a general procedure for computing analogues of Young’s seminormal representations of the symmetric groups . The method is to generalize the Jucys – Murphy elements in the group algebras of the symmetric groups to arbitrary Weyl groups and Iwahori – Hecke algebras . The combinatorics of these elements allow one to compute irreducible representations explicitly and often very easily . In this paper we do these computations for Weyl groups and Iwahori – Hecke algebras of types A n , B n , D n , G 2 . Although these computations are within reach for types F 4 , E 6 , and E 7 , we shall , in view of the length of the current paper , postpone this to another work . In reading this paper , I would suggest that the reader begin with § 3 , the symmetric group case , and go back and pick up the generalities from §§ 1 and 2 as they are needed . This will make the motivation for the material in the earlier sections much more clear and the further examples in the later sections very easy . The realization that the Jucys – Murphy elements for Weyl groups and Iwahori – Hecke algebras come from the very natural central elements in (2 . 1) and Proposition 2 . 4 is one of the main points of this paper . There is a simple concrete connection (Proposition 2 . 8) between Jucys – Murphy type elements in Iwahori – Hecke algebras and Jucys – Murphy elements in group algebras of Weyl groups . I know that the analogues of the Jucys – Murphy elements in Weyl groups of types B and D will be new to some of the experts and known to others . These Jucys – Murphy elements for types B and D are not new ; similar elements appear in the paper of Cherednik [ 7 ] , but I was not able to recognize them there until they were pointed out to me by M . Nazarov . I extend my thanks to him for this . Some people were asking me for Jucys – Murphy elements in type G 2 as late as June 1995 . In July 1995 I was told that it was not known how to quantize the elements of Cherednik , that is , to find analogues of them in the Iwahori – Hecke algebras of types B and D . Of course , this had been done already in 1974 , by Hoefsmit . I have chosen to state my results in terms of the general mechanism of path algebras which I have defined in § 1 . This is a technique which I learned from H . Wenzl during our work on the paper [ 30 ] . It is a well-known method in several fields (with many dif ferent terminologies) . I shall mention here only a few of the